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Connecting Engineers

Publication of the Chamber of Engineers November 2012 | Issue 43

Aspects of Speech Technology p7

Interview with Ing. Jeanelle Cassar p19

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Contents

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Publication of the Chamber of Engineers

Cover Image Engineers are working towards automated mining systems to improve safety and increase productivity.

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November 2012 www.coe.org.mt

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From the Editor

From the President

Editor Ing. John Pace

Aspects of Speech Technology

Mems meets ASICs

Interview with Ing. Jeanelle Cassar

Middlesea’s New SME Insurance Policy

Grid Connected Brushless DC Drive for Micro-Wind Turbines

Graphic VU Meter Display for Sound Mixers

Intelligent Solutions for Today’s Modern Educational Facilities

Materials in Aircraft Design Over the Years

An Engineering Restoration Project

Editorial Board Ing. John Pace Ing. Paul Refalo Ing. Ray Vassallo Prof. Robert Ghirlando Chamber of Engineers, Professional Centre, Sliema Road, Gzira, GZR 1633, Malta Tel: +356 2133 4858 Fax: +356 2134 7118 Email: info@coe.org.mt Web: www.coe.org.mt

Note: The winners of the CoE Best Mechanical Final Year Students’ Project 2011 will be published in the next issue.

© Chamber of Engineers 2012. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopy, recording or otherwise, without the prior permission of the Chamber of Engineers – Malta. Opinions expressed in Engineering Today are not necessarily those of the Chamber of Engineers – Malta. All care has been taken to ensure truth and accuracy, but the Editorial Board cannot be held responsible for errors or omissions in the articles, pictographs or illustrations.

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From the editor by Ing. John Pace

wind farms in the north of Malta. At that time electricity rates were heavily subsidised and the proposed feed in price was higher that what Enemalta charged the public and the proposals died a natural death.

In early October it was announced that a temporary floating wind turbine was to be installed in order to study the impact of the planned large wind farm on seabirds. The cost of this turbine was estimated at â&#x201A;Ź15 million. Naturally many people asked questions on whether the cost was justified, if the money could be better spent elsewhere, or, possibly, if government was having second thoughts on the wind farm project and was looking for a way to exit gracefully. The wind farm project has had a turbulent history. It is Maltaâ&#x20AC;&#x2122;s biggest project towards meeting its obligations for a share of its energy use being supplied by renewable sources. Public opinion was divided, with some staunch advocates who viewed it as an essential move away from the present total dependence on a single fuel, while the opposite camp was equally emphatic that wind farms would do irreparable damage to the environment. Serious interest in wind energy started in the late nineties, when a number of private entrepreneurs carried out studies for possible

November 2012

Issue No. 43

As Malta was preparing to join the EU, it was obliged have a renewable energy policy, so government issued Legal Notice 186 in 2004 which stated that the national indicative target for electricity produced from renewable sources would be 5% of gross national electricity consumption by 2010. (this was later changed to 10% by 2020). The legal notice also directed the Malta Resources Authority to report every five years how such targets would be achieved and, if necessary updated. As wind installations proliferated in Europe and America, people started discussing wind farms. Landowners were worried that such wind towers would lower the value of their property and an opposition Member of Parliament from the north of Malta asked for an assurance that no wind farms would be permitted. The minister, while personally agreeing with the questioner, stated that he would not impose his private view. The MRA meanwhile engaged energy consultants Mott Macdonald to prepare a report on Renewable Energy Exploitation in Malta, which was published in 2005, and stated that the 5% target could only be reached if large land based wind farms were constructed. The report also examined various available onshore and offshore sites and their potential. Opposition to land based wind farms rose to a peak and the MRA stated in the 2006 Energy Policy consultation paper that â&#x20AC;&#x153;Government is concerned of the high visual and landscape

impacts within the urban Maltese environment associated with large-scale on-shore and nearshore wind-farms and such developments will not be authorised for construction.” Instead a deep sea wind farm was advocated, and formal requests for information from interested suppliers were issued. I have never met an engineer who was in favour of this proposal, which no doubt originated from some very persuasive salesman, and the idea was eventually abandoned. This situation was unsustainable and in the 2009 Energy Policy document it was stated that “The Government is in favour of onshore and offshore wind farm developments, as long as any planning and environmental impacts are acceptable.” Then in April it was announced that a wind farm would be built at the Sikka L-Bajda, off the north west of Malta at a depth of 10 to 35 metres. The maximum generating capacity was to be 95 megawatts and the cost estimated at €300 million. The MRA started gathering data and a high mast was erected at Armier to gather wind data. Geological investigations were also made and the presence of caves under the sea bottom was discovered. Objections were also received from Birdlife Malta that the site would possibly disturb the colony of Shearwaters that breed on the Rdum tal-Madonna at the nearest land from the proposed wind farm.

This site was designated a special protection area listed in the Annex of the Birds Directive. This is a serious objection and may well result in the abandonment of the project. This is why the government invested the huge sum of €15 million to ascertain the impact on the birds. It is not known what the chances are that the result of the study will be favourable for the wind farm. There is a strong lobby against renewable sources of any kind, not always disinterested. Solar panels are regarded by many people as ugly and, if used on a large scale will take up agricultural land or otherwise spoil the countryside. I find this attitude unreasonable, as is the opinion that Malta should get most of its electricity through a cable from Europe so that the pollution associated with electricity generation, whether conventional or from renewable sources, would be suffered by other countries. The dangers of this attitude are seldom appreciated as it ignores the price we will have to pay for such a privilege. In the meantime the wind farm project is at a standstill. What if the birds study is unfavourable? The solution will then be to go for photovoltaics. It will probably be more cost effective as the economics have changed profoundly since 2009. Personally, I think this is the way to go. ET

Ing. John Pace Editor, Engineering Today

November 2012

Issue No. 43

From the President by Ing. Saviour Baldacchino

Dear Colleagues... Following the meeting with the Prime Minister, on the 20th of August 2012 the Executive Council met with the Leader of the Opposition, Dr Joseph Muscat at his office in Hamrun. The topics discussed during the meeting included updating of the Engineering Act, regulation of technology supported by MRA incentives, quality in higher education institutions, the need for foreign direct investment in technology based and engineering industries, research and innovation in industry and the socio-economic development of the engineer in society. A memorandum highlighting further details on the topics discussed was presented to Dr Muscat for consideration in the drafting of the electoral programme.

Recently, the Chamber announced the 11th edition of the Malta Engineering Excellence Awards 2012, which are now open for nominations. The award ceremony will be held during the first week of December and it will be followed by the end of year membersâ&#x20AC;&#x2122; reception. Paid up members and their guests will be invited free of charge but it is strongly encouraged to contribute a small donation for charity. Further details will be communicated to members in due course as the event approaches. During September 2012, the Chamber participated in the World Engineering Forum 2012 held in Ljubljana, Slovenia. The event was organised by the Slovenian Chamber

UNESCO representative delivering his presentation during the World Engineering Forum in Ljubljana

November 2012

Issue No. 43

From left Johan A. Psaila, CoE International Secretary, Saviour Baldacchino, CoE President, John Power Engineers Ireland Director General, Damien Owens, Engineers Ireland Registrar

of Engineers and the World Federation of Engineering Organisations (WFEO). The main theme of the forum was Sustainable Construction for People. It was divided into four main sections: Cities and Urban Development, Sustainable Infrastructure, Green Buildings and Disaster/Risk Management. During the event the Chamber’s delegation participated in WFEO’s technical committee meetings on Disaster Risk Management, Information and Communications Technology and Engineering Capacity Building. It was also invited to participate in the 5th Presidents’ meeting of European Engineering Organisations and in the WFEO Executive Council meeting as observer. During Freshers’ Week at the University of Malta, Daniel Micallef, the Chamber’s Events Secretary, addressed new engineering students and encouraged them to join the Chamber as student members. I take this opportunity to thank UESA (the University Engineering Students’ Association) for their collaboration during this event.

During last week, UESA hosted the European Young Engineers (EYE) 2012 Autumn Conference, EYE@Malta. Chamber VicePresident, Ray Vassallo address the conference. In October, the Chamber’s delegation lead by the undersigned participated in the FEANI General Assembly in Rome. During this event, nine countries signed the cooperation agreement regarding issuing the Engineering Card, the professional card for engineers. Germany, the Netherlands, Poland, Portugal, Croatia, Ireland, Luxemburg, Slovenia and the Czech Republic will from next year onwards start issuing this mobility card for engineers who apply for it. FEANI president Mr Lars Bytoft, announced his resignation during the meeting in Rome for reasons related to increased responsibilities at his home country, during an emotional farewell address. Dr Rafael Fernandez Aller, Vice-President of FEANI will assume the role of president for the coming year.

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From the President (cont.)

The Chamber encourages members to follow its activities on line at www.coe.org.mt, through its monthly E-newsletter and fortnightly â&#x20AC;&#x153;Events & Opportunitiesâ&#x20AC;? e-mail communication. If you are missing any of these communications, please let us know on info@coe.org.mt. Your feedback on anything you would like to see happening at the Chamber is greatly appreciated.

Johan A. Psaila CoE International Secretary and Saviour Baldacchino, CoE President during the FEANI General Assembly

Last week, the Chamber held a bilateral meeting with Engineers Ireland at their headquarters in Dublin. During the meeting various topics of mutual interest were discussed including continuous professional development, membership grades, engineers licensing and course and employer accreditation.

We wish all students who started their engineering academic formation all the best of luck in their studies. ET

25th October 2012

Yours Sincerely,

Ing. Saviour M. Baldacchino President, Chamber of Engineers

Ing. Saviour Baldacchino President, Chamber of Engineers

November 2012

Issue No. 43

president@coe.org.mt http://www.coe.org.mt

Aspects of Speech Technology by Prof. Paul Micallef

The area can be broadly classified into speech analysis, speech synthesis, speech coding, speech and speaker recognition, and speech enhancement in noisy backgrounds. The reason for this article is to describe an area of reserach and application which is a typical multidisciplinary environment in this case spanning engineering and linguistics, with applications involving also people with disabilities and therefore requiring particular studies in the human machine interface. I will be talking here about two areas where I have been active together with engineering students. These are speech synthesis and speech recognition. The primary aim has been to adapt existing tools , theories and algorithms, to handle Maltese speech synthesis and Maltese speech recognition. Text to Speech Synthesis Basically written text from any source spanning emails, newspapers and books, is converted to spoken discourse. Figure 1, [1] gives a block diagram of one type of system that has been popular for many years and forms the basis of many workling systems.

Figure 1 There is first the linguistic part that analyses input text. In computer terms, it means that computer text, which traditionally used 8-bit ASCII but has now moved to UTF8 with its

variable length character code, thereby allowing non English characters to be typed and used, is analysed by an algorithm in a process that is called grapheme to phoneme. This means that the words in the text are turned into how the word will be spoken within the context of the sentence to which it belongs. Every spoken language has a set of phonemes that form the spoken words. Usually there is a small subset in every language characteristic of the language. The classical word to ask non Maltese to say is ‘Tal-Qroqq’. The transformation also follows some particular rules. Two examples in Maltese – the word “ktibt” is spoken as ‘k’ ‘t’ ‘i’ ‘p’ ‘t’ ; the word “lagħab” is spoken ‘l’ ‘a’ ‘p’. Surnames and names are also a law unto themselves. Other pitfalls are acronyms, numbers such as dates and time, and all form of implied text that has to be analysed and converted to a set of phonemes. While doing all this, information regarding word stress, as well as intonation markers need to be generated and kept. Intonation is the way a speaker elocutes. A sentence such as - “Good morning, may I help you” said by a Maltese, Italian, French, Indian or Texan person is understood by everyone but is totally different in its rendering. The linguistic side of intonation is well understood, but its practical synthesis in computer speech is still very difficult, [2]. While all languages have grapheme to phoneme rules that can be used to obtain the phonemic output, with today’s massive memory and computing power, word to phoneme dictionaries are used directly. Any new word, surname or acronym is simply added there, together with its associated phonemic structure. The rules are used only if the word is not found in the dictionary.

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Aspects of Speech Technology (cont.)

The next stage is to look at the phonemic series generated and decide how to produce it as speech. Up to a few years ago the main line of development involved taking one particular personâ&#x20AC;&#x2122;s voice as the basis for the synthesis. This voice was recorded speaking a specific set of sentences and text that capture all the possible phonemic sequences of the language. The speech was then spliced in various ways, the most used being the diphone which is a piece of speech roughly from the middle of one phone to the middle of the next phone. From a signal processing point of view this method considers that diphones will be joined at the most stable part, the middle, putting all the articulation effects due to movement from one phone to another within the diphone itself. The result is the creation of a massive database of diphones with knowledge of various parameters such as pitch, energy and duration, in such a way that during concatenation of diphones for synthesis

November 2012

Issue No. 43

the nearest pairs, in terms of some algorithmic distance are chosen to obtain the sequence of diphones that results in the best natural speech output [3]. The trick here is how to handle the search through the huge synthesis segments database to operate in real time. These systems are tied to the particular voice that has been used to splice the speech and obtain the diphones. At present a Maltese company is finishing the development of such a system in a contract from MITA, using also EU funds. There is now a new idea which is giving outputs almost as good as the single voice but allows for easy transformation from one voice to another. This uses Hidden Markov Model (HMM) techniques, (more on these in the section on speech recognition) and a dictionary of annotated speech [4]. The unit selection is not based on time waveforms, but spectral

and prosody parameters that represent the phonemes. HMM’s are responsible for selecting those parameters which most precisely represent the synthesis model of the text to be read. A vocoder generates the synthesized voice from these parameters. Every mobile has a vocoder, and therefore the speech quality can potentially be as good as from a mobile. It is also possible to adapt the voice characteristics in the database from short recordings of 5 to 8 minutes, so that the TTS voice can now be personalised. Further research in this area aims at considering multilingual extensions and the ability to handle emotions in the synthesized speech.

use all these frames with their coefficients, to distinguish the phonemes that result in words.

