13.IJAEST-Vol-No-7-Issue-No-1-Capacity-Enhancement-of-MIMO-system-using-water-filling-Model-092-097 by ISERP ISERP

Kuldeep Kumar* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 7, Issue No. 1, 092 - 097

Capacity Enhancement of MIMO system using water-filling Model

Abstract

(OFDM), is used it is sought as a one of the solution for increasing the capacity and data rate of a system in a environment where the communication take place in a frequency-selective fading environments and there can be probably more chances for data corruption. It has been found out from the research and the result that MultipleInput and Multiple-Output (MIMO) [2] can be effectively used to increase the capacity of the system by a factor of the minimum number of transmitter and receiver antennas attached in the MIMO system as compared to a Single-Input Single-Output (SISO) system that has flat fading or frequency selective fading environment or narrowband channels, OFDM can also increase diversity gain and minimize the inter-symbol interference on a time-varying multi-path fading channel. When we know the parameters of the channel both at the transmitter end and at the receiver end, we can further increase the capacity of MIMO OFDM systems by assigning power at the transmitter according to the water-filling algorithm to the channels . At transmitters, we send the transmitted signals of all the carriers that are Eigen beamformed independently to orthogonal modes of all the channels in a MIMO OFDM systems. In a MIMO OFDM we use a improved power allocation scheme called water filling as compared to the classical water-filling power allocation scheme .In a MIMO system the results show that there is increase in the MIMO capacity with water filling. In order to apply the water filling algorithm we need to know the channel parameters . The capacity can further be increased by assigning the power to the transmitted signals to the orthogonal eigen values according to the water filling rule.We transmit the different signals at the transmitter which are eigen beamforced independently to a orthogonal mode in space time and is sent simultaneously by multiple carriers .A novel eigen modes are used to directly start the adaptive power allocation to the data bits allocation as such this is usually done with a single carrier transmitter .Here we have two type of water

Here we discuss the a general theoretical framework for power budget allocation in the wireless cellular network ,where multiple access points or small base stations send independent coded information to multiple mobile terminals through orthogonal Code division multiplexing channels. Here in we study, the outage probability of a MIMO system with the change in the power budget also we would analyze the mean capacity of a system and the effect with the water filling algorithm. For independent continuous fading channels, we show that as we increase the power budget in the water filling algorithm the mean capacity of the system increases also the outage capacity of a system reduces for the water filling algorithm as the power budget in increased. Hence efficiently Power budget allocation at the transmitter increases the system capacity and reduces the outage probability.

Manwinder Singh Rayat Institutes Of Engg. and IT, Ropar. Punjab Email: Singh.manwinder@gmail.com

Kuldeep Kumar Rayat Institutes Of Engg. and IT, Ropar, Punjab Email: kuldeep_heer@yahoo.com

Key Wordsâ&#x20AC;&#x201C; water filling, outage probability, Multi Input Multi Output (MIMO), system capacity .Signal to Noise Ratio (SNR), Power Budget

1. INTRODUCTION

It has been shown that Multiple-Input Multiple-Output (MIMO) systems is used to support higher data rate in the MIMO system we keep the transmit power budget and performance requirement same as compared to a SingleInput Single-Output (SISO) systems. If we compare the MIMO system with a SIMO system the MIMO system require lesser transmit power than the SISO system. If we consider the energy consumption in total which will includes the Transmit energy and the circuit energy then we need to find out which system is more energy efficient as there is a huge complexity involved in the circuit of a Multiple-Input Multiple-Output (MIMO) systems .The MIMO system has multiple transmit and receive antennas and Orthogonal Frequency-Division Multiplexing

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filling models one is the classical water filling model and other is the improved water filling model. Of all the power allocation schemes the greedy power allocation scheme is the optimal one to transmit the bits. The article is organized as follows. In section 2, discusses the MIMO capacity and apply the water filling algorithm to increase the power. Section 3, we describe the water filling In section 4, we discuss the outage probability of a MIMO system and section 5, we conclude our discussion with the results.

In this equation,” deter” means the determinant, I is a identity matrix and “†” means conjugate transpose of the matrix.

