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International Journal of Computer & Organization Trends –Volume 3 Issue 10 – Nov 2013

A Review on Fractional Delay FIR Digital Filters Design and Optimization Techniques Amritpal Singh#1, Dr. Naveen Dhillon*2, Sukhpreet Singh Bains @3 1

MTECH ECE, RIET Phagwara, India 2

3

HOD ECE RIET, Phagwara, India

Assistant Prof. ECE, RIET, Phagwara, India

Abstract

Communication systems to systems for medical applications to

In this paper various methods to design variable fractional-delay

name just a few. They can be implemented in hardware or

(FD) FIR filters are proposed. First in the study, the Kaiser

software

Window function parameters used many application was found

And can process both real-time and offline (recorded) signals.

with GA. This values obtained were used in FIR filter design A

Digital filters in hardware form can now routinely perform

method for developing VFD filters is also an essential technique

tasks that were almost exclusively performed by analog

for the fractional linear discrete-time systems. Theoretically speaking, the design of variable digital filters under optimal sense is more complicated and difficult than the design of fixed delay filters, since the impulse response or the poles and zeros of the

systems in the past whereas software digital filters can be Implemented using low-level or user-friendly high-level programming languages. Nowadays digital filters can be used

filters are some type of functions in the variable parameter are

to perform many filtering tasks which in the not so distant past

generally assumed to be polynomial functions. Therefore sub

were performed almost exclusively by analog filters and are

optimal approaches for the design of variable digital filters should

replacing the traditional role of analog filters in many

be investigated for the purpose of reducing the computation

applications. Beside the inherent advantages such as high

complexity. Recently advances have been made on the design of

accuracy and reliability, small physical size, and reduced

some type of VFD filters, such as finite-impulse response (FIR)

sensitivity

VFD filters and infinite-impulse response (IIR) all pass VFD filters. Large numbers of coefficients should be designed; related iteration algorithms still feature considerable computation complexity. Examples illustrate our proposed method and

to

component

tolerances

or

drift,

digital

implementations allow one to achieve certain characteristics not possible with analog implementations such as exact linear phase and multi-rate operation. Digital filtering can be applied

Comparisons, to various earlier designs show a reduction of the

to very low frequency signals, such as those occurring in

arithmetic complexity. A linear phase FIR filter is designed with

biomedical and seismic applications very efficiently. In

Farrow Structure and minimization of filter coefficients is done

addition, the characteristics of digital filters can be changed or

using Parks-McClellan (PM) algorithm and Genetic algorithms

adapted by simply changing the content of a finite number of

(GA).it is found that GA outperforms PM method in various

registers, thus multiple filtering tasks can be performed by one

design cases considered.

programmable digital filter without the need to replicate the

Keywords: Farrow Structure, Park McClellan Algorithm,

hardware. With the ever increasing number of applications

Genetic algorithm

involving digital filters, the variety of requirements that have

1. Introduction

to be met by digital filters has increased. As a result, design techniques that are capable of satisfying the required design

Digital Filters are used in numerous applications from control

requirements are becoming an important necessity.

systems, systems for audio and video processing, and

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International Journal of Computer & Organization Trends –Volume 3 Issue 10 – Nov 2013 2. Window function and Digital FIR filters.

The frequency spectrum of window is as below

The method the most used in digital filter design is Fourier series method. However, there is a problem in this method. The problem is that Fourier series method causes to Gibb’s oscillations at cut-off frequency region. This case was illustrated as below.

Fi gure2. 3, shows spe c t r a l par a me t e r s ( Br , r a nd s)

3. Finite Impulse Response Filters FIR filters are digital filters with finite impulse response. The order of FIR filter is remarkably higher as compared to an IIR filter with the same frequency response. An FIR filter has a Figure 2.1: The undesired oscillations for different filter lengths

number of useful properties which sometimes make it

The resultant oscillations are reduced by using window

preferable to an infinite impulse response (IIR) filter.FIR

function (or simply window). A typical window amplitude

filters are digital filters with finite impulse response. They are

response with length of N =31 is given in Fig. 2.

also known as non recursive filters as they do not require any feedback. This means that any rounding errors are not compounded by summed iterations. The same relative error occurs in each calculation. This also makes implementation simpler .They are inherently stable. This is due to the fact that, because there is no required feedback, all the poles are located at the origin and thus are located within the unit circle which is the required condition for stability in a discrete, linear-time invariant system. They can easily be designed to have a precise linear phase by making the coefficient sequence symmetric, linear phase, or phase change proportional to frequency, corresponding to equal delay at all frequencies.

This property is sometimes desired for phase-sensitive applications, for example data communications, crossover

Fig.2.2 A typical window

filters, and mastering where there is a need for a linear phase For various new cases, suitable window coefficient values are calculated and they apply to filter. The various methods have been improved to avoid calculation process in design.