Speech Recognition The flip side of text to speech is ofcourse, speech to text. In this case an individual can speak into a microphone and have his speech come out as text. In practice this has been achieved in some limited environments but only for major languages such as English. The process starts from digital sampling at a rate of about 8000Hz for telephone speech and 16,000 samples for recorded speech. The samples are broken down into frames of about 20ms, analysed, using various algorithms and eventually come out as a set of what are known as MFCC’s - Mel Frequency Cepstral Coefficients. A set of 13 coefficients per frame has been found to be sufficient. From the MFCC’s two other sets are derived, the difference, and the second difference, to finish up with a set of 39 coefficients per frame. There are other possible speech paramters to use including wavelets, but MFCC’s and their derivatives are still the working horse in speech recognition. With an overlap of 10ms per frame there are a hundred frames per second. The main task is how to

Pattern recognition is one of the most studied and researched areas. Various ideas, all based on mathematical statistics, are around including neural networks, support vector machines and hidden markov models. In speech recognition it is the HMM’s that have been found to operate in an optimal way, because a speech recognition system is essentially based on a dictionary of words that encompass the knowledge of the system, and is used in a circular bottom-up, top-down system before a decision is made, [5]. A short note on the Hidden markov Model. A Markov process is a probabilistic system where the movement from a state to another, and the probability of it happeneing, is explicitly known. Let us consider a weather system consisting of three states – rainy, sunny, cloudy. One can monitor the system directly and over time build sufficient information to derive the probability of a particular sequence eg RRR or SSCR etc. If now a person is shut in a windowless room, but has available readings from an anemometer, a

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Aspects of Speech Technology (cont.)

thermometer and a hygrometer and the fact of the three states, is it possible for him to know the state sequence outside, based on the readings being given by the instruments in the room. This is a Hidden Markov Process. In the case of speech the available observations are the MFCC vectors, the states are usually an artifact to move from the beginning of a phoneme to its end, so usually three states (start, middle, end) are sufficient, and the guesswork involves finding the start and end of the most probable phoneme , from the sequence of observation MFCC’s. The classical tutorial on HMM is [6]. Anybody who tries to use the speech recognition tool in Windows 7, for Maltese, will soon find

out that all that comes out are English words, since that is what the dictionary has. One can try to add to the dictionary Maltese words, but since the underlying phonemes are English ones, the result is usually very unsatisfactory. At present, based on the previous work of my students, an annotated system for Maltese is being built to be able to eventually have a sufficiently large database for training the Hidden Markov Models for the various phones. In some cases triphones, have to be trained. For what are known as LVCSR (large vocabulary continuous speech recognition) systems this is a huge undertaking in terms of the dictionary as well as in terms of the

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sophistication necessary in each HMM. Suffice it to say that given 30 phonemes, to be able to capture the articulation of phonemes such as ‘r’, triphones are needed so that instead of having one HMM for ‘r’, 900 HMM’s are needed to capture the articulation from the previous phone to the next phone. For thirty phonemes, this can theoretically mushroom to thirty cubed. In practice a few thousand HMM’s are trained and used in good English systems. The basis of recognition consists of a guessing game between the HMM’s for the different phonemes or triphones, and the dictionary of words that have associated with each word a set of phonemes. The guessing game has an underlying mathematical algorithm known as the Viterbi algorithm. Between the outcome from the HMM’s and the probability of the phonemes to fit into a given word in the dictionary, a good result should come out, Figure 2.

Figure 2 Research in speech recognition today has two main aims. The first is research leading to robust recognition in noisy environments to be able to use speech recognition in cars, offices and virtual conferences, [7]. The second is recognition of emotive speech and spoken dialogues aiming at being able to control and command any machine, producing the

November 2012

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Aspects of Speech Technology (cont.)

equivalent of ‘Hal’ in 2001 space odyssey. While huge advances have been made, the keyboard is not going to vanish overnight, and Hal is still a cinematic production. The synergy between computer design and VLSI has resulted in an explosion of computational power. This has resulted in formulation of algorithms to handle processes that could only be dreamed about a decade ago. Computational power will continue to grow and it is therefore reasonable to assume that within the next decade we will probably have achieved both aims mentioned above. One hopes that Maltese engineers will contribute with their research to this advancement, and will develop the tools to maintain the technology applications associated with our language at par with those in other languages. ET

References 1. Micallef Paul, “A Text to Speech Synthesis System for Maltese”, Ph.D. Thesis, University of Surrey, 1997 2. Taylor Paul, “Text to Speech Synthesis”, Chapter 6, Cambridge University Press, 2009 3. Hunt A.J., Black A.W., “Unit selection in a concatenative speech synthesis system using a large speech database”, Proceedings of the International Conference on Speech and Language Processing 1996, pp 373-376 4. Tokuda K., Yoshimura T., Masuko T., Kobayashi T, Kitamura T., “Speech Parameter Generation Algorithms for HMM-based speech synthesis”, Proceedings ICASSP 2000. 5. Young Steve, “HTK Hidden Markov Model Toolkit”, Cambridge University Engineering Department, 1989 6. Rabiner Lawrence R., “A Tutorial on Hidden Markov Models and Selected Applications in Speech recognition”, Proceedings IEEE Vol 77 No 2, February 1989, pp 257-286 7. Nakatani Tomahiro, Sehr Armin, Kellermann Walter, “Reverberation Speech Processing for Human Communication and Automatic Speech Recognition”, Tutorial, Proceedings ICASSP 2012.

Prof. Paul Micallef

Associate Professor, Department of Communication and Computer Engineering, University of Malta

November 2012

Issue No. 43

MEMS meet ASICs by Dr. Ing. Owen Casha

The critical physical dimensions of MEMS may vary from a few microns, all the way to several millimetres. A microsystem in its most basic form consists of a Micro-Electro-Mechanical System (MEMS) which develops an electrical signal that is then processed by an Application Specific Integrated Circuit (ASIC).

including micro-valves for gas and liquid flow control, optical switches and mirrors such as those used in a video projectors which have independently controlled micro-mirror arrays.

MEMS technologies enable the design and fabrication of miniaturised mechanical and electro-mechanical devices and structures that are produced using micro-fabrication techniques. MEMS can be further described as functional machine systems with micrometerscale components. MEMS have attracted interest mainly because miniaturised mechanical structures can be developed utilising semiconductor fabrication technology. It is important to note that the design of innovative microstructures such as capacitive sensors, electrostatic actuators, energy scavengers and microfluidic test structures were all made possible through MEMS technology. ASICs are microelectronic circuits designed using micro and nanoscale semiconductor technologies. In the context of a microsystem the ASIC in most cases carries out some initial analogue signal processing followed by analogue-to-digital conversion which makes possible any additional digital signal processing that may be required. Over the past years, MEMS researchers developed a large number of microsensors for almost every possible sensing modality including: temperature, pressure, inertial forces, chemical species, magnetic fields and radiation. These microsensors outperformed their macro-scale counterparts not only in accuracy, but also in production costs. It was no surprise that these structures have become successful commercialised products. Recently, MEMS research also included micro-actuators

Students Researching Back in 2007, the Department of Microelectronics and Nanoelectronics at the University of Malta started studying the use of microsystems technology in the design of radio frequency applications in collaboration with CEA LETI, Grenoble, France. In particular, the application of a MEMS based piezomagnetic variable inductor for frequency tuning purposes in voltagecontrolled oscillators (VCO) was investigated [1,2]. Although traditionally a MEMS variable inductor is considered as a means to extend the frequency tuning range, in this work it was shown that with correct inductor design it is also possible to facilitate and improve the VCO design in terms of phase noise response and power consumption in comparison to a design based on standard capacitive tuning [2]. This led to other research investigations such as, the design of reconfigurable radio frequency matching networks [3] and low noise amplifiers for multi-standard applications [4]. Currently more design work is being carried out in the department in collaboration with ST Microelectronics (ST-M), which is funding

November 2012

Issue No. 43

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Mems meet ASICs (cont.)

postgraduate studies in both MEMS and ASIC design. In particular, inertial microsensors such as accelerometers intended for gaming console applications are being investigated [5]. The ASIC design is being focused on analogue to digital converters utilising successive approximation and sigma-delta techniques. The MEMS devices are being designed using ST-M propriety ThELMA (Thick Epitaxial Layer for Microactuators and Accelerometers) process. ASICs are designed using CMOS processes available through Europractice (an EU-funded initiative intended to promote research in Universities in advanced integrated circuit design) to which the department has access.

the ASIC for the signal processing) placed in a single package. The way forward in such a design is to improve the interfacing and integration of these two chips. The research also focuses on reduction of power consumption and the chip area cost. These are crucial features in handheld battery-operated devices, in which these sensors are mainly applied. Research in this area is being carried out by Mr Ansel Briffa, Mr Jean Marie Darmanin and Mr Kristian Grixti, as part of their Master of Science in Microelectronics and Nanoelectronics at the Faculty of Information and Communication Technology. They are being supervised by Dr Ing. Edward Gatt, Dr Ivan Grech, Dr Ing. Owen Casha and Prof. Ing. Joseph Micallef. Training in this field will hopefully entice industry in building research and design teams oriented towards this rapidly expanding field [6]. This proves that the development of microsystems, where MEMS meet ASICs, is an attractive and exciting field to work in. It is a multiphysics field that provides new and innovative ways of implementing smart micro devices with functionalities that previously one would not have dreamed of. ET

These accelerometer microsystems require two main functions: sensing and signal processing which are made possible by using two separate chips (the MEMS chip for the microsensor and

Dr. Ing. Owen Casha

Lecturer, Department of Microelectronics and Nanoelectronics, Faculty of ICT, University of Malta

November 2012

Issue No. 43

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Interview with Ing. Jeanelle Cassar by Ing. John Pace

Jeanelle Cassar is young and very modest about her personal achievements, which are considerable.

When Ing. Jeanelle Cassar asked to submit an article for Engineering Today I thought it would be interesting to interview her. I met her at her office at Marsa where she is the Commercial Manager of the family owned company Cassar Ship Repairs. The aspects that impressed me most were her self confidence, and the fact that she enjoyed challenges. ET: You are commercial manager at Cassar Ship Repair. Tell me something about the company. JC: The company was founded by my grandfather, who originally was in the business of transporting tourists to Gozo and at first carried out repairs to his own boats. The company expanded mostly when my father, Anthony Cassar, took over, assisted by his two brothers. Originally the company was engaged mostly with afloat repairs, but later on acquired a floating dock which can accommodate small size vessels. The dock has a length of 126 meters and a width of 21 meters and can take ships up to 5000 tons. The shipyard also has slipways and workshops at the extreme end of the Grand Harbour. However the forte of the company is afloat repairs, and in this case the scope is unlimited. ET: What are the capabilities of the company? JC: We undertake all sorts of work, steelwork, mechanical works, the main engine room, cleaning, and of course when more specialized works such as automation or electronics are required we involve also our subcontractors.

ET: What was your relationship with the former Malta Drydocks? JC: At that time Cassar Ship repair was not so well known internationally, and was regarded as the smaller undertaking compared with the drydocks. We used to take on works when the drydocks were full and they also used to pass on jobs to us. What distinguished us was that we undertook afloat repairs, which is still our specialization. Here we built our reputation as we undertake emergency repairs outside the yard, unlike the drydocks which required the ship to dock. We have vessels which can go out when needed, for example for anchor replacement outside territorial waters. We also have a self propelled crane barge, and many ship owners got to know about us from this facility, where we are the number one. ET: What are the capabilities of your workforce? JC: Half of our workforce was formerly at the drydocks, and the rest have been with us for a long time. Our welders are fully certified and many of our fitters have gained experience through long years with the company. Some of them have been with the company since it was set up and have grown with it. ET: What is the present state of the company? JC: Formerly we used to wait for work to come to us, but in these past years we have been facing strong competition from ship repairers

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Interview with Ing. Jeanelle Cassar (cont.)

in the East Mediterranean, and so we are concentrating more on marketing. Our CEO, Mr Spiro Aquilina, former General Manager of Malta Drydocks has a vast experience in marketing and both of us joined the company at the same time. We have agents in several countries so that through the agents Cassar Ship Repair is known all over the world, so that, for example if an American ship owner has some needs in the Mediterranean, he will know through our agent in the US of our capabilities, and the agent will contact us when such repairs are needed. ET: What other services does Cassar Ship Repair offer? JC: We have a company in the group that has its own fleet of vessels which does conveyances, that is transportation of persons and provisions to ships; we also have a water barge to supply ships outside territorial waters. We also do towing by means of a tugboat which operates

anywhere in the high seas. Another capability is oil spill containment and cleaning and another section deals with tank cleaning, both on board ships and ashore. ET: Being born in an engineering family was it natural for you to take up engineering? JC: No, in actual fact until I entered university I never considered an engineering career and my father never dreamed of me becoming an engineer. I considered going in for architecture, but finally decided on engineering. While I occasionally visited the works here, I did not consider the company as my future career. I did the summer jobs elsewhere and was thinking of working outside Malta. Having graduated in Mechanical Engineering in Malta, I decided on a masters course in Marine Engineering at Southampton University. As my first degree was of a general nature I was given a two week crash course in Marine Engineering

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Interview with Ing. Jeanelle Cassar (cont.)

to enable me to work with students who had a first degree in the field. Actually the Masters course consisted in applying engineering principles to the marine environment. I was then offered a job with Global Maritime, which is a consultancy firm based in London. I did five months in the design department where we were designing an offshore wind installation vessel.

position sensors to achieve this. This is a complex system and our job was to examine the system and ensure that it maintained its stability in case of failure of any of its components. Fifty percent of my time was spent in the office examining the drawings and making calculations, but then I would go on the vessel and carry out tests. This was done both at the commissioning stage at the shipyard and also on the actual site when the charterers requested a failure analysis report. My life the was out of a suitcase and I used to travel a lot to Asia, to major shipyards in China, to Malaysia, Singapore, where I spent two months, Canada and Norway, which is a big offshore hub. So I spent two years touring the world, writing reports and enjoying the experience. I met a lot of people, though in the offshore field you tend to meet the same group of people, as the number of specialized persons is quite small. After two years I decided to come back and take up the job at the shipyard, which is entirely different, as I had no experience of ship repair. This was two years ago and I learnt, and am still learning a lot from Spiro Aquilina who has a wealth of experience from the drydocks.

This was a great experience, but I yearned for the life of a marine engineer, away from the computer desk, going to all sorts of places around the world. So, after five months of design I asked to be transferred to the marine engineering department. We specialized in the dynamic positioning of vessels: oil production rigs and supply vessels need to maintain their position and have a system of thrusters and

Lately we have succeeded in bringing a luxury passenger vessel for repairs, which is a new experience, as we tend to concentrate on commercial vessels. We have also done floating dock repairs on a superyacht, and we are now looking at expanding our operations by making international contacts. We attend international events where we meet ship owners and agents to bring work to the shipyard. Competition is getting stronger and nowadays we have to knock on doors to get work that previously came by itself. We do a lot of local work, but to survive we need to do international work.

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Interview with Ing. Jeanelle Cassar (cont.)

have to arrange a berth for the ship, and keep in contact with the agents to organize the work. Once the ship comes in I hand over the work to the ship repair manager and come in again after the job is finished. I also do some consultancy work in my free time, where I am involved with another company. ET: You have had a varied career in the last four years. Do you think that women engineers finish doing an office job while it is the men who get their hands dirty? JC: It depends on what the person wants. I still believe that in an office, when you are doing analysis and calculations it is still an engineering job. However hands-on experience is vital and avoids certain mistakes which would not be obvious to a designer without field experience. Finally it depends on the character of the person. ET

ET: What do you spend your day at the shipyard? JC: We are constantly contacting agents to present our yard and spreading our contacts. When a now job comes in we quote and see how we can cut costs to beat the competition. Ship repair is a 24 hour job. Last Christmas we had a ship which had to come in for emergency repairs and we had to organize the work with urgency. When a ship comes in for repairs we

Ing. John Pace Editor, Engineering Today

November 2012

Issue No. 43

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A Grid Connected Brushless DC Drive for Micro-Wind Turbines

by Josef Mizzi

Micro wind turbines can be a prospective source of energy that could further enhance the efforts being done by our country to reach the European set target of 10% renewable energy by 2020. This can be achieved by developing an efficient wind turbine that should also be visually appealing so that it can be used in urban areas. Further improvements in the design of similar micro wind turbines in conjunction with incentives to install micro wind turbines in several residential and rural areas, can provide a significant renewable source of electrical energy generation which can assist the output from photovoltaic panels. Cogging is one of the major problems with permanent magnet alternators used in micro wind turbines, as it tends to hinder the generation of power during light winds. This problem causes the rotor to â&#x20AC;&#x2DC;lockâ&#x20AC;&#x2122; in certain positions and requires larger forces to start turning the rotor. This effect is caused due to the summation of forces required to drive a magnetic pole across the air gap. Therefore the

cut in speed of the wind turbine is increased since higher wind speeds are required to overcome this and provide the required minimum voltage output. The objective of this research was to design, develop and implement a brushless drive system for a micro wind turbine. Moreover, reducing cogging and maximising the voltage output from the alternator will reduce the cut in speed of the turbine. In doing so, it would be possible to draw a small amount of power everyday from the gusty winds, and the peak power at higher winds, instead of limiting the operation of the turbine to those few windy days. Moreover, the aim is to develop the drive in the most cost-effective way possible, and hence present a cheap, effective and attractive solution to electrical power generation from wind energy.