2. MIMO CHANNEL CAPACITY

The capacity formula for where the transmit and receive antennas are the same i.e (nr= nt = n) is given as [5].

 2   2 k b / s / Hz  



(4)



r  w* s  e

n   C   log 2 1   k 1    t

(3)

C  log 2 1   . 22n b / s / Hz

(5)

Here we describes some of the very important properties of channel matrix H for a MIMO system . We describe in this section some statistical properties of the channel matrix H. The channel matrix HH† using eigen value decomposition can be rewritten as follows

Lets us have a MIMO system with M ,N transmitter and receiver antennas respectively . The space time codes are used to encode the data along with the space time direction with T data symbols. The model of communication system is given as

    C  log 2 det er  I nr    HH † b / s / Hz   t   

(1)

Where r is the received signal and s is the sent signal .* is the convolution and e is the error function . w is the channel matrix .When M=N=1 then is a SISO .We will

HH† = E^E†

(6)

Where E is eigenvector matrix of orthonormal columns and ^ is a diagonal matrix having the Eigen values on the main diagonal. Using this notation above , the capacity of MIMO channels can be written as below :

= 1, 2, ..., n r) Receive antenna and jth (j = 1, 2 ..., n t) transmit antenna,The MIMO channel matrix is given by H(  , t)

In general channel matrix rank is given by is given by

discuss about a MIMO system where M=N  1here it is understood that we can increase the capacity of a MIMO system without changing the bandwidth and transmit power rather we can put more antennas at transmitter and receiver side to increase the capacity.One of the important part of a MIMO system is the channel matrix denoted by

H ( , t ) .Consider the impulse response between the ith (i

 h1,1  , t  h1, 2  , t   h  , t  h  , t  2, 2  2,1 .  . H ( , t )   . .   . . hn ,1  , t  hn , 2  , t  r  r

....... h1,nt  , t   ....... h2,nt  , t   ..... .   ...... .  ....... .  ....... hnr ,nt  , t 

    C  Eh log 2 det I nr  E  E †  t   

 (7)

rank ( H )  k  min nr , nt 

(8)

Using the equation (6), with the fact known that the determinant of a unitary matrix is 1, the capacity expression can be written as: (2)

In frequency domain, the channel is made of a complex matrix having the independent and identically distributed entries with unit variance and zero mean. The generalized capacity formula for a MIMO system is given by

 k    C  E H  log 2 1  t   i 1  nt       k  E H  log 2 1   t  nt     i 1

(9)

(10)

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values and

 i are the eigen

 i are the squared singular values of the

. diagonal matrix  . When the channel matrix is known at the transmitter, the maximum capacity of a MIMO channel can be achieved by using the water-filling algorithm [3] on the transmit covariance matrix. The capacity is then given by:

 k    C  E H  log 2 1   i t  nt   i1 

(11)



Where i is the ith channel and i is a scalar that shows the portion of the available transmit power that is available in the outgoing ith channel. So, by using the water-filling algorithm we can meet the total power constraint.

As we have to maximize the total number of bits to be transported , The results shows that improved waterfilling power allocation scheme is given on the base of classical water filling schemes. As per the scheme, the adaptive power and bit allocation are conducted in two steps. Firstly, the initial power allocation is given by classical water-filling scheme, and then is, the first step is is to allocate the power for all the orthogonal eigen modes according to the classical water-filling scheme to all the eigen values. Then, after finding the transported bits at channel eigenmodes,we allocate the residual power by reallocated among these eigenmodes to transport additional bits.Here we see that the Mimo system with water filling has a greater mean capacity as compared to a MIMO system without water filling .Also as per the results in Fig.1,2, 3 the mean capacity of a MIMO system increases with the increase in the power budget at the input of the transmitter. Following figures are the result that we get for the capacity of a 4X4 MIMO system by varying the power budget.