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characteristic within the pass band of the filter.FIR filters have only zeros and no poles in their transfer function and therefore it does not oscillate and have a constant delay. Therefore the

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International Journal of Computer & Organization Trends –Volume 3 Issue 10 – Nov 2013 FIR filter has finite length. The phase shift of the filter will be

The Farrow Structure shown in fig4.1 is composed of fixed

product of the time delay and frequency.

linear phase finite length impulse response (FIR) sub filters Sk(z),k=0,1…….,L,of order Nk as well as the variable

4. Proposed design of Fractional Delay Filters

multipliers µ.The transfer function is

Fractional-delay (FD) filters are a type of digital filter

H(z,µ)=∑

designed for band limited interpolation. Band limited interpolation is a technique for evaluating a signal sample at an arbitrary point in time, even if it is located somewhere between

Sk(z)µk

,|µ|≤0.5

………. 4.1

In the transfer function of the Farrow Structure, different sub filters are weighted by different powers of the FD Sub filters of even order are considered.

two sampling points. The value of the sample obtained is exact because the signal is band limited to half the sampling rate (Fs/2). This implies that the continuous-time signal can be exactly regenerated from the sampled data. The major steps to be followed in the design of FIR filters are as follows.

4.2: Parks-McClellan method Parks-McClellan method (also known as the Equiripple, Optimal, or Minimax method) with the Remez exchange algorithm is used to find an optimal equiripple set of coefficients to design an optimal linear phase filter. This is a

4.1 Farrow structure

standard method for the FIR filter design which minimizes the Variable Fractional delay FIR Filter is designed using the Farrow Structure, a popular method for implementing timevarying FIR FD filters. To compute the output of a fractional delay filter, we need to estimate the values of the input signal between

the

existing

discrete-time

samples.

Special

interpolation filters can be used to compute new sample values at arbitrary points. To design a filter means to select the coefficients such that the system has specific characteristics.

The advantage of the Farrow structure over a Direct-Form FIR resides in its tunability. Each of the impulse response coefficients are modelled as Mth order polynomials of the delay variable which implemented the variable filter as a linear combination of M+ 1 filters as shown below

filter length for a particular set of design constraints. This method is used to design linear phase, symmetric or antisymmetric filters of any standard type. Better filters result from minimization of maximum error in both, the stopband and the passband of the filter which leads to equiripple filters.

Such filters are an optimum approximate and can be achieved using algorithmic techniques. In this algorithm to design FIR filters, some of its parameters such as the filter length (M), passband and stopband normalized frequencies (wp, ws), maximum of the absolute ripple in the passband and stopband (δp, δs) are fixed and the remaining parameters are to be optimized. Parameters M, δp, and δs are fixed while the remaining parameters are optimized. The Parks–McClellan (PM) algorithm is the most popular approach for optimum FIR filters design due to its flexibility and computational efficiency. In the PM algorithm, an approximate error function is defined by E (ω) =G (ω) [Hd(ejω)–H(ejω)]

…………..4.2

Where Hd(ejω) and H(ejω) are the frequency responses of the desired and the approximate filters respectively. G(ω) is the weighting function.

Fig 4.1: Farrow Structure with fixed sub filters Sk(z) and a variable FD of µ

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International Journal of Computer & Organization Trends –Volume 3 Issue 10 – Nov 2013 5. OPTIMIZATION

5.1. GENETIC ALGORITHM

Generalized optimization techniques are then used to minimize

In our design, we utilize genetic algorithm to optimize the

(or maximize) a given function, known as the objective

design of the digital filter. Genetic Algorithms are searching

function, or cost function. A linear Optimization problem is

and optimization techniques inspired by two biological

the one whose objective function is a linear function of the

principles, namely the process of “nature selection” and the

input. Optimization algorithms generally take a starting guess

mechanics of “natural genetics”. The genetic algorithm is an

point and change the variables subjected to the constraints in

artificial genetic system based on the process of natural

such a way that it decrease (or increase) the objective function.

selection and genetic operators. It is also a heuristic algorithm

Some sort of termination condition is then required. The first

which tries to find the optimal results by decreasing the value

step of any optimization routine is having a basic knowledge

of objective function (error function) continuously. GA, one of

of the problem and its limitations. In the case of filter

the Evolutionary calculating methods, has been appeared by

optimization the designer must be familiar with the electrical

using Darwin’s evolutionary theory when there are any data

performance of each filter design. This method of approach

about solution, starting population is obtained from that

requires the performance of each new design to be calculated

data. The number of chromosome of starting population is

and compared against the ideal optimum specification.

selected by user. Chromosomes in the creating population are

Generalized optimization techniques minimize (or maximize) a

used in objective-fitness function, different from problem to

given function, known as the objective function, or cost

problem, to measure the fitness. Due to the objective-fitness

function. A linear Optimization problem is the one whose

value, for creating next generation, chromosome in the

objective function is a linear function of the input.

population is reproduced and crossed and mutated if necessary.

Optimization algorithms generally take a starting guess point and change the variables subjected to the constraints in such a way that it decrease (or increase) the objective function. Some sort of termination condition is then required. First, a starting point x0 is chosen and f(x) is evaluated at that point. Then, a new value of x is chosen to reduce the objective function. This process is repeated until some termination condition is met.