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A Grid Connected Brushless DC Drive for Micro-Wind Turbines (cont.)

In most design procedures â&#x20AC;&#x2DC;Finite Element Techniquesâ&#x20AC;&#x2122; are used to model the flux in the motor in order to design the optimum configuration of magnets. However, in this study, a test rig was designed and constructed so that within a few hours of work the configuration of magnets of the alternator could be changed and tested. The approach adopted was to make a pre thought configuration and obtain the actual results right away.

The design methodology adopted in this study will be explained below. First an analysis on existing hub motors in electric bicycles and electric hub motor were carried out to get a better insight of the existing low speed brushless d.c. and brushless a.c motors on the market. Afterwards the testing rig previously mentioned was constructed. Consequently several configurations of alternators were tested and analysed on this rig. After testing several configurations of magnets and different topologies, results showed that the use of magnets with particular width that will exploit the advantage of the slot in the middle of the stator pole, results in a fairly constant reluctance around the alternator. Cogging is significantly reduced without much loss in voltage.

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Furthermore the use of flat magnets rather than arched magnets also proved to reduce cogging. When a number of flat magnets are placed to form a circular rotor, the resulting airgap is not constant. This is of an advantage since when the magnetic pole is passing across the salient stator pole, the magnet is first moved closer and then further away from the salient pole gradually due to the variation in the airgap around the rotor as shown in figure 1.

Figure 1 â&#x20AC;&#x201C; Cross-section of the alternator, the magnets are shown in Red and Blue, while the stator windings are shown in orange and green. The non uniform air gap is clearly shown Apart from using the optimum magnet configuration deduced from the tests, in the final design of the turbine, two alternators were used. Their angular position with respect to each other were aligned in such a way, that when one is at maximum reluctance the other is at minimum reluctance. This was done to compensate and reduce the overall combined cogging. In this way, no voltage is lost to reduce cogging since this is eliminated mechanically rather than by manipulating the flux.

Two asynchronous ceiling fan motors were converted to a two phase permanent synchronous generators using rear earth magnets which were specifically ordered based on the research work and results. In order to obtain the highest output voltage from the two phase output of the alternator, the voltage was rectified separately and added at d.c. level, to avoid any cancellation of voltage at a.c. level due to phase shifts and unwanted internal circulating currents inside the alternators. The above mentioned drive was assembled in a specially designed vertical axis wind turbine as shown in figure 2. The ideal design of a turbine for such an application was concluded to be a Savonius rotor, inside a set of stationary-fixed wings that directs the wind to particular areas on the rotor to reduce the drag of the turbine. ET

Figure 2 - The wind turbine constructed, including the two alternators

Josef Mizzi

Reading for an MSc by research in Engineering, University of Malta The paper is the winner of the Best Electrical Final Year Studentsâ&#x20AC;&#x2122; Project 2011

November 2012

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Graphic VU Meter Display for Sound Mixers Implemented using FPGAs

by Roberto Drago and Dr. Edward Gatt

This paper describes the implementation of hardware which allows visualisation of the sound amplitude being received from a total of sixteen channels. Abstract - This paper focuses on how a visual display of sixteen separate input channels was achieved, together with the possibility of performing a real-time Fast Fourier Transform (FFT) using the Cooley-Tukey algorithm on one channel selected by the user. These algorithms are realised using VHSIC Hardware Description Language (VHDL) and are implemented on a Field Programmable Gate-Array (FPGA). This paper goes into detail of how a system-on-chip architecture was achieved. This effectively reduces delays, allowing for increased clock speeds and for high parallelism given the design architecture of FPGAs.

on sound signals, it was necessary to obtain sampling speeds of at least 48kHz to correctly resolve frequency components up to 24kHz and avoid aliasing.

Index Terms - Analgue to Digital Converion, Fast Fourier Transform, Field Programmable Gate Arrays, Hardware Description Languages.

Given all the aforementioned considerations, the ADC from Linear Technology (LTC1407A-1) was chosen. It is a high-speed ADC with the capability of outputting 1.5 mega samples per second per channel. The disadvantage of this ADC is that it provides only two channels. Thus, some form of multiplexer had to be used since a total of sixteen input channels were required. This device is also able to convert â&#x20AC;&#x201C;1.25V to 1.25V bipolar inputs differentially and uses the serial SPI interface giving the result of the two sampled channels in 34 clock cycles. An added advantage of this chip is that the sampling rate can be reduced by lowering the speed of the SPI algorithm, effectively reducing the number of conversion commands sent. Thus, the number of samples per second may be varied dynamically according to the need. For example, while in FFT mode, a higher sample per second count is necessary than when displaying the meter display of channel amplitudes. [1]

1. Introduction Such meters are called graphical VU meters, and can accept input from either mixing consoles or from any other sound device. On high-end costly sound mixers, this visualisation is normally embedded on the device itself, but this is not the case with common average priced sound mixers and devices. Additionally, an added visualisation of a real-time FFT was incorporated to the hardware to further aid the sound engineer. 2. Ananlogue To Digital Conversion A) Choice of ADC Integrated Chip The first consideration was the resolution needed. Given that most ADCs give a resolution ranging between 8-bits and 16-bits, the decision to opt for 14-bit minimum resolution ADCs was taken. This gives very small steps of analogue voltage ranges which are necessary for the FFT since it must work within very limited error ranges. The second consideration involved the actual sampling speed. Since this artefact focuses

The third consideration is the ease with which the chip lends itself to the application. To reduce the amount of output pins, most ADCs implement a serial protocol to latch the data from the ADC (slave) to the controller (master). Although this is slightly more complex and slower when compared to a parallel-out ADC, it is preferred since it leads to a chip with a small package and footprint. B) Linear Technology LTC6912 and LTC1407A

Additionally, this circuit was improved by adding a dual serial-programmable gain amplifier chip between the channel inputs and the inputs to the ADC. The IC was again chosen from Linear Technology (LTC6912). This chip allows the ability to program a fixed gain setting or dynamically changing gain ranging from -1 to -100. This chip also offers linear gain response for sound inputs having frequencies up to the range of 48kHz.

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Graphic VU Meter Display for Sound Mixers Implemented using FPGAs (cont.)

3. Sound Input It was necessary to design a circuit to receive the analogue sound input from sixteen separate channels. As discussed earlier, the aim was to be able to receive sound signals from either an XLR connector or a balanced insert jack cable. Thus, the connector from Neutrik NCJ6F-AH-0, shown in Figure 1, was chosen as it houses a combo connector for both XLR inputand balanced jack.

Figure 1 - Neutrik NCJ6F-AH-0 Combo Connector [2] For this artefact, insert cables were used to simply tap the audio signal into the FPGA without introducing any effects on the audio signal. This is achieved by using a male to male balanced jack cable and soldering together the Tip and Ring section of the female combo Neutrik connector. Using this technique, the signal is routed from the mixer to the FPGA and back to the mixer without any intervention on the signal. The disadvantage of such system is that the audio signal is single-ended. Given this, the signal coming from the inverting input of the XLR had to be ignored so that the circuitry after the connector deals with the same type of single-ended signals. 4. Flash Storage The flash memory chosen is a ST25P16 chip which was configured on a PmodÂŽ by Digilent. The main features of this chip include: 16Mbit

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storage space, a large data retention rate, multiple erase/program cycles, and works through the SPI communication protocol described above. This flash ROM was needed to save the user text which labels each meter. 5. Meter Display The continuously sampled channels are displayed on the screen as VU meters as shown in Figure 2.

Figure 2 - Meter Display Screen The top-section of the screen is reserved to channel labelling. Alterations in the text are achieved through the PS2 keyboard. This was achieved using a keyboard scanner block and a keyboard de-bouncer block to ensure that only one key command is sent per actual key pressed. The next section of the screen displays the actual sixteen VU meters of the sixteen separate sound input channels. During this mode, the multiplexer constantly selects the channels one by one for eventual conversion to a digital representation. Given that this occurs at a fast rate, the user is led to believe that the channels are being sampled simultaneously. 6. FFT Processing The FFT processor implemented is able to convert 1024 sampled data points from the

II.

block

METER DISPLAY

continuously sampled channels are displayed on the s VU meters as shown in Figure 2.

Figure 3 - 8-point FFT Example [5]

For each iteration, the following equations were computed for the butterfly pair shown in Figure 4:

time domain into the frequency domain in real-time. This is achieved through the use of the Cooley-Tukey algorithm [3], together with the use of various True Dual Port Block RAMs generated through the Xilinx Core Generator Utility [4]. Figure 2 - Meter Display Screen

Three nested-loops were programmed to keep track of the number of butterflies processed, number of blocksto processed top-section of thethescreen is reserved channel and the stages processed. g. Alterations in thenumber text are of achieved through theFor PS2example, for an eight-point FFT, stage 1 has four blocks of one butterfly pair each, stage 2 has two blocks of two butterfly pairs each, while stage 3 has one block with four butterfly pairs. An example follows in Figure 3. Figure 1 - Butterfly Pair [5]

Figure 4 - Butterfly Pair [5]

Figure 4 - Butterfly Pair [5] Figure 1 - Butterfly Pair Pair [5] [5] Figure 1 - Butterfly

Once thethe pr Once

For each butterfly pair in the network shown, the the output from output fr Once1 1-the processing is finalised, the algorithm copies all Figure Figure -Butterfly Butterfly Pair Pair[5] [5] Once OnceOnce the the proces proce Figure 1 - Butterfly Pairperformed [5] theth the following computations were in Throughout Throughout the output from thePair last RAM Figure Figure 1 1- -Butterfly Butterfly Pair[5] [5] block used to RAM_final_data. the theOnce output output from from the the Once the the proces proce the outputcom froc imaginary For each butterfly pair pair in the network shown, following imaginary For each butterfly inprocess, theFigure network shown, the following succession: Throughout this the realthecomponents the Throughout Throughout this this p theand output output from from the the 1 - Butterfly Pair [5] the