Here  is the diagonal matrix and

3. Water filling Power allocation algorithm

Mean Capacity vs SNR

IJ 2 

2

 P  P (c )  Q  e  (2e  1)  3   4  1

(12)

Q is called the as complementary error function. Then, for given total transmit power of the MIMO system is given by , the number of bits transported by the AWGN channel, can be derived by the Eq. 12, as shown in the following formula 2   3P   P    c  f loorlog 1  2  Q 1  e        4     

4x4 MIMO at Pt=0.2 4x4WF MIMO at Pt=0.2

Mean Capacity bps/Hz --->

The adaptive modulation can be implemented in two aspects , the adaptive power allocation technique is such that the transmitted power maximize the number of the transported bits. Here in this study we study the SNR and the Capacity of a 4x4 MIMO system. We would find out the mean capacity of the system with and without the water filling algorithms .Here the power is distributed in the orthogonal eigenmodes in order to maximize transmit bits to get the maximum capacity of the MIMO system.The transmit power required to send the c bits for an additive white Gaussian noise (AWGN) channel with M-QAM modulation, is given by the following formula.

0 -10

-5

10 SNR (dB)

--->

Fig. 1.Mean Capacity at Pt=0.2 with water filling algorithm The figure 1,2,3,4 shows the mean capacity of the system when the power budget Pt 0.2,0.3,0,5 and 1 with and without the water filling model ..Figure 4 shows the mean capacity of a 4x4 MIMO system for all the water filling . By observing the different figures we see that as the power budget is increased the capacity of the system also increases

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4. OUTAGE PROBABLITY

Mean Capacity vs SNR 35 4x4 MIMO at Pt=0.3 4x4WF MIMO at Pt=0.3

Mean Capacity bps/Hz

--->

The outage probability is defined as that an outage will occur within a given time period. It can also be defined as temporary suspension of the operation The signal outage probability is calculated if one knows the probability distribution of the fading either Rayleigh or Racian .The outage will take place if the signal drops below the signal of the noise power level.

Calculation of the outage probability involves that we find the probability that the signal-to-interference ratio that drops below a certain threshold. As all signals that are fading have fluctuating signal powers, It mean that in order to find the outage we must integrate the probability density functions of all signals involved in fluctuating.

-5

10 SNR (dB)

--->

0 -10

Fig. 2.Mean Capacity at Pt=0.3 with water filling algorithm

The Outage formula is given by

Mean Capacity vs SNR 35

---> Mean Capacity bps/Hz

Where SNR is Signal to noise ratio, z is the minimum

SNR required , f is the probability distribution function

-5

10 SNR (dB)

0 -10

P(out)  1  P(SNR  z)  1  Laplace( f ; z /  ) (14)

4x4 MIMO at Pt=0.5 4x4WF MIMO at Pt=0.5

--->

Fig. 3.Mean Capacity at Pt=0.5 with water filling algorithm 40

MIMO MIMO MIMO MIMO MIMO

at at at at at

Pt=0.2 Pt=0.3 Pt=0.4 Pt=0.5 Pt=1.0

4x4WF 4x4WF 4x4WF 4x4WF 4x4WF

Mean Capacity bps/Hz ---->

35 30

of the power,  is the mean power of the wanted signal. In order to find the outage probability of a MIMO network we take a 4x4 MIMO and would try to analyze the variation of the outage probability with the variation of power budget. It has been shown in the results that with the increase in th power budget the outage probability decreases. Also it is found from the graphs that the outage of the waterfilling model is more in a MIMO than the MIMO system. The outage probability in a MIMO system is given by ,

  SNR Pr (outage)  Pr log 2 det(I  HH   Nt  (15) 

25 20

Where H is the channel matrix,I is the information

transmitted,

H  ( H * H ) 1 H *

-5

10 15 SNR (dB)---->

is the number of the transmit antennas



0 -10

and Following are the graphs showing the outage capacity (probability) variation of a SISO,SIMO, MISO, MIMO system with SNR

Fig. 4.Mean Capacity at different Pt with water filling algorithm

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Kuldeep Kumar* et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 7, Issue No. 1, 092 - 097