The minima or maxima of functions occur at points where the derivative of that function is equal to zero. In filter design problems, the cost function is generally an error function. Thus the cost function must derive the magnitude response of the

The calculated copy numbers of each chromosome determine which chromosome will be used at reselection, crossing or mutation operations. Chromosome having higher possibility to be selected is crossed more times and it is in the next generation more times. Chromosome having less possibility is rarely taking places in the next generation or maybe it is no in that.

The step process of GA is as such below: Step 1:

Initial population resulting in chromosome that

provides solutions is created to reach results.

given input x, and subtract it from a desired response. This generates an error signal that can be minimized. The two most

Step 2: Chromosomes in the creating population are used in

common types of objective functions used in filter design are

objective-fitness function and values of that are calculated

least squares and minimax .The least-squares optimization

One part used in GA program and run especially is objective-

problem is concerned with minimizing the sum of the squared

fitness function.

error vector, while minimax optimization problems are concerned with minimizing the maximum value in the error vector. A weighting function is generally a constant function that applies a weight to each value in the error vector

Step 3: After chromosomes are selected between each other randomly, copying and mutation processes are applied.

Step 4: Fitness values of new chromosomes obtained from crossing and that in the first population are recalculated.

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International Journal of Computer & Organization Trends –Volume 3 Issue 10 – Nov 2013 Chromosomes having the best objective-fitness and being up

calculated after using PM and Genetic Algorithm. So the

to population number are created initial population of new

average

generation. Thus number of population is fixed

coefficient values is given by:

Step 5:

% average error after PM = (0.8543 – 0.4342)/ 0.8543 =

This pr oc ess i s co n t i n u ed un t i l val u e o f

d esi red convergence is reached

error calculated through PM algorithm filter

49.17% The percent average error calculated after using GA optimal filter coefficients is given by: % average error after GA = (0.4342 – 0.3802) /0.4342 = 12.3% The percent average error calculated using GA has been minimized to a great extent and gives lower absolute error as compared to that of using PM algorithm which has higher ripple in the passband. Better filters result from minimization of maximum error in both the passband and stopband of the filter which leads to equiripple filters. Such filters are an optimum approximate and it shows that the GA outperforms PM and is achieving the design requirement.

7. Conclusion With this study, the FIR digital filter design was realized by using of window that its parameters were calculated with GA. At this work, the coefficient values of Kaiser Window used in many applications because of having parameters were found.

By using of this window, the amplitude response for desired FIR filter was plotted and successful results were taken. With helping of GA used at programme, the calculation operations encountered in design was reduced and coefficient calculation for various new case was easily realized.

Fig Block diagram of genetic algorithm function used in this design.

6. Comparison of Average Error calculation using PM and GA Algorithms: The design objective is to find the approximate filter coefficients that result in the optimum filter. The filter is optimum in the sense that the maximum weighted error is minimized.

After

obtaining

the

three

different

filter

coefficients values obtained by Ideal Filtering, through Fractional delay and PM algorithm and finally after the filter

References [1] Soo-Chang Pei, “Design of variable comb filter using FIR variable fractional delay element,” Elsevier, vol. 92, pp. 2409–2421, March 2012.

[2] Tian-Bo Deng, “ Bi-Minimax Design of Even-Order Variable FractionalDelay FIR Digital Filters,” IEEE Trans. Circuits Sys .I, Reg. Papers, vol. 59, no. 8, pp. 1549-8328, Aug. 2012 [3] Amir Eghbali, Hakan Johansson and TapioSaramaki, “A method for the design of Farrow- structure based variable fractional-delay FIR filters,” Elsevier, vol. 93, pp. 1341–1348, Nov. 2012.

[4] Journal of Theoretical and Applied Information Technology, vol. 48 No.1, Feb. 2012 [5] Marcelo Basilio Joaquim, Carlos A.S. Lucietto, “A nearly optimum linearphase digital FIR filters design,” Elsevier, vol. 21, pp. 690–693, Jan. 2011.

[6] Soo Chang Pei1, Jong-Jy Shyu, Cheng-Han Chan and Yun-Da Huang, “ A new structure for the design of variable fractional delay FIR Filters,” 19th European Signal Processing Conference (EUSIPCO), pp. 2076-1465, Sept.

2011.

optimization using Genetic Algorithm, percent average error is

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International Journal of Computer & Organization Trends –Volume 3 Issue 10 – Nov 2013 [7] Chien Cheng Tseng and Su-LingLee, “Design of linear phase FIR filters using fractional derivative constraints,” Elsevier, vol. 92, pp. 1317-1327, Dec.2011.

[8] Jong Jy Shyu, Soo-Chang Pei, Cheng-Han Chan, Yun-Da Huang and ShihHsin Lin, “ A New Criterion for the Design of Variable Fractional-Delay FIR Digital Filters,” IEEE Trans. Circuits Sys .I, Reg. Papers, vol. 57, no. 2, pp.

1549-8328, Feb. 2010.

[9] Tian-Bo Deng, “Minimax Design of Low-Complexity All pass Variable Fractional-Delay Digital Filters,” IEEE Trans. Circuits Sys .I, Reg. Papers, vol. 57, no. 8, pp.1549-8328, Aug. 2010.

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