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as per the following equations: 5 and Fi đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; (đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2) as per the equations: ďż˝2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; upper butterfly, 21) đ?&#x2018; đ?&#x2018; (đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x2C6;&#x2019;đ?&#x2018; đ?&#x2018; 1) sharp changes in = = ++ cos cos ďż˝ďż˝ each ďż˝2ďż˝butterfly â&#x2C6;&#x2019; jsin jsinďż˝ďż˝in the ďż˝ďż˝2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; for for the the first first oo2 upper đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 ďż˝(đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; â&#x2C6;&#x2014; đ?&#x2018; đ?&#x2018; ďż˝ +following â&#x20AC;˘ Compute for the butterfly pair confirms that theth đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x160;đ?&#x2018;&#x160; butterfly, 10kHz square-wa đ?&#x2018; đ?&#x2018; butterfly, 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2019;= ďż˝ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;ďż˝ + ďż˝jsin +cos jsin for the o đ?&#x2018;&#x160;đ?&#x2018;&#x160; time FFT processo ďż˝ďż˝ ďż˝â&#x2C6;&#x2019; o oupper đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 = đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 +ďż˝+ ďż˝ ďż˝cos = â&#x2C6;&#x2019;=cos ďż˝cos ďż˝= for the đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018; đ?&#x2018; butterfly, upper = + ďż˝ jsin ďż˝ ďż˝ for the first o đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Figure 5 and Fi đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ďż˝ ďż˝ â&#x2C6;&#x2019; jsin ďż˝ ďż˝ for the o đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018; đ?&#x2018; 2 đ?&#x2018; đ?&#x2018; sharp changes in 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; being taken into consideration. đ?&#x2018; đ?&#x2018; theđ?&#x2018; đ?&#x2018; â&#x2C6;&#x2019; (đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 1) â&#x2C6;&#x2014; đ?&#x2018; đ?&#x2018; ďż˝ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1) upper butterfly, đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; butterfly, for the2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; each butterfly in butterfly pair that theth otheupper đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 = ďż˝(đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? + đ?&#x2018; đ?&#x2018; confirms đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 10kHz square-wa 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; o2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 â&#x2C6;&#x2019; 1) â&#x2C6;&#x2014;and đ?&#x2018; đ?&#x2018; ďż˝ + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; ďż˝ + jsin ďż˝ ďż˝ â&#x20AC;˘ for Compute o đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; = â&#x2C6;&#x2019; cos ďż˝ second ďż˝= ďż˝(đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? o đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 =butterfly. đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 ďż˝ for timeFigure FFT second lower butterfly, đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 5processo Fi đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; =đ?&#x2018;&#x160;đ?&#x2018;&#x160; =â&#x2C6;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; cos cos ďż˝â&#x2C6;&#x2019;ďż˝lower ďż˝upper ďż˝butterfly. + +jsin jsin ďż˝ďż˝upper ďż˝+ ďż˝ butterfly the the thesharp oo butterfly, 2 for being taken into consideration. changes in đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; (đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; 1) â&#x20AC;˘ Compute đ?&#x2018;&#x160;đ?&#x2018;&#x160; for the each butterfly in the pair đ?&#x2018; đ?&#x2018; = cos ďż˝ ďż˝ + jsin ďż˝ ďż˝ for o confirms that theth đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; (đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; + â&#x2C6;&#x2019;the 1) 10kHz square-wa đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; ďż˝+ second lower butterfly. đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2019; for the first đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; time FFT processo ďż˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 = đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 ďż˝ o o đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x160;đ?&#x2018;&#x160; = = â&#x2C6;&#x2019; â&#x2C6;&#x2019; cos cos ďż˝ ďż˝ ďż˝ ďż˝ + + jsin jsin ďż˝ ďż˝ ďż˝ for for the đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; obtained above with the â&#x20AC;˘ Multiply the two values of đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; into second second lower lower butterfly. butterfly. 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; being taken consideration. obtained above with the â&#x20AC;˘ Multiply the two values of đ?&#x2018;&#x160;đ?&#x2018;&#x160; 2 ďż˝ + jsin đ?&#x2018; đ?&#x2018; sharp changes in đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; the 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x20AC;˘ Compute đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; second for the each butterfly in butterfly pairďż˝ + jsin đ?&#x2018; đ?&#x2018; ďż˝ ďż˝â&#x2C6;&#x2019; ďż˝ ďż˝ for the o butterfly. đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2019; confirms that theth lower cos ďż˝ first for o= đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; ďż˝ 10kHz square-wa đ?&#x2018; đ?&#x2018; =butterfly, +lower cos ďż˝= ďż˝ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x2C6;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; jsin ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;cos đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018; đ?&#x2018; according đ?&#x2018; đ?&#x2018; the đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ďż˝ = upper ďż˝ the othesecond đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 + ďż˝for đ?&#x2018; đ?&#x2018; second lower butterfly. lower butterfly. đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; + â&#x20AC;˘ Multiply the two values of đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; obtained above with second input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ďż˝ butterfly tothe đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 +đ?&#x2018; đ?&#x2018; changes ďż˝ ďż˝ đ?&#x2018; đ?&#x2018; ođ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; being taken into consideration. 2= second lower input + ďż˝ according to ďż˝ sharp in 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Compute đ?&#x2018;&#x160;đ?&#x2018;&#x160; for the each butterfly in the pair đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; obtained obtained above above with with the the â&#x20AC;˘ â&#x20AC;˘ Multiply Multiply the the two two values values of of đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x160;đ?&#x2018;&#x160; second lower butterfly. 2 confirms that theth second lower butterfly. 2 đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; of =butterfly, + cos ďż˝2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; ďż˝2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; ďż˝ forabove the first đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; upper obtained with the â&#x20AC;˘ othe Multiply the two values đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; to đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;jsin đ?&#x2018; đ?&#x2018; ofďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + đ?&#x2018; đ?&#x2018; ďż˝2ďż˝ according to đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; above equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; compute two temporary đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; â&#x20AC;˘â&#x20AC;˘â&#x20AC;˘ Multiply the two values of obtained obtained above with with the the Multiply the two values being taken into consideration. đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; sharp changes in th 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; to compute two temporary = â&#x2C6;&#x2019; cos ďż˝ ďż˝ + jsin ďż˝ ďż˝ for oequation: đ?&#x2018;&#x160;đ?&#x2018;&#x160; ďż˝ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Compute đ?&#x2018;&#x160;đ?&#x2018;&#x160; for the each butterfly in the butterfly pair 2 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; second second lower lower input input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + + ďż˝ ďż˝ according according to to the the đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ďż˝ ďż˝ đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = â&#x20AC;˘ Multiply đ?&#x2018;&#x160;đ?&#x2018;&#x160; for butterfly in the butterfly pai 2Compute above with thewith the values ofeach đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; â&#x20AC;˘ ďż˝đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018; đ?&#x2018; ďż˝ two +â&#x20AC;˘The cos ďż˝input â&#x2C6;&#x2019; jsin ďż˝the for the first o second đ?&#x2018;&#x160;đ?&#x2018;&#x160; 2+ 2two upper butterfly, obtained above Multiply the values oftođ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; the â&#x20AC;˘ second Multiply two values of obtained above lower according ďż˝algorithm đ?&#x2018; đ?&#x2018; ďż˝2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; obtained đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; the đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; values. multiplication of these two equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + đ?&#x2018; đ?&#x2018; ďż˝2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; to compute two temporary đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ďż˝ offor 2 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;+ being taken into consideration. second lower lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ďż˝into according to tođ?&#x2018; đ?&#x2018; the thetwo ďż˝jsin ďż˝2ďż˝algorithm =+lower â&#x2C6;&#x2019;input cos ďż˝ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; ďż˝+ + ďż˝according ovalues. đ?&#x2018;&#x160;đ?&#x2018;&#x160; The multiplication these đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; second butterfly. being taken consideration. đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; equation: equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; to to compute compute two two temporary temporary ďż˝ ďż˝ 2 đ?&#x2018; đ?&#x2018; ďż˝ đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; lower input + ďż˝ +according to theFigure ďż˝ lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; toFig đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; upper butterfly, ďż˝2ďż˝ according + cos ďż˝isđ?&#x2018; đ?&#x2018; ďż˝ second â&#x2C6;&#x2019; jsin ďż˝below: for the temporary first ocomplex đ?&#x2018;&#x160;đ?&#x2018;&#x160; with the= second rx(k+Nâ &#x201E;2) 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; complex islower outlined 2+ 2second equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; to input two ďż˝ đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2the values. The multiplication algorithm of these numbers outlined below: đ?&#x2018; đ?&#x2018; numbers đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; two 2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; = â&#x2C6;&#x2019; cos ďż˝ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; ďż˝tođ?&#x2018; đ?&#x2018; algorithm +đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; jsin ďż˝compute ďż˝ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; for otwo đ?&#x2018;&#x160;đ?&#x2018;&#x160; second lower butterfly. 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + + to compute compute two temporary temporary ďż˝values ďż˝2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; values. values. The The multiplication multiplication algorithm of of these these two two 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; ďż˝đ?&#x2018; đ?&#x2018; đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ofequation: đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;+ 2 Figure 2 FFT Display Screen â&#x20AC;&#x201C; 10kHz Sine Wave obtained above with the â&#x20AC;˘ equation: Multiply the two đ?&#x2018;&#x160;đ?&#x2018;&#x160; = + cos ďż˝ ďż˝ â&#x2C6;&#x2019; jsin ďż˝ ďż˝ for the first o values. đ?&#x2018;&#x160;đ?&#x2018;&#x160; upper butterfly, equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; to compute two temporary equation: + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; to compute two tempo ďż˝ according to the x(k+Nâ &#x201E;2) W_ complex numbers is outlined below: 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; The multiplication algorithm of these two đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; o =+ cos ďż˝these ďż˝two jsin ďż˝ ďż˝ forFigure the 2firs oďż˝below: đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2019; + 2of (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) +đ?&#x2018; đ?&#x2018; 2 đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018; đ?&#x2018; + đ?&#x2018; đ?&#x2018; â&#x2C6;&#x2019; Figure 2- FF â&#x2C6;&#x2019;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? ďż˝+ đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) +đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; ďż˝â&#x2C6;&#x2019; ďż˝+đ?&#x2018; đ?&#x2018; these for the o osecond đ?&#x2018;&#x160;đ?&#x2018;&#x160; lower butterfly. values. The The multiplication multiplication algorithm algorithm of complex complex numbers numbers iscos isoutlined outlined below: (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;=two + (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; +two đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?)đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; jsin Figu above with the â&#x20AC;˘ values. Multiply the values đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018; đ?&#x2018; of đ?&#x2018; đ?&#x2018; values. numbers isđ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; outlined below: values. The multiplication algorithm these values. The algorithm ofđ?&#x2018; đ?&#x2018; theseofFigure two second lower input += ďż˝multiplication according to ďż˝obtained upper to complex compute two temporary The đ?&#x2018; đ?&#x2018; upper 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; butterfly, Figure 2 2- FF butterfly, 2 o (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) complex complex numbers numbers is is outlined outlined below: below: đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = â&#x2C6;&#x2019; cos ďż˝ ďż˝ + jsin ďż˝ ďż˝ for the o đ?&#x2018;&#x160;đ?&#x2018;&#x160; second lower butterfly. o Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192;according = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?) đ?&#x2018; đ?&#x2018; complex numbers is outlined below: đ?&#x2018; đ?&#x2018; complex numbers is+ outlined below: omultiplication o (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = =value: (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; +đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018; đ?&#x2018; above with theđ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) â&#x20AC;˘ Multiply the two values of đ?&#x2018;&#x160;đ?&#x2018;&#x160; o(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; Work đ?&#x2018;&#x192;đ?&#x2018;&#x192; = â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?)â&#x2C6;&#x2019; second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) tođ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) ďż˝obtained đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; + equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + temporary ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; compute two temporary ďż˝đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) algorithm of these two 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; o+ (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + =â&#x2C6;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) + + 2 đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)tođ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = â&#x2C6;&#x2019; cos ďż˝ ďż˝ + jsin ďż˝ ďż˝ for the o đ?&#x2018;&#x160;đ?&#x2018;&#x160; o Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) second lower butterfly. đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; o (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? +values +đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = =(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) + +đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; ++ đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018; đ?&#x2018; Figure 3 - 8-point FFT Example [5] đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; =đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; cos ďż˝=these ďż˝â&#x2C6;&#x2019;+ jsin ďż˝â&#x2C6;&#x2019; đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; for + the đ?&#x2018;&#x160;đ?&#x2018;&#x160; ođ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? Work value: = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) above with the â&#x20AC;˘ Multiply the two đ?&#x2018;&#x160;đ?&#x2018;&#x160; second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) according tođ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018; đ?&#x2018; ofo(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018; đ?&#x2018; oďż˝obtained (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;+ đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?)ďż˝ + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? đ?&#x2018; đ?&#x2018; ovalue: + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; to compute two temporary ďż˝2temporary values. The multiplication algorithm of two ocomplex o Work temporary temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018; đ?&#x2018; = = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) đ?&#x2018;?đ?&#x2018;?) numbers isđ?&#x2018;&#x192;đ?&#x2018;&#x192; outlined đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; oWork temporary đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? +â&#x2C6;&#x2019;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) 2đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; đ?&#x2018; đ?&#x2018; value: o Work Work temporary đ?&#x2018;&#x192;đ?&#x2018;&#x192;== đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?)below: đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; second lower butterfly. đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; value: đ?&#x2018; đ?&#x2018; Figure 2 đ?&#x2018; đ?&#x2018; o Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) â&#x20AC;˘ Multiply second lower butterfly. complex numbers is outlined below: second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝ according to ďż˝ o o Work Work temporary temporary value: value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x192;đ?&#x2018;&#x192; = = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) đ?&#x2018;?đ?&#x2018;?) obtained above with the the two values of đ?&#x2018;&#x160;đ?&#x2018;&#x160; values. The multiplication algorithm of these two equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; to compute two temporary ďż˝ o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018; đ?&#x2018; o Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) đ?&#x2018; đ?&#x2018; 2 o Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) 2 value: oo Work temporary temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; =đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + +đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) oWork Work temporary đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019;+đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; value: Figure 2 đ?&#x2018; đ?&#x2018; = onumbers Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;== đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; values complex is outlined below: above with the the two of đ?&#x2018;&#x160;đ?&#x2018;&#x160; second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) according ďż˝obtained values. multiplication algorithm ofđ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; these two equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; to compute two temporary ďż˝â&#x20AC;˘result: o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) â&#x20AC;˘ Multiply (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;The + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; +đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) + (a+jb)(c+jd) = (ac-bd) + j(ad+bc) obtained with the Multiply the two values ofto đ?&#x2018;&#x160;đ?&#x2018;&#x160; oo Work Work temporary temporary value: value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? ++ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018; đ?&#x2018; = 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? 2 o Real đ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; o Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;+=đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;?above + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)Figure o Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) =đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? 2oo Work Work temporary temporary value: value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; below: đ?&#x2018; đ?&#x2018; o Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; + complex numbers is outlined đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝ according to the ďż˝ values. The multiplication algorithm of these two o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; to compute two temporary ďż˝ o (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) Work temporary value: P=c(a-b) second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝ according to the ďż˝ đ?&#x2018; đ?&#x2018; 2 o Real đ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; equations were 2 value: đ?&#x2018;&#x192;đ?&#x2018;&#x192;= đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?) For each iteration, theresult: following oo Work Work temporary temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) 2đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;â&#x2C6;&#x2019;=đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)Figure oresult: Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; â&#x2C6;&#x2019;â&#x2C6;&#x2019;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) o=temporary Work temporary value: 2đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; o Work value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? complex numbers is outlined below: đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; o o Real Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x192;đ?&#x2018;&#x192; + + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; values. The multiplication algorithm of these Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018;&#x192;đ?&#x2018;&#x192; to compute temporary (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) ďż˝2đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) Work temporary value: Q=a(c+d) đ?&#x2018; đ?&#x2018; = equation: +â&#x2C6;&#x2019;+đ?&#x2018; đ?&#x2018; two ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; totwo compute two temporary ďż˝đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; o Real result: + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; o o Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?) computed for othe Imaginary butterfly result: pair shown đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; in Figure 4: 2 Figure 2 Work temporary đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) oo Real Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192;is đ?&#x2018;&#x192;đ?&#x2018;&#x192;+ +đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;temporary đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;value: complex numbers outlined below: oresult: Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192;then đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;added The above two values must with of these o Real đ?&#x2018;&#x192;đ?&#x2018;&#x192;â&#x2C6;&#x2019; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; be The multiplication algorithm ofđ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; these two oâ&#x20AC;˘The (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; +result: đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) =â&#x2C6;&#x2019; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) ++ ++ đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) Work temporary value: R=b(c-d) oo Imaginary Imaginary result: result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x192;đ?&#x2018;&#x192; values. The multiplication algorithm two Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?) â&#x20AC;˘values. above two temporary values must be added with othe Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; o Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; =đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) then Figure 2Fig upper input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to produce the final complex numbers is outlined below: â&#x20AC;˘ The above two temporary values must be then added with Figure o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) ootheImaginary Imaginary result: result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;o= complex numbers isđ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; outlined below: Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019;final đ?&#x2018;&#x192;đ?&#x2018;&#x192; (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + result: đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; =â&#x2C6;&#x2019;Imaginary (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) + + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) upper input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to produce the Real P+R o result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Work temporary value: đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) đ?&#x2018; đ?&#x2018; Figure 3 FFT Display Screen â&#x20AC;&#x201C; 10kHz Square Wave â&#x20AC;˘ â&#x20AC;˘ The The above above two two temporary temporary values values must must be be then then added added with with o Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . These are the ďż˝ the upper input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to produce the đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018; đ?&#x2018; o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; â&#x20AC;˘ final The above two temporary values must be then added with đ?&#x2018; đ?&#x2018; 2 o (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; +two đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . temporary These are Figure Figure 3 3- FF -F ďż˝đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) Imaginary Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192;result: ++ đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;value: đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x20AC;˘đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; The above values must be then oQ-P (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) =đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;the đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; +added đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) đ?&#x2018; đ?&#x2018; two â&#x20AC;˘â&#x20AC;˘ The The above above two temporary temporary values values must must be be then then added added with with the the upper upper input input complex complex number number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to to produce produce the the final final o temporary đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) 2 â&#x20AC;˘ one The above two temporary values must be then with Figurw o Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) output of butterfly pair. The addition algorithm of +added computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + đ?&#x2018; đ?&#x2018; ďż˝2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; . These are the theWork upper input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to produce the final o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) Figure Figure3the 3- FF - Ff đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; of one butterfly pair. The addition algorithm of the upper input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to produce đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; ooutput Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; the theupper upper input input complex complex number number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to to produce produce the the final final the upper input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to produce the final đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; computation: computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . . These These are are the the ďż˝ ďż˝ o Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) these temporary two complex outlined đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; is Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; o temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;?đ?&#x2018;?) The fina output of one butterfly pair. The addition algorithmo of Work value: =outlined đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? +đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) 2Work 2= computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) +numbers đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . below: These are the ďż˝đ?&#x2018; đ?&#x2018; I2 MPLEMENTATION I. đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; these two complex numbers is below: đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . These are ďż˝ o Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192;+computation: +đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; computation: computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ++ ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; These These are the ďż˝The ďż˝2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018; đ?&#x2018; đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; +are ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; These are the ďż˝2the output of ofone one butterfly butterfly pair. pair. The addition algorithm algorithm of of The final đ?&#x2018; đ?&#x2018; addition đ?&#x2018; đ?&#x2018; . .+ these two complex numbers is outlined below: output o output Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019;+ đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018; đ?&#x2018; .2 2 Digilent Spag o Work value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; oftemporary one butterfly pair. The addition algorithm of addition otwo + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?file + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? +with đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) â&#x20AC;˘ output The above temporary values must be then added The final generated program was implemented aalgorithm oisđ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; =The đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)on o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;Work = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) output of one butterfly pair. these these two two complex complex numbers numbers is outlined outlined below: below: output of of one one butterfly butterfly pair. pair. The The addition addition algorithm algorithm of of Digilent Spartan (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; o Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; o + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) output of one butterfly pair. The addition algorithm of oupper Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192;these Figure 3 - Fi this artefact these two complex numbers is outlined below: The The final final genera gene Digilent Spartan Starter Board The design summary for the input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to final o (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + two complex numbers is outlined below: â&#x20AC;˘ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) The above temporary values must be then added with The final otwo Real result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;ois3E +outlined đ?&#x2018;?đ?&#x2018;?Work o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019;produce đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) [6]. these these two complex complex numbers numbers is outlined below: below: this artefact is g these two complex numbers isthe outlined value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; =below: đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) final o two Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Digilent Digilent Spartan Spartan 3E (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; The The final genera gene Figure 33E -F oo +input + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + +đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) (đ?&#x2018;?đ?&#x2018;? (đ?&#x2018;?đ?&#x2018;? ++ + (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + +đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?temporary + ++ đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? ++đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) đ?&#x2018; đ?&#x2018; this artefact is given in Table III. Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; o(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; Real result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + đ?&#x2018;?đ?&#x2018;?== the upper complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to produce the final computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . These are the ďż˝ Digilent Spart (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x20AC;˘ The above two temporary values must be then added with o + + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = đ?&#x2018;?đ?&#x2018;? + + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) đ?&#x2018; đ?&#x2018; o Real result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; 2 o + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) o Imaginary result: đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; this this artefact artefact is is given given Digilent Digilent Spartan Spartan 3E 3E (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; o o + + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + + (đ?&#x2018;?đ?&#x2018;? (đ?&#x2018;?đ?&#x2018;? + + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + + đ?&#x2018;?đ?&#x2018;? đ?&#x2018;?đ?&#x2018;? + + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; o + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) đ?&#x2018; đ?&#x2018; Figure 3 o Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; oupper Imaginary result: â&#x2C6;&#x2019;đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x192;đ?&#x2018;&#x192;+ this artefact -isF computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; These are final the the complex number to the ďż˝ đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) of input one butterfly pair. The addition algorithm of oo Real result: result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;++ đ?&#x2018;?đ?&#x2018;?đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; oReal result: â&#x20AC;˘ output The above two temporary values be. produce then with TABLE I. đ?&#x2018; đ?&#x2018; D ESIGNadded Sđ?&#x2018;&#x17D;đ?&#x2018;&#x17D;UMMARY this thisartefact artefactisisgiven given o Imaginary Real result: + đ?&#x2018;?đ?&#x2018;?đ?&#x2018; đ?&#x2018; 2ođ?&#x2018;&#x2018;đ?&#x2018;&#x2018;must o Imaginary result: đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; Real result: + đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;below: Figure 3 - F these two complex numbers is outlined â&#x20AC;˘ The final result was scaled by a scaling factor of two. ooupper Real Real result: result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;++đ?&#x2018;?đ?&#x2018;?đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;?đ?&#x2018;?ođ?&#x2018;&#x201E;đ?&#x2018;&#x201E; result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;. produce + đ?&#x2018;?đ?&#x2018;?algorithm Imaginary result: â&#x2C6;&#x2019;Real đ?&#x2018;&#x192;đ?&#x2018;&#x192;The of input one butterfly pair. addition of computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; These are the ďż˝Imaginary the complex number o result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;the â&#x2C6;&#x2019; final đ?&#x2018;&#x192;đ?&#x2018;&#x192; TAB TA đ?&#x2018; đ?&#x2018; to â&#x20AC;˘ â&#x20AC;˘output The above two temporary values must be added with The final gene November 2012 |then Issue No. 43 2đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) 37 ooThe Imaginary Imaginary result: result: đ?&#x2018;?đ?&#x2018;? đ?&#x2018;?đ?&#x2018;? + + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; final result was scaled by a scaling factor of two. đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; isđ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) necessary since the FFT results keep scaling oThis Imaginary result: đ?&#x2018;?đ?&#x2018;?outlined +ođ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; addition đ?&#x2018; đ?&#x2018; these two complex numbers is= below: Imaginary result: đ?&#x2018;?đ?&#x2018;?the + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; Figure 3TAB -F â&#x20AC;˘ The final result was scaled by a scaling factor two. output of one butterfly pair. The algorithm of TA computation: + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . These are ďż˝ the upper input complex number to produce the final Digilent Spartan 3E (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) o Imaginary result: đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; oo Imaginary Imaginary result: result: đ?&#x2018;?đ?&#x2018;? đ?&#x2018;?đ?&#x2018;? + + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018; đ?&#x2018; The final gene â&#x20AC;˘ of The above two temporary values must be then added with Thisupwards is necessary since the2 large FFT results keep scaling â&#x20AC;˘ scaled The above two temporary values must be then added with resulting inby very numbers. Given the size đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;below: This is necessary since the FFT results keep scaling â&#x20AC;˘â&#x20AC;˘ The The final final result result was was scaled by a a scaling scaling factor factor of of two. two. these two complex numbers is outlined đ?&#x2018; đ?&#x2018; Figure 3 - Ft this artefact is given output of one butterfly pair. The ofđ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) Spartan (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; computation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; .scaling These are the ďż˝(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; the upper input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) produce the final + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? =The +addition đ?&#x2018;?đ?&#x2018;?ato+ đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) â&#x20AC;˘o The final result was scaled by factor of two. â&#x20AC;˘fixed final result was scaled by aDigilent factor of3E resulting in very large numbers. Given the size The final đ?&#x2018; đ?&#x2018; the upper input complex number to produce thegene fina limitations and the width ofalgorithm the RAM blocks, itscaling 2data oupwards Real result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;+++ đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)