Outage probability vs SNR for 4 bps/Hz 1

1 1x1 SISO at Pt=1.0 1x4 SIMO at Pt=1.00 4x1 MISO at Pt=1.0 4x4 MIMO at Pt=1.0 4x4WF MIMO at Pt=1.0

Outage Capacity

0.7

0.8

0.6 0.5 0.4 0.3 0.2

--->

0.8

0.7 0.6 0.5 0.4 0.3 0.2

0.1 0

4x4 MIMO at Pt=0.4 4x4WF MIMO at Pt=0.4

0.9

Outage Probability

0.9

0.1

SNR

Fig. 5.Outage probability at Pt=1 for different systems

10 12 SNR in dB --->

Fig. 8.Outage probability of MIMO system at Pt=0.2 1

0.8

0.9

0.6

0.8

0.5

0.7

0.4

Outage Capacity

0.7

1x1 SISO at Pt=0.5 1x4 SIMO at Pt=0.5 4x1 MISO at Pt=0.5 4x4 MIMO at Pt=0.5 4x4WF MIMO at Pt=0.5

0.9

0.3 0.2

MIMO MIMO MIMO MIMO MIMO

at at at at at

Pt=0.2 Pt=0.3 Pt=0.4 Pt=0.5 Pt=1.0

0.6 0.5 0.4 0.3

0.1 0

4x4WF 4x4WF 4x4WF 4x4WF 4x4WF

0.2

0.1

SNR

Fig. 6.Outage probability at Pt=0.5 for different systems

Fig.9.Outage probability of MIMO WF system

Outage probability vs SNR for 4 bps/Hz 1

4x4 MIMO at Pt=0.2 4x4WF MIMO at Pt=0.2

0.9

--->

0.7

Outage Probability

0.8

0.6 0.5 0.4 0.3 0.2 0.1

10 12 SNR in dB --->

Fig. 7.Outage probability of MIMO system at Pt=0.2

5. CONCLUSION This paper we have developed an understanding and described the Mean capacity allocation in a wireless cellular network based on the water filling power allocation in order to enhance the capacity of a MIMO systems with different channel assumptions. Here each transmitter decide the distribution of power to the several independent fading channels. We studied the change in the Mean Capacity of the system with the power budget of a system. Also, an improved water-filling scheme is proposed for increase in the system capacity. Results indicates that the water-filling scheme has better capacity than without water filling at greater value of power budget. We also discussed the variation of the outage probability of the system .

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6. REFERENCES

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[3] S. Ariyavisitakul and C. Li Fung, "Signal and interference statistics of a CDMA system with feedback power control," IEEE Transactions on Communications, vol. 41, pp. 1626-1634, 1993.

[9] S. Zhang, Y. Chen, and S. Xu, “Improving Energy Efficiency through Bandwidth, Power, and Adaptive Modulation,” in IEEE 72nd VTC 2010Fall, 2010

[4] J. S. Evans and D. Everitt, "Effective bandwidth-based admission control for multiservice CDMA cellular networks" IEEE Transactions on Vehicular Technology, vol. 48, pp. 36-46, 1999.

[10] Fabien, Hoshyar, Reza and Tafazolli, (2009) “ A closed-form approximation of the outage probability for distributed MIMO systems” In: IEEE 10th Workshop on Signal Processing Advances in Wireless Communications, 2009. SPAWC '09, 21-24 June 2009 [11] M. Kaynia, G. E. Oien, A. J. Goldsmith, and N. Jindal, "Outage probability in wireless networks", Proc. IEEE Information Theory Winter School (ITWS), Loen, Norway, March 2009.

[5] T. S. Rappaport, Wireless Communications: Principles and Practice, 2nd Edition ed.: Pearson Education, 1996.

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[7] B. Hassibi and B. M. Hochwald. “How much training is needed in multiple-antenna wireless links” IEEE Transactions on Information Theory, 48(4):951 – 963, April 2003.

[6] B. Hashem and E. S. Sousa, "Reverse link capacity and interference statistics of a fixed-step powercontrolled DS/CDMA system under slow multipath fading," IEEE Transactions on Communications, vol. 47, pp. 1905-1912, 1999.

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