- Butterfly [5]Figure Once islastfinalised, algorithm the processing outputthe from the RAM blockthe used to RAM_final_data. 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; components are combined tocopies give the 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; the 1 - Butterfly Pair [5]2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; imaginary đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;Pair Once processing is finalised, thetogether algorithm copies all all đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; tterflyFigure pair in1 the shown, following cos ďż˝ o the ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;+ the ďż˝ onetwork đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; = â&#x2C6;&#x2019; = ďż˝â&#x2C6;&#x2019; cos ďż˝ forďż˝ + jsin ďż˝ Throughout for the this process, the real components and the đ?&#x2018; đ?&#x2018; jsin đ?&#x2018; đ?&#x2018; 1 - Butterfly đ?&#x2018; đ?&#x2018; [5] đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; the output from the last RAM block used to RAM_final_data. Figure Pair the output from the last RAM block used to RAM_final_data. Once the processing is finalised, the algorithm copies all magnitude response. This is achieved through the following ere performed in succession: lower butterfly. lower shown, butterfly. imaginary components are the combined together to give Forsecond each butterfly pair insecond the network the following Throughout thistheprocess, realused components and the output process, from last the RAM block to RAM_final_data. Throughoutthe this real components and the equation: magnitude response. This achieved through đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; computations were performed in succession: đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;following imaginary components areis the combined together the to following give the For each butterfly pairofinđ?&#x2018;&#x160;đ?&#x2018;&#x160;the network shown, the Throughout this process, real components and the obtained above with the â&#x20AC;˘ Multiply the two values obtained above with the â&#x20AC;˘ Multiply the two values of đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018; đ?&#x2018; and đ?&#x2018; đ?&#x2018; imaginary equation: the indexes đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; (đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1) đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + components are combined butterfly pair in computations the network following magnitude response. This achieved together through following wereshown, performed in imaginary components areis combined together the to to givegive the the VU Meter for Sound Mixers FPGAs (cont.) For each butterfly pair inthe network shown, đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;ďż˝succession: đ?&#x2018; đ?&#x2018; following â&#x20AC;˘ Graphic Compute the indexes (đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1) and đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;Implemented + second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;Display +the to second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; the + to theusing ďż˝the 2 is 2 â&#x20AC;Ś (3) 2ďż˝ according equation: 2ďż˝ according |đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; | ďż˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; magnitude response. This achieved through the following = ) + đ??źđ??źđ??źđ??ź(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; ) magnitude response. This is achieved through the following computations were performed in succession: sđ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2) were performed in succession: as per the following equations: 2 2 đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; ďż˝ equation: â&#x20AC;˘ Compute indexes đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;the (đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1) and đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + two temporary |đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; | = ďż˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ) + đ??źđ??źđ??źđ??ź(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ) â&#x20AC;Ś (3) đ?&#x2018; đ?&#x2018; ďż˝ two as percompute equations: ďż˝2(đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2) equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + the +following to compute equation: 2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; to đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; temporary equation: (đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2) as per following equations: ďż˝2The â&#x20AC;˘ đ?&#x2018; đ?&#x2018; Compute the values. indexes đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;the (đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1) and đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + of these two |đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; | = ďż˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; )2 + đ??źđ??źđ??źđ??ź(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; )2 â&#x20AC;Ś (3) values. multiplication algorithm of these two The multiplication algorithm e the indexes đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; (đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1) and đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + o đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 = ďż˝(đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; 1) â&#x2C6;&#x2014; đ?&#x2018; đ?&#x2018; ďż˝ + 210kHz đ?&#x2018; đ?&#x2018; ďż˝ numbers đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 = complex ďż˝(đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019;asbelow: 1) â&#x2C6;&#x2014; đ?&#x2018; đ?&#x2018; ďż˝ Figure 25 -and FFT Display Screen |đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; |Figure 2+- đ??źđ??źđ??źđ??ź(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; FFTSine Display â&#x20AC;&#x201C; 10kHz Sine Wave = ďż˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; )2taken â&#x20AC;ŚScreen (3)from (đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2) per the following equations: is outlined complex numbers is+ outlined below: đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; Wave Figure 62Figure aređ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; )â&#x20AC;&#x201C;screen-shots the real(đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? 2 taken â&#x2C6;&#x2019; 1) o2 following đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 = ďż˝(đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; 1) â&#x2C6;&#x2014;Figure đ?&#x2018; đ?&#x2018; ďż˝ + 5 and Figure 6 are screen-shots from thea real|đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; | ďż˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; = ) + đ??źđ??źđ??źđ??ź(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; ) â&#x20AC;Ś (3) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2) as per the equations: đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; timeFigure FFT processor for an input of 10kHz sine-wave and đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?o â&#x2C6;&#x2019; (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; 1) + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)o = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? 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The latter screen-shot time FFT processor for an input of 10kHz sine-wave time FFT processor for an input of 10kHz sine-wave and o đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 = đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 + ďż˝ ďż˝ Figure 5 and Figure 6 are screen-shots taken from the real-aand a (đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? 1) â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) value:đ?&#x2018; đ?&#x2018; 2đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) confirms o ďż˝(đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? Work temporary o value: đ?&#x2018;&#x192;đ?&#x2018;&#x192;1) =â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018; đ?&#x2018; that the frequency components from Work temporary 10kHz The generated latter screen-shot đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 = â&#x2C6;&#x2019; â&#x2C6;&#x2014; đ?&#x2018; đ?&#x2018; ďż˝ + time FFTsquare-wave processor forrespectively. an input ofThe 10kHz sine-wave andthea 10kHz square-wave respectively. latter screen-shot ďż˝ o đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 = đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 + ďż˝ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; ďż˝ butterfly in the butterfly đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 = â&#x20AC;˘đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 +theďż˝ each 2 sharp changes in the square wave were also detected. đ?&#x2018; đ?&#x2018; Compute đ?&#x2018;&#x160;đ?&#x2018;&#x160; for pair confirms that the frequency components generated from the realFigure 5 and Figure 6 are screen-shots taken from the 10kHz square-wave respectively. The generated latter screen-shot 2=đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 đ?&#x2018; đ?&#x2018; o â&#x2C6;&#x2019; Work temporary value: =temporary đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) value: o Work đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;ďż˝ = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) o đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 + ďż˝confirms (đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? that the frequency components the đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2 sharp changes in thefrequency square wave were also generated detected. fromfrom being1) taken into consideration. â&#x20AC;˘ Compute đ?&#x2018;&#x160;đ?&#x2018;&#x160; for the each butterfly in the butterfly pair confirms that the components đ?&#x2018; đ?&#x2018; time FFT processor for anwave inputwere of 10kHz sine-wavetheand a đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; consideration. Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) sharp changes in the square also detected. being taken into sharp changes in the square wave were also detected. đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; for the eachâ&#x20AC;˘ obutterfly in the butterfly pair Compute đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x160;đ?&#x2018;&#x160; for the 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; each the butterfly pair 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; butterfly in ďż˝ ďż˝ ďż˝ďż˝ â&#x2C6;&#x2019; jsin ďż˝ ďż˝ for10kHz the first square-wave respectively. The latter screen-shot o đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; = + cos đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;2 =o đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 đ?&#x2018; đ?&#x2018; result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; + being 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Realtaken result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; +consideration. đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;o+ 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Real đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; into n into consideration. 2 + cos ďż˝ ďż˝ â&#x2C6;&#x2019; jsin ďż˝ ďż˝ forconfirms the first that the frequency components generated from the o upper đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; =butterfly, đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019;2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; o â&#x20AC;˘ Imaginary đ?&#x2018;&#x192;đ?&#x2018;&#x192;temporary o Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; = + cos ďż˝ ďż˝ â&#x2C6;&#x2019; jsin ďż˝ ďż˝ for the first o đ?&#x2018;&#x160;đ?&#x2018;&#x160; The above two values must upper butterfly, 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; sharp changes in the square wave were also detected. đ?&#x2018; đ?&#x2018; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; e đ?&#x2018;&#x160;đ?&#x2018;&#x160; butterfly the đ?&#x2018; đ?&#x2018; ďż˝ + jsinpair ďż˝ butterfly ďż˝ đ?&#x2018; đ?&#x2018; ďż˝ for the ojsin đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; ďż˝ =inâ&#x2C6;&#x2019; cos =đ?&#x2018; đ?&#x2018; + for cos the ďż˝â&#x20AC;˘ each ďż˝above â&#x2C6;&#x2019; ďż˝above for the first đ?&#x2018; đ?&#x2018; with đ?&#x2018; đ?&#x2018; upper upper butterfly, be then added the input 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; The two temporary values must be then added with đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; â&#x20AC;˘ The two temporary values must be with đ?&#x2018; đ?&#x2018; ken into consideration. â&#x2C6;&#x2019; cosbutterfly. ďż˝ ďż˝ + jsin ďż˝ ďż˝ for thethen added o second đ?&#x2018;&#x160;đ?&#x2018;&#x160; =đ?&#x2018; đ?&#x2018; lower Figure â&#x20AC;&#x201C; 10kHz Wave Figure 3 - FFTSquare Display Screen â&#x20AC;&#x201C; 10kHz Square Wave đ?&#x2018; đ?&#x2018; input đ?&#x2018; đ?&#x2018; number inputđ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; complex đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;)complex totoproduce the final the number upper đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;)the to produce the final3 - FFT Display Screen complex number produce 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; per butterfly, the upper â&#x2C6;&#x2019; cosbutterfly. ďż˝ đ?&#x2018; đ?&#x2018; ďż˝ +đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; jsin ďż˝ ďż˝ đ?&#x2018; đ?&#x2018; for đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;the o second đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; =lower đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018; đ?&#x2018; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; with â&#x20AC;˘ Multiply the two values đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) computation: +computation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝of2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . +These đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; above +areďż˝2the ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . These are the đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; final đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) computation: x(k)+x(k+Nâ &#x201E;2) Wđ?&#x2018; đ?&#x2018; the đ?&#x2018; đ?&#x2018; obtained ďż˝ output ďż˝of â&#x2C6;&#x2019;one jsin ďż˝two ďż˝ of for the first đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; = + cos2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; second lower butterfly. 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; obtained above with the algorithm of â&#x20AC;˘đ?&#x2018; đ?&#x2018; Multiply the values ofone đ?&#x2018;&#x160;đ?&#x2018;&#x160; butterfly pair. The addition algorithm of output butterfly pair. The addition second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝ according to ďż˝ đ?&#x2018; đ?&#x2018; I. IMPLEMENTATION These are the output of one butterfly I. IMPLEMENTATION đ?&#x2018; đ?&#x2018; 2the = â&#x2C6;&#x2019; cos ďż˝ ďż˝ two + jsin ďż˝ numbers ďż˝ two for đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; ďż˝obtained đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; complex isđ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; outlined below: complex numberstwo isofoutlined below: đ?&#x2018; đ?&#x2018; values đ?&#x2018; đ?&#x2018; +these above with â&#x20AC;˘ Multiply the of đ?&#x2018;&#x160;đ?&#x2018;&#x160; upper butterfly,đ?&#x2018; đ?&#x2018; these second lower input + to the pair. The addition these đ?&#x2018; đ?&#x2018; compute equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;two toalgorithm temporary ďż˝2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; The final generated The program was implemented a implemented on a final file generated program fileonwas 2ďż˝ according đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; + đ?&#x2018; đ?&#x2018; ďż˝ according to the ond lower butterfly. đ?&#x2018; đ?&#x2018; o second lower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ďż˝is+ Digilent Spartan 3E Starter Board [6]. The design summary for (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; o + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) Digilent Spartan 3E Starter Board [6]. The design summary for (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; equation: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + to compute two temporary ďż˝ values. The multiplication algorithm of these two + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) two complex numbers outlined below: 2 đ?&#x2018; đ?&#x2018; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;2 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; below: Figurein 2 Table - FFT Display Sine Wave đ?&#x2018; đ?&#x2018; isďż˝ outlined III. isScreen this artefact givenâ&#x20AC;&#x201C;in10kHz Table III. complex numbers ďż˝Real + jsin ďż˝đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + ďż˝ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; for the đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; = â&#x2C6;&#x2019; cos ďż˝ o equation: values. The multiplication of these twothis artefact is given đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; to algorithm compute two temporary đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ?&#x2018; đ?&#x2018; result: + đ?&#x2018;?đ?&#x2018;? o Real result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? 2 đ?&#x2018; đ?&#x2018; above with the he two values ofđ?&#x2018; đ?&#x2018; đ?&#x2018;&#x160;đ?&#x2018;&#x160;values. Figure 2 - FFT Display Screen â&#x20AC;&#x201C; 10kHz Sine Wave complex numbers is below: đ?&#x2018; đ?&#x2018; o obtained multiplication algorithm these (a+jb)+(c+jd) == (a+c+j(b+d)) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;The + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? +outlined đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) + ofđ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?)two second lower butterfly. TABLE DESIGN SUMMARY TABLE I. SineDWave ESIGN SUMMARY o complex Imaginary result: đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; o Imaginary result: đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; Figure 2 - I.FFT Display Screen â&#x20AC;&#x201C; 10kHz is outlined below: ower input đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + đ?&#x2018; đ?&#x2018; ďż˝numbers according to ďż˝Real o (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; +result: đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? +a+c đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)value: = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;the đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) o đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; Work temporary đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;+đ?&#x2018;?đ?&#x2018;?)đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) 2 â&#x20AC;˘ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; The final scaled ab+d scaling factor of The final result was scaled by two. a+ scaling factor of two. o result (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; +â&#x20AC;˘was đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)by = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) +đ?&#x2018;?đ?&#x2018;?)đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) obtained above with the yđ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; the+two Imaginary result: đ?&#x2018; đ?&#x2018; ďż˝ values Work value: = â&#x2C6;&#x2019; ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; Thisoftoisđ?&#x2018;&#x160;đ?&#x2018;&#x160;oođ?&#x2018; đ?&#x2018; necessary compute two temporary Work temporary temporary đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;?keep + the đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) scaling since the FFT đ?&#x2018;&#x192;đ?&#x2018;&#x192; results This is value: necessary since FFT results keep scaling 2 đ?&#x2018; đ?&#x2018; ďż˝ o Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; â&#x2C6;&#x2019; input upwards đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + ďż˝ in according to= the resulting very large numbers. Given the size upwards resulting inđ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; very large numbers. Given the size o Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; =two đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;?đ?&#x2018;?) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) helower multiplication algorithm of these 2the o and Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; the = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) â&#x20AC;˘ The final result was scaled by a scaling factor limitations fixed data width of RAM blocks, itof the RAM blocks, it limitations and the fixed data width đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; o Work temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018; đ?&#x2018; Figure 2 - FFT Display Screen â&#x20AC;&#x201C; 10kHz Sine Wave o Work temporary đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;to =introduce đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; truncation đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) n: đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; +is outlined to compute two ďż˝2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; umbers below: was to introduce thistemporary scaling and was necessary thisresults scaling and truncation of This is đ?&#x2018;&#x192;đ?&#x2018;&#x192; necessary since the FFT o two. Real result: + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;value: đ?&#x2018; đ?&#x2018; necessary Figure 5 - FFT Display Screen â&#x20AC;&#x201C; o Work temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) system. system. o Real result:result: đ?&#x2018;&#x192;đ?&#x2018;&#x192;of + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;these scaling upwards resulting The multiplicationkeep two in very large 10kHz Sine Wave o algorithm â&#x2C6;&#x2019; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)After = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;Imaginary đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?)result: + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; +đ?&#x2018;&#x192;đ?&#x2018;&#x192;size đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) limitations othe Real đ?&#x2018;&#x192;đ?&#x2018;&#x192;have + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;the Figure - FFT Display Screen â&#x20AC;&#x201C; 10kHz Sine Wave allnumbers. butterflies been the been finaland Given After all the have traversed, the 2final x numbers is outlined below: o Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; butterflies đ?&#x2018;&#x192;đ?&#x2018;&#x192; traversed, above twoconversion temporary values must be then added with outputsâ&#x20AC;˘ areThe the resultant the time-domain into the outputs arewidth the from resultant conversion from time-domain into the fixed data of the RAM blocks, it o Imaginary result: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; Figure 3 - FFT Display Screen â&#x20AC;&#x201C; 10kHz Square Wave ork temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192; =â&#x2C6;&#x2019;đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019;temporary đ?&#x2018;?đ?&#x2018;?) domain. the upper input complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) the final â&#x20AC;˘ =The above twofrequency values must to beproduce then added with frequency domain. the (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;)(đ?&#x2018;?đ?&#x2018;? +theđ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) â&#x20AC;˘

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; necessary introduce scaling thewas upper complex number to the final computation: +tođ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + đ?&#x2018; đ?&#x2018; ďż˝ đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160;đ?&#x2018; đ?&#x2018; this These areand the The aboveinput twođ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) temporary values must be. produce then added with

2 đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; truncation system. ork temporary value: đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; + computation: + + đ?&#x2018; đ?&#x2018; The ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; These are final the the upper input complex the ďż˝2đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) output one butterfly pair. addition algorithm of đ?&#x2018; đ?&#x2018; to. produce Work temporary value: đ?&#x2018;&#x192;đ?&#x2018;&#x192;=of =đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) â&#x2C6;&#x2019;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018;?đ?&#x2018;?)đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;number đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;below: these two numbers output of complex oneđ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) butterfly The addition algorithm of computation: + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;pair. +isđ?&#x2018; đ?&#x2018; outlined are the ďż˝2ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; đ?&#x2018; đ?&#x2018; . These ork temporary value: đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; = đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019;butterfly After all the butterflies have been traversed, theof these two complex numbers is outlined below: output of one The addition algorithm Work temporary value: = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? +đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) pair. (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

Figure 3 - FFT Display Screen â&#x20AC;&#x201C; 10kHz Square Wave Figure 3 - FFT Display Screen â&#x20AC;&#x201C; 10kHz Square Wave I. IMPLEMENTATION

I. program IMPLEMENTATION The final generated file was implemented on a I. program IMPLEMENTATION Digilent Spartan 3E Starter Board [6]. summaryonfora o + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) The final generated file The was design implemented these two complex numbers is=outlined below: final outputs are the resultant conversion from this artefact is generated given in Table III. [6]. Digilent Spartan 3E Starter Board The design summaryonfora (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; o + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) The final program file was implemented al result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; value: o đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; Real result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) + đ?&#x2018;?đ?&#x2018;? Work temporary =+đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019;(đ?&#x2018;?đ?&#x2018;? the o time-domain into the=frequency domain. this artefact is given in TableBoard III. [6]. The design summary for Digilent Spartan 3E Starter (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + +đ?&#x2018;?đ?&#x2018;?đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) o Real result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + TABLE I. DESIGN SUMMARY Imaginary result: đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; this artefact is given in Table III. aginary result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; ooo Real result:result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; TABLE I. DESIGN SUMMARY Real result: đ?&#x2018;&#x192;đ?&#x2018;&#x192; +đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;â&#x2C6;&#x2019;Once Imaginary the processing is finalised, thefactor algorithm â&#x20AC;˘ The final result was scaled by a scaling of two. TABLE I. DESIGN SUMMARY o Imaginary result: đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; This isallnecessary thebyFFT results keep block scaling the from the last RAM The final result wassince scaled awith scaling factor of two. two temporary must beoutput then added maginary result: â&#x20AC;˘values đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;copies â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; upwards resulting insince very large numbers. size Figure 3 - FFT Display Screen â&#x20AC;&#x201C; 10kHz Square Wave This final istonecessary the resultsGiven keep the scaling RAM_final_data. this â&#x20AC;˘ used The result was scaled byFFT a Throughout scaling factor of two. nput complex number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) to the produce the final Once the resulting processing is finalised, the algorithm copies limitations and fixed data width of the RAM blocks, itall upwards in very large numbers. Given the size This is necessary since the FFT results keep scaling the real components and the imaginary ove two temporarytheprocess, values must be then added with đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;the output from last RAM block used to RAM_final_data. đ?&#x2018; đ?&#x2018; was necessary to introduce this scaling and truncation limitations and the fixed data width of the RAM blocks, it on: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + .process, These are the ďż˝ ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; upwards resulting in very large numbers. Given thethe size 3 - FFT Display Screen â&#x20AC;&#x201C; 10kHz Square Wave đ?&#x2018; đ?&#x2018; components combined together to give this are the the real components and the Figure er input complex Throughout number đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) final system. was2necessary to produce introduce this scaling and truncation Figure 6 - FFT Display Screen â&#x20AC;&#x201C; limitations andto the fixed data width of the RAM blocks, it imaginary components are combined together through to give the onethebutterfly The addition algorithm of đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; to introduce own, following pair. magnitude response. This is achieved I. IMPLEMENTATION đ?&#x2018; đ?&#x2018; ďż˝all was necessary this scaling and truncation After theđ?&#x2018; đ?&#x2018; butterflies been traversed, final 10kHz Square Wave ation: đ?&#x2018;Ľđ?&#x2018;Ľ(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) + đ?&#x2018;Ľđ?&#x2018;Ľďż˝đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; +system. ďż˝đ?&#x2018;&#x160;đ?&#x2018;&#x160; . This These are the magnitude response. ishave achieved through the the following 2the the following equation: complex numbers is outlined below: system. outputs are resultant conversion from the time-domain into After all the butterflies have been traversed, the final equation: The final generated file was implemented on a of oneand butterfly The resultant addition algorithm oftime-domain I. program IMPLEMENTATION thepair. frequency đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1) đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; + outputs areallthedomain. conversion into After the butterflies have from been the traversed, the final 2 2 Digilent Spartan 3E Starter Board [6]. The design summary for wo complex numbers is outlined below: + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) + (đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) |đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; ) â&#x20AC;Ś (3) the frequency ng equations: đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; | = ďż˝đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ) + đ??źđ??źđ??źđ??ź(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; 7. Implementation outputs are thedomain. resultant conversion from đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;the time-domain into final The generated program file was implemented on a this artefact is given in Table III. the frequency domain. Digilent Spartan 3E Starter Board [6]. The design summary for (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? 1) â&#x2C6;&#x2014;+ + đ?&#x2018;?đ?&#x2018;?+ đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)) +â&#x2C6;&#x2019;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;) (đ?&#x2018;?đ?&#x2018;? al result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018; đ?&#x2018; ďż˝+ Figure andFigure Figure are screen-shots Figure 5 5 and 6 are6screen-shots taken fromtaken the real- The final generated program file was this artefact is given in Table III. time FFTthe processor for an input of 10kHzfor sine-wave and a implemented TABLEon I. a Digilent DESIGNSpartan SUMMARY from real-time FFT processor an input 3E Starter Real result: đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;? aginary đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018; đ?&#x2018; result: đ?&#x2018;?đ?&#x2018;? + 10kHz square-wave respectively. The latter screen-shot đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;1 + ďż˝ ďż˝ of 10kHz sine-wave and a 10kHz square-wave Board [6]. The design summary for this artefact 2 confirms that the frequency components generated from the TABLE I. DESIGN SUMMARY maginary result: đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; respectively. The latter screen-shot confirms result was scaled by a scaling factor of two. sharp changes in the square wave were also detected. is given in Table III. ET the butterfly pair

thatFFT the results frequency components generated ecessary sincescaled the keep al result was bythe a scaling factor scaling of two. from sharp changes the square wave đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; esulting in very large numbers. Given thein size necessary since the FFT results keep scaling ďż˝ for the first were also detected. đ?&#x2018; đ?&#x2018; the fixed data large widthnumbers. of the RAM blocks, it s and resulting in very Given the size sary to introduce this scaling and truncation đ?&#x153;&#x2039;đ?&#x153;&#x2039;â&#x2C6;&#x2014;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; and the fixed data width of the RAM blocks, it ons ďż˝ for the đ?&#x2018; đ?&#x2018; cessary to introduce this scaling and truncation he butterflies ed above with the have been traversed, the final resultant time-domain ccording toconversion the the butterflies havefrom beenthetraversed, the into final omain. e two temporary he resultant conversion from the time-domain into of these two ym domain. Figure 2 - FFT Display Screen â&#x20AC;&#x201C; 10kHz Sine Wave + đ?&#x2018;&#x2014;đ?&#x2018;&#x2014;(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?)

â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?)

+ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;)

â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;) 38

November 2012

Issue No. 43

Design Summary Device Utilization Summary Logic Utilization

Used

Available

Utilization

Number of Slice Flip Flops

1,839

9,312

19%

Number of 4 input LUTs

6,582

9,312

70%

Number of occupied Slices

4,342

4,656

93%

4,342

100%

4,342

6,838

9,312

73%

Number of Slices containing only related logic Number of Slices containing unrelated logic Total Number of 4 input LUTs Number used as logic

6,566

Number used as a route-thru

256

Number used for Dual Port RAMs

Number of bonded IOBs

232

14%

Number of RAMB16s

60%

29%

75%

Number of BUFGMUXs Number of MULT18X18SIOs Average Fanout of Non-Clock Nets

3.51

References

1. Linear Technology. [Online]. http://cds.

linear.com 2. Octopart. [Online]. http://cdn.sigma. octopart.com/5352908/image/NeutrikNCJ6FA-H.jpeg 3. C. Sidney Burrus. (2009, September) The Cooley-Tukey Fast Fourier Transform Algorithm. [Online]. http://cnx.org/content/ m16334/latest/

Roberto Drago

4. Xilinx Core Generator. 5. Lyons R.G. Flylib. [Online]. http://flylib.com/ books/en/2.729.1.50/1/

6. Digilent. [Online]. http://www.digilentinc.

Department of Communications & Computer Engineering, University of Malta The paper is the winner of the Best Computer Engineering Final Year Student Project 2011

com

Dr. Edward Gatt

Head of Department of Microelectronics & Nanoelectronics, Faculty of Information & Communication Technology, University of Malta

November 2012

Issue No. 43

project profile our experi-

Intelligent Solutions for Today’s Modern Educational Facilities Client

London Borough of Hillingdon

Electrical design

Faber Maunsell

The London Borough of Hillingdon’s Stockley Academy provides a truly intelligent environment in which to learn.

Project Stockley Academy database is prevented by security encryption. This Main contractor Costain

Prolojik

to learn. The lighting controls installed provide not only huge All lighting has been supplied with DALI compatible energy savings, by utilising absence detection as well as

As specialists in the commercial building sector,

savings, but also allow complete control for today and control geardaylight with emergency lighting additionally the future. Prolojik products utilised:

Prolojik’s core business is the design and supply

Overide facilities have been provided where requested and the being supplied with DALI emergency inverters, > PC900 2DI/2DO modules client now manages and schedules all tasks relating to the

of fully networked addressable lighting control

essential for monitoring and control.

Electrical Contractor Shepherd Engineering provides a reliable and comprehensive report ofServices Limited

the status of the academy’s emergency lighting and Intelligent solution for one of today’s modern safety credentials. educational facilities. Stockley Academy provides a truly intelligent environment

systems. The lighting controls installed at the Stockley Academy provide not only huge energy savings, by utilising absence detection as well as daylight savings, but also allow complete control for today and the future.

lighting installation in an innovative and controlled fashion. Each classroom has prescence detection with the additional benefit of manual dimming control. Corridor hold is also utilised.

The security system interfaces with the Prolojik lighting controls Lighting Control Modules and activates circulation lighting.

Key features: > Infra Red override > Scene setting > Daylight Link control > PIR presence detection > Security interface >

PN400 Network controllers

> > > > > > >

PC100 IR controllers PL521 1-10V LCM’s PS300 Multi Sensor PC500 Area switch concentrators PV360 Supervisor PC610 4 scene wallplates with blue LED backlit buttons PL423 Analogue Hardwired 1-10V LCM’s

Utilising peer to peer communication over an open Each Lighting Control Module (LCM) is connected protocol network.

to the ‘Open System’ intelligent node network www.prolojik.com Prolojik Limited 85 Cressex Enterprise Centre Lincoln Road High Wycombe Buckinghamshire HP12 3RL UK using an unscreened twisted pair cable. This cable Tel: +44 (0)1494 515 100 Fax: +44 (0)1494 400 343 info@prolojik.com

is wired in a ‘free topology’ radial allowing T and

Every luminaire in the building is individually

spur connections. Each LCM has a LonWorks™ node

addressed as are the many multi-sensors, switches

which allows ‘peer to peer’ communication without

and 4 Scene Wall plates utilised throughout the

it having to route through an area controller. These

campus. Each classroom has presence detection with

LCMs store their own copy of the operational data

the additional benefit of manual dimming control.

base. In this way the LCMs are not dependent on the

Override facilities have also been provided thus the

area controller for the ‘corridor hold’ feature or its

client now manages and schedules all tasks relating

local operations such as presence detection and user

to the lighting installation in an innovative and

on/off/dimming control. Such features are required

controlled manner.

when localised control is required to override central

The system utilises peer to peer communication over an open protocol network, this is thanks to Prolojik’s historic commitment to interoperability. Because of this the Academy was able to interface the security system with Prolojik’s lighting controls to activate circulation lighting for patrolling guards when needed. The

Prolojik

control without interruption and delay.

Key Features Utilised at the Stockley Academy • Infra Red remote override facility • Local scene setting utilising Prolojik’s 4 scene wall plate • Daylight Link control for daylight harvesting

Perspective™

graphics

supervisor

software shows every luminaire along with their current lighting level, switching/dimming schedules and configuration properties. Each and every luminaire is monitored for lamp and ballast condition with emergency light fittings being tested for battery autonomy remotely via the software supervisor. All results are logged for maintenance reporting, tampering of the recorded

• Passive infrared presence & absence detection • Security interface for circulation lighting For more information about Prolojik’s intelligent lighting solutions or to arrange a consultation, please contact:

Mr. Michel Le Brun, Technical Sales Executive at Hydrolectric Ltd Tel: 2124 1111 or 7985 5983 e-Mail: michel.lebrun@hydrolectric.com.mt or visit www.prologik.com

Materials in Aircraft Design Over the Years

by Ing. A. Micallef Grimaud

Aircrafts have been with us since the first flight of the Wright brothers in 1903 and is commonly acknowledged that it has come a long way since. Due to the nature of the working characteristics of the machine, one very important aspect to be considered is weight, which greatly affects the performance, while having to keep the structural integrity. The Wright brothers had chosen the best materials available at the time to provide the required strength with lowest weight, having opted for wood, or more specifically, spruce, which is a strong and lightweight wood. Fabric was chosen to provide the covering for the surfaces. As more demands were made on aircraft performance, operating speed increased, and with it the stresses induced on the airframe structures, at which point metal had to be considered instead of wood. Aluminium was chosen as this had a high strength to weight ratio with reasonable cost. Aluminium is a very versatile material that can be cast, extruded, forged, produced in sheets with a variety of surface finishes available, flexible, lightweight and also corrosion resistant. Due to higher performance requirements, alloys have continuously developed to meet such challenges, having Copper, Magnesium or Zinc as alloying elements to provide the necessary qualities of performance, weight and flexibility.

November 2012

Issue No. 43

Up till the 1940â&#x20AC;&#x2122;s, aircraft continued to be improved and evolved from the basic design of the Wright brothers with petrol driven piston engines geared up with propellers. The great leap in commercial aircraft design came along with the onset of the jet engine, as the power source, with the De Havilland Comet in 1952, flying at cruising speed of 490mph. To gain such a higher speed, conventional wisdom dictated operating at a higher altitude, where the air pressure, and therefore density is lower, therefore offering less resistance to the aircraft and the associated power required to overcome air friction. In turn, due to the limitation of the human body that cannot survive without an adequate oxygen supply as well as for passenger comfort and safety, the aircraft had to be fitted with a system that provided conditioned air at a pressure above that of the surrounded atmosphere. This resulted in having a pressurized vessel with wings, that had to fly at 40 000 ft, while maintaining a cabin pressure equivalent to that at 8 000 ft, subjecting the fuselage to a differential pressure of 8 psi. Unfortunately this aircraft type suffered from a couple of accidents and it resulted that the stresses induced on the

structure had brought on a new problem into the already complex equation: fatigue. This new requirement brought about the need for aircraft designers and engineers to take fatigue into consideration, whereby the effect is minimized to a safe level. As air travel was becoming more accessible to the general public and aircraft sizes increased in order to keep up with the demand, coupled with the fuel price increases further weight reductions were needed to make aircraft more efficient. Apart from more improved metals, composites, which are the most important materials to be adapted for aviation since the use of Aluminium in the 1920s, started to enter the scene. Fiberglas was first used in the Boeing 707 passenger jet in the 1950s, where it comprised about two percent of the structure, but Carbon fibre is becoming more popular, where it is composed of layers of carbon fiber that are

impregnated with a polymer.. The benefits that composite materials offer are that they can be both lightweight and strong, typically about 20 per cent lighter than Aluminium. The heavier an aircraft weighs, the more fuel it burns, so reducing weight is important to aeronautical engineers for better performance, to accountants for lower fuel consumption and hence cost, and to environmentalists as this means less air pollution. The production of composite structures is more complex than manufacturing using traditional metal processes, where the composite material, in tape or fabric form, is built up in layers (laminated)to create the desired thickness and shape of the applicable structure. The assembly is then cured through a cycle of high heat and pressure over a time period that lasts several hours. The resin matrix material flows and when the heat is removed, it solidifies to the required

November 2012

Issue No. 43

Edinburgh Business School (EBS) of Heriot-Watt University is home to some of the most challenging, most rewarding business education programmes in the world. EBS offers flexibility and choice, especially in the renowned distance learning postgraduate programmes.

You can begin your studies at any time and you can finish them when you like. You take exams when you’re ready – not when we say so. You can also pay for your courses as you go along – one at a time.

EBS offers, the MBA programme, a range of postgraduate certificates, postgraduate diplomas and MSc programmes in a number of specialist subjects being Marketing, Finance, HRM and Strategic Planning.

The MIM EB-LEARN Platform. The MIM EB-Learn is a sophisticated e-learning system providing interactive communication tools between students and the MIM by means of the Internet. Moving along with new technologies and innovative ideas the (MIM) has now launched Online Sessions which qualify for continuous professional development through its e-learning platform.

Online programmes available in various subjects such as:

Marketing SaleS Finance ManageMent

Contact MIM for a free introductory online session at nbianco@maltamanagement.com or visit www.maltamanagement.com > Educational Programmes > MIM EB-LEARN for more information.

Materials in Aircraft Design Over the Years (cont.)

shape. One useful feature of composites is that they can be layered, with the fibres in each layer running in a different direction. This allows materials engineers to design structures that behave in certain ways. For instance, they can design a structure that will bend in one direction, but not another. Modern airliners use significant amounts of composites to achieve lighter weight, with about ten percent of the structural weight of the Boeing 777. For the latest Boeing model, the 787 Dreamliner, Boeing claims that 50 percent by weight of its structure is made of composite materials, most notably the fuselage. The core material is normally low strength material, but its higher thickness provides the sandwich composite with high bending stiffness with overall low density. Other composites are comprised of various materials that, when integrated together, form a heterogeneous material with advantageous structural properties.

The other major aircraft manufacturer, Airbus, claims that 25% of the A380 is produced using composite materials and it is projected that over 50% by weight of the A350 will be composite-based, including elements such as the wings and fuselage. This jetliner also represents the first Airbus aircraft that utilizes a higher percentage of composite technologies than that of metallic applications. Its fuselage panels, frames, window frames, clips, and doors are made from carbon fibre reinforced plastic (CFRP), with a hybrid door frame structure consisting of this material and titanium. In this brief overview, we have seen that composite materials offer the direct benefit of being a lighter, simpler structure and therefore consume less fuel, making them more efficient and environmentally friendly. Apart from these direct benefits, they offer other advantages over the traditional metal structures.

November 2012

Issue No. 43

E-Power An enery enery saving saving device device system system E-Power -- An The Case of AMG Cold Stores -- An enery E-Power The Case of AMG Cold AnStores enery saving saving device device system system E-Power

The Case of AMG Cold Stores The Casebeen of one AMG Cold AMG have of the firstStores to adopt the E-Power energy saving device by Energia Europa. The next generation energy efficiency

AMG have been one of the first to adopt thethe E-Power energy saving device Europa. The and voltage harmonisation device is saving company over €27 per day by onEnergia their electricity bills.next generation energy efficiency and voltage harmonisation device is saving the company over €27 per day on their electricity bills.next generation energy efficiency AMG have been one of the first to adopt the E-Power energy saving device by Energia Europa. The “Eachhave year been our electricity tremendously. The system has given us the promised estimated savings asenergy of dayefficiency one! My AMG one of thebills firstincreased to adopt the E-Power energy saving device by Energia Europa. The next generation and voltage harmonisation device is saving the company over €27 per day on us their electricity bills. “Each yearbit, our electricity increased tremendously. system has given thein promised estimated day one! My favourite besides the bills obvious savings, isthe that I canThe monitor consumption real-timebills. online, it’ssavings great!”as- of Gordon Spiteri and voltage harmonisation device is saving company over €27the per day on their electricity favourite the bills obvious savings, is that I canThe monitor thehas consumption real-timeestimated online, it’ssavings great!”as- of Gordon Spiteri “Each year ourbesides electricity increased tremendously. system given us theinpromised day one! My Director atbit, AMG Cold Stores Ltd. “Each year our electricity bills increased tremendously. The system has given us the promised estimated savings as of day one! My Director AMG Cold Stores Ltd. favouriteatbit, besides the obvious savings, is that I can monitor the consumption in real-time online, it’s great!” - Gordon Spiteri favourite bit, besides the obvious savings, is that I can monitor the consumption in real-time online, it’s great!” - Gordon Spiteri Director at AMG Cold Stores Ltd. Director at The AMG AMG Cold Stores CaseLtd. Study

The AMG Case Study • Product: The AMG Case Study • Product: The AMG Case Study • Installation: • • • • • • • • • • • • • • • • • • • • •

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E-Power E-Power E-Power Compact 414 kVA (3ph+n) 400/230 E-Power Compact 414 kVA (3ph+n) 400/230 a cold storage plant – AMG Cold Stores E-Power a cold storage plant – AMG Cold Stores E-Power Compact 414 kVA (3ph+n) 400/230 854.5 kWh E-Power Compact 414 kVA (3ph+n) 400/230 854.5 a cold kWh storage plant – AMG Cold Stores 16.22% a cold storage plant – AMG Cold Stores 16.22% 854.5 kWh 139 kWh 854.5 kWh 139 kWh 16.22% € 27.8 16.22%€ at €0.20 / kWhr € 27.8 € at €0.20 / kWhr 139 kWh 83.4 kg 139 kWh 83.4 kg € 27.8 € at €0.20 / kWhr € 27.8 € at €0.20 / kWhr 83.4 kg 83.4 kg

Energia Europa, the E-Power suppliers themselves ran a test on No additional software is needed because access to the Energia Europa, theinstalled E-Poweratsuppliers themselves ran a test on No additional software is needed isbecause access such to the the E-Power device AMG to ascertain its effectiveness. information pages and configuration via a browser, as the E-Power device installed at AMG to ascertain its effectiveness. information pages and configuration is via a browser, such as Energia Europa, the E-Power suppliers themselves ran a test on No additional software is needed because access to the The results above may be substantiated by the report provided. Internet Explorer or Firefox. No additional software is needed because access to the Energia Europa, the E-Power suppliers themselves ranprovided. a test on The results above be substantiated by the report Internet Explorer orand Firefox. the E-Power devicemay installed at AMG to ascertain its effectiveness. information pages configuration is via a browser, such as E-power really is tomorrow’s technology and should information pages and configuration is viatoday a browser, such be as the E-Power device installed at AMG to ascertain its effectiveness. The results above may be substantiated by the report provided. Internet Explorer or Firefox. E-power really is tomorrow’s technology today and should be What is E-Power? aInternet consideration large consumers of electricity. The KWH Explorerfor orall Firefox. The results above may be substantiated by the report provided. What is E-Power? asavings, consideration all large consumers of today electricity. The KWH E-power really also isfortomorrow’s technology and be E-Power, the commercial version of the E-Box, it is an which translate into Co2 savings, are of should particular E-power which really also is tomorrow’s technology todayare and should be What is E-Power? E-Power, the commercial version of the E-Box, it is an savings, translate into Co2 savings, of particular a consideration for are all large of electricity. The KWH electromagnetic conversion apparatus with energy recycling and relevance in what both consumers economically and environmentally What is E-Power? arelevance consideration for are all large consumers of electricity. The KWH electromagnetic conversion apparatus with energy recycling and in what both economically and environmentally E-Power, the commercial versionparts, of the E-Box,to itoptimize is an savings, which also translate into Co2 savings, are of particular harmonics control, without moving designed challenging times. E-Power, the commercial versionparts, of the E-Box,to itoptimize is an savings, which also translate into Co2 savings, are of particular harmonics control, without apparatus moving challenging times. electromagnetic conversion withdesigned energy recycling and relevance in what are andguarantee environmentally energy circuits. E-Power is backed upboth by aeconomically no nonsense of an electromagnetic conversion apparatus with energy recycling and relevance in what are both economically and environmentally energy circuits. harmonics control, without moving parts, designed to optimize challenging E-Power is times. backed up bysaving. a no nonsense guarantee of an It is an electronically controlled static 3 Phase autotransformer instantaneous 10% energy harmonics control, without moving parts, designed to optimize challenging times. energy It is an circuits. electronically static 3 Phase instantaneous 10% energy E-Power is backed up bysaving. a no nonsense guarantee of an with recyclingcontrolled and harmonics control.autotransformer It is installed energyenergy circuits. E-Power is backed up by a no nonsense guarantee of an with energy recycling and harmonics control. It is installed It is an electronically controlledsupply static lines 3 Phase instantaneous 10% energy saving. E-Power Benefits downstream of the electrical for autotransformer control of both It is an electronically controlledsupply static 3 Phase instantaneous 10% energy saving. downstream thelight electrical for autotransformer control of both E-Power Benefits with energy recycling and harmonicslines control. It is installed motive powerof and power. with energy recycling and harmonics control. It is installed motive power and light power. • Reduces energy use & costs by up to 20% by optimising the E-Power Benefits downstream the electrical supply lines for control of both • Reduces energy use & costs by up to 20% by optimising the It operates byof downstream of the electrical supply lines for control of both E-Power Benefits electricity supply voltage motive power It by and light power. electricity supply voltage • operates Current Optimisation motive power and light power. energy emissions use & costs to 20% by optimising the • Reduces carbon byby upup to 20% Current Optimisation • Reduces carbon energy emissions use & costs to 20% by optimising the It•• operates by of energy transmission: (L1 = 80A; L2 = 65A; L3 = byby upup to Re-balancing supply voltagedemand • electricity Reduces your maximum by20% up to 15% It• operates by of energy transmission: (L1 = 80A; L2 = 65A; L3 = Re-balancing electricityyour supply voltagedemand by up to 15% • 40A). Current Optimisation •• Reduces maximum The system re-establishes balance as much as possible, Reduces carbon emissions by up toequipment 20% Protects electrical and electronic from transients • 40A). Current Optimisation The system re-establishes balance as much as possible, Reduces carbon emissions by up toequipment 20% • thus Re-balancing energy transmission: (L1 =phases 80A; L2work = 65A;much L3 = •• Protects electrical and electronic from transients avoidingofthe situation where some Reduces your maximum demand by up to 15% up to 25,000V • thus Re-balancing ofthe energy transmission: (L1 =phases 80A; L2work = 65A;much L3 = situation where some • up Reduces your maximum demand by up to 15% 40A).avoiding The system re-establishes balance as much as possible, to 25,000V harder than others, also avoiding accelerated consumption. Protects maintenance electrical and costs electronic equipment from and transients • Lowers on motors, lighting other 40A). The system re-establishes balance as much as possible, than others, also avoiding accelerated consumption. Protects maintenance electrical and costs electronic equipment from and transients thus avoiding the situation where some phases workand/or much • Lowers on motors, lighting other • harder Control of output voltage to avoid voltage peaks up to 25,000V electrical equipment thus avoiding the situation where some phases work much • Control of output voltage to avoid voltage peaks and/or up to 25,000V harder than others, avoiding accelerated electrical equipment current which could also damage consumers of theconsumption. load. Lowers maintenance costs motors, lighting and power other • Improves power quality andonsubstantially improves harder than others, avoiding accelerated which could also damage consumers of theconsumption. load. • Lowers maintenance costs motors, lighting and power other ControlFactor of output voltage to avoid voltage peaks and/or Improves power quality andonsubstantially improves • current Power Correction electrical equipment factor • Control of output voltage to avoid voltage peaks and/or Power Factor Correction electrical equipment current which could damage consumers of the load. • Reduction of Harmonics (3rd , 5th and 7th ) • factor Improves power temperatures quality and substantially improves Lowers operating of motors and lighting power current which could damage of )the load. •• Reduction of Harmonics (3rd consumers , 5th and 7th Improves power temperatures quality and substantially improves power Power Factor Correction •• Lowers operating ofcomponents motors and lighting factor E-Power contains a patented bypass system which makes Prolongs the life of electrical by optimising • Power Factor Correction factor • Reduction of Harmonics (3rd , 5th and 7th ) E-Power contains a patented bypass system which makes • Prolongs the life of electrical components by optimising • voltage Lowers operating temperatures motors and lighting it• Reduction possible toof “unplug” in7th the) case of system which reduces electrical of stress Harmonicsthe (3rdsystem , 5th and • voltage Lowers operating temperatures of motors and lighting it possible to “unplug” theofsystem the case of sosystem electrical stress E-Power contains a the patented bypassin system which makes Prolongswhich the reduces life of electrical components byefficiency optimising malfunction without loss power. The function is quick • Reduces phase voltage imbalance, improving the of E-Power contains a the patented bypassThe system which makes Prolongs phase the life of electrical components byefficiency optimising malfunction without loss of power. function is so quick • Reduces voltage imbalance, improving the of it possible to “unplug” the voltage case ofoptimiser system voltage which reduces electrical stress that it is transparent to thethe endsystem user. Noinother AC motors it possible to “unplug” the system inother the voltage case ofoptimiser system voltage which reduces electrical stress that it is transparent to the end user. No AC motors malfunction without the loss of power. The function is so quick Reduces phase voltage imbalance, improving the efficiency of has an automatic bypass. • Suppresses harmonics that can damage sensitive electronic malfunction without the loss of power. The function is so quick • Reduces phase voltage imbalance, improving the efficiency of has an automatic bypass. Suppresses harmonics that can damage sensitive electronic that it is transparent to the end user. No other voltage optimiser AC motors equipment, improving HV to LV transformer efficiency A harmonic filter is integrated into the designvoltage of theoptimiser E-power, that it is transparent to the end user. No other AC motors improving LVtotransformer efficiency has an automatic • equipment, Suppresses harmonics that can damage sensitive electronic A harmonic filterbypass. isinintegrated into theDistortion design of (THD) the E-power, Reduces reactive powerHV bytoup 45% giving attenuation Total Harmonic and a has an automatic bypass. Suppresses harmonics that can damage sensitive electronic •• Reduces reactive power by up totransformer 45% equipment, improving HV to LV efficiency giving attenuation in Total Harmonic Distortion (THD) and a Fully controllable via the intelligent e-controller A harmonic filter is into design of As theharmonics E-power, marked reduction in integrated 3rd, 5th and 7ththe harmonics. equipment, improving HVintelligent to LV transformer efficiency •• Fully controllable via the e-controller A harmonic filter is into design of As theharmonics E-power, Reduces reactive power by up to 45% marked reduction ininintegrated 3rd, 5thelectronic and 7ththe harmonics. giving attenuation Total Harmonic Distortion and (THD) and toa are potentially damaging to equipment known • Reduces reactive power by up to 45% giving attenuation in Total Harmonicequipment Distortion and (THD) and toa • Fully controllable via the intelligent e-controller are potentially damaging to known marked reduction in 3rd, 5thelectronic and 7th harmonics. reduce the efficiency of HV transformers, this is As an harmonics important E-Power is represented locally by ElectroFix of Valletta Road, • Fully controllable via the intelligent e-controller marked reduction in 3rd, 5thtransformers, and 7th harmonics. As harmonics reduce the efficiency of HV this is an important E-Power is represented locally by ElectroFix Road, are potentially damaging to electronic equipment and known to function that delivers tangible energy-saving benefits. Qormi. ElectroFix has established itself as oneofof Valletta the leaders in are potentially damaging to electronic equipment and known to function that delivers tangible energy-saving benefits. Qormi. ElectroFix has established itself as one of the leaders in reduce the efficiency of HV transformers, this is an important E-Power is represented locally by ElectroFix of Valletta Road, the energy efficiency market in Malta. They represent leading E-powerthe is fully controllable via the intelligent It reduce efficiency of HV transformers, this ise-controller. an important E-Power is represented locallyin by ElectroFix of Vallettaleading Road, the energy efficiency market Malta. They represent function that delivers tangible energy-saving benefits. Qormi. ElectroFix has established itself as one of the leaders in E-power is fully via the intelligent e-controller. It global brands including German Conergy, Masdar PV, Stiebel is designed logcontrollable all consumption data which can be viewed in function thattodelivers tangible energy-saving benefits. Qormi. ElectroFix has established itself as one of thePV, leaders in global brands including German Conergy, Masdar Stiebel the energy efficiency market in Malta. They represent leading is designed to log all consumption data which can be viewed in Eltron, Lorentz and leading solar water-heater brand, Solahart. E-power is fully controllable via the intelligent e-controller. It real-time. the energy efficiency market in Malta. They represent leading Eltron, Lorentz and leading solar water-heater brand, Solahart. E-power is fully controllable via the intelligent e-controller. It global brands including German Conergy, Masdar PV, Stiebel real-time. is designed to log all consumption data which can be viewed in global brands including German Conergy, Masdar PV, Stiebel is designed to log all consumption data which can be viewed in Eltron, Lorentz and leading solar water-heater brand, Solahart. real-time. Eltron, Lorentz and leading solar water-heater brand, Solahart. real-time. ElectroFix | Valletta Road, Qormi | Tel: 2167 5353 | www.electrofixenergy.com ElectroFix | Valletta Road, Qormi | Tel: 2167 5353 | www.electrofixenergy.com ElectroFix | Valletta Road, Qormi | Tel: 2167 5353 | www.electrofixenergy.com ElectroFix | Valletta Road, Qormi | Tel: 2167 5353 | www.electrofixenergy.com

Materials in Aircraft Design Over the Years (cont.)

They are known to be more reliable in their operation with a longer lifespan than other traditional metallic materials, they do not suffer from fatigue problems and since they are not metals, they are considered to be corrosion free. With such better reliability, aircraft manufacturers have increased their service intervals, which significantly reduces maintenance costs in terms of labour man hours during the life of the aircraft as well as increasing the proportion of aircraft availability. Being corrosion-free, they need less quantities of chemicals necessary to protect the structural components from the negative effects of the harsh environment in which they operate, and thereby reducing the relative costs. Using less chemicals has another advantage in environmental terms as the need of hazardous waste disposal is reduced.

In a little over 100 years, aircraft have evolved greatly, with increases in size, performance, payload, range, comfort and speed. A specially modified aircraft has also taken mankind to outer space and flew back to earth, only to repeat the feat many times over. Engineers have always met the new challenges posed to them by more stringent demands from industry and operating parameters. Aircraft design over all these years is just one of many such examples and will continue to develop to meet future demands of cost, fuel efficiency and environmental friendliness. ET

Ing. A. Micallef Grimaud Director, Mechanical Engineering, MCAST

November 2012

Issue No. 43

Engineering Restoration Project by Michael J. Turner

Restoration of an Icon - Artistic Engineering. On 28th September 2012, Maltese and British experts began dismantling the historic pipe organ at St. Paul’s Anglican pro-Cathedral in Valletta. Organ builders are a fascinating mixture of engineers, architects, musicians and electricians, which skills are best combined in a pot of long experience. After several years of careful analysis the work of restoration has been entrusted to K. Jones and Associates, of Wicklow, Ireland, now owned by CEO Derek Byrne, whose mother is Maltese, and is a frequent visitor with many Maltese cousins. Derek was not chosen for this connection but because his report and assessment was considered to be the best on offer. The Maltese connection is just an added beneficial co-incidence. Furthermore Derek has agreed to sub contract work to Ing. Robert Buhagiar of Zabbar, who is Malta’s only properly qualified organ builder and restorer. Robert has been responsible for the maintenance of the St. Paul’s organ for several years, and is largely responsible for keeping it working as the parts failed through old age. He has built and restored several small organs in Malta, including Mosta. Derek has the required level of experience, having built and restored organs in South Africa and Ireland for many years, and he is particularly interested in St. Pauls organ because of his Maltese origins and the artistic challenge it represents. The organ case itself is an original “Father” Bernard Smith case built for Chester Cathedral circa 1680, while he was Royal Organ Builder, and which was brought to Malta in 1854. It is now only one of two outside the UK, the other being in Vienna.

Historic Pipe Organ restored Photography by Ray Attard

The dismantling, which took one week, was a complicated and delicate job, as well as a hard physical slog. Engineers will understand the difficulties of taking down and logging of 2000 pipes, varying in size from a few inches to 16 feet, some of which are doubled and bent, plus the complicated sounding boards, bellows etc. All of which have to be meticulously recorded because it will all have to be rebuilt and repositioned in the cathedral at end 2013. The electrical work is highly complicated, connecting all the working parts, especially as a new console ( where the organist sits and plays) will be movable to three different positions.

November 2012

Issue No. 43

Engineering Restoration Project (cont.)

Chester, and Dr. Hugo Agius-Muscat, has built up an international reputation as head organist. The world renowned conductor, composer and organist, Wayne Marshall, has played it and is waiting impatiently to play the restored instrument. The restorers are working on a mission of love and the casing itself, the really historic component, with its beautiful, complicated and delicate woodwork, will be worked on in Malta to restore it to its former glory. Because of the multiple engineering skills involved, mechanical, electronic, architectural and construction, we would welcome donations from companies, individuals or even the Engineers Association, to preserve this iconic part of Malta’s heritage. Organ at St. Paul’s Anglican proCathedral in Valletta

Photography by Ray Attard

Melfar Ltd and Joinwell have both played a part in supplying material and Parnis England are entrusted with the shipment of a container to Ireland, for the parts which are to be worked on in Wicklow. The rest will stay in Malta.

Sponsorships for this worthy restoration project start from just Eur 20.00 and you can get in touch by sending an email on organ@ stpaulsorgan.org, or by pointing your browser to www.stpaulsorgan.org. ET

Over the years, St. Paul’s organ has been played by distinguished musicians, there is a strong tradition that Handel himself played it while in

Michael J. Turner

Longest serving member of the Organ Appeal Committee, after Musical Director and Head Organist, Dr. Hugo Agius Muscat

November 2012

Issue No. 43

malta engineering excellence

2012

awards

11th Ediďż˝on

The Chamber of Engineers is receiving nominaďż˝ons in the following three categories:

Industrial Excellence

Leadership

Lifetime Achievement

Criteria for Nomina on â&#x20AC;˘

Any member of the Chamber of Engineers may nominate a candidate

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The nominaďż˝on must be accompanied by documentaďż˝on that must eďŹ&#x20AC;ecďż˝vely highlight the engineering contribuďż˝ons of the candidate

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The nominaďż˝on must include a detailed rĂŠsumĂŠ of the nominee

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The nominaďż˝on shall be submiďż˝ed with a covering leďż˝er addressed to the Chairperson, Awards Selecďż˝on Commiďż˝ee at the address below

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Nominaďż˝ons must be submiďż˝ed by 9th November 2012

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Nominees must be of Maltese naďż˝onality and residents of Malta, or companies registered in Malta

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Nominaďż˝on forms can be downloaded from the Chamber of Engineers website www.coe.org.mt

Awards ceremony will be held on 7th December 2012 at St. Johnâ&#x20AC;&#x2122;s Cavalier, Embassy of the Sovereign Military of the Order of St. John, Valleďż˝a under the Disďż˝nguished Patronage of H.E. Dr George Abela, President of Malta.

Further informaďż˝on may be obtained from: Chamber of Engineers 127, Professional Centre, Sliema Road, Gzira GZR 1633 Tel: 21334858 | email: info@coe.org.mt

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