Songklanakarin J. Sci. Tecnol. 33 (3), 335-340, May - Jun. 0 ttp://www.sjst.psu.ac.t Original Article Performance analysis and comparison of m x n zero forcing and MMSE equalizer based receiver for mimo wireless cannel N. Satis Kumar and K. R. Sankar Kumar Department of Electronics and Communication Engineering, SriRamakrisna Engineerign College, Coimbatore, TN,India -640 Received 5 February 0; Accepted 8 June 0 Abstract Wireless transmission is affected by fading and interference effects wic can be combated wit equalizer.te use of MIMO system promises good improvement in terms of spectral efficency,link relaibility andsignal to Noise Ratio (SNR). Te effect of fading and interference always causes an issue for signal recovery in wireless communication. Equalization compensates for Intersymbol Interference (ISI) created by multipat witin time dispersive cannels. Tis paper analyses te performance of Zeroforcing and MMSE equalizer for MIMO wireless caneels. Te simulation results are obtained using MatLab tool box version 7.0 at RF signal processing lab.te Bit Error Rate (BER) caracteristics for te various transmitting and receiveing antennna is simulated in matlab tool box and many advantages and disdvantagesof te system is descrbed. Te simulation results sow tat te equalizer based zero forcing receiver is good for noise free cannel and is successful in remving ISI,but MMSE is a better coice tan ZF in terms of BER carateristics and under Noise performance. Keywords: MIMO, Zero forcing Equalizer, ISI, BER, linear equalization, MMSE. Introduction Wireless communication systems as sown tat using multiple antennas at bot transmitter and receiver and tus provides te possibility of iger data rates compared to single antenna systems (David Gesbert et al., 003; Zang et al., 006; Tse et al., 005). Te system wit multiple antennas at te transmitter and receiver is commonly referred is to as multiple input multiple output (MIMO) systems. Te multiple antennas are used to increase data rates troug multiplexing or to improve performance troug diversity. Tis tecnique offers iger capacity to wireless systems and te capacity increases linearly wit te number of antennas and link range wit out additional bandwidt and power requirements.mimo can reduce fading and improve iger spectral efficiency and link reliability or diversity (Jindal, Corresponding autor. Email address: nsk0000@gmail.com 005). In MIMO wireless communication, an equalizer is employed wic is a network tat makes an attempt to recover a signal tat as suffers wit an Inter symbol Interference (ISI).. Zero Forcing Equalizer Matematics Zero Forcing Equalizer is a linear equalization algoritm used in communication systems, it inverts te frequency response of te cannel,wic was proposed by Robert Lucky. Te Zero-Forcing Equalizer applies te inverse of te cannel to te received signal, to restore te signal before te cannel. Te name Zero forcing corresponds to bringing down te Inter Symbol Interference (ISI) to zero in a noise free case. Tis will be useful wen ISI is more predominant wen comparing to te noise (Zang et al., 006; yi Jiang et al., 0). For a cannel wit frequency response F (f) te zero forcing equalizer C (f) is constructed by C (f) = / F(f). Tus te combination of cannel and equalizer gives a flat frequency response and linear pase F (f) C (f) =.
336 N. S. Kumar & K. R. S. Kumar / Songklanakarin J. Sci. Tecnol. 33 (3), 335-340, 0 Consider a x MIMO cannel, te received signal on te first receive antenna is, is, y x + n x. x, x n, Te received signal on te Second receive antenna y Were y, y x + n x.,x, x n,,,,, x, x n, n are te received symbol on te first and second antenna respectively, is te cannel from st transmit antenna to st is te cannel from nd transmit antenna to st is te cannel from st transmit antenna to nd is te cannel from nd transmit antenna to nd are te transmitted symbols and are te noise on st and nd receive antennas. Te equation can be represented in matrix notation as follows: y, = y x,, x Equivalently, n + n.3 Y = HX + N.4 Were, Y = Received Symbol Matrix. H = Cannel matrix. X = Transmitted symbol Matrix. N = Noise Matrix. To solve for x, we need to find a matrix W wic satisfies WH = I. Te Zero Forcing (ZF) detector for meeting tis constraint is given by, W = (H H H) - H H.5 Were W - Equalization Matrix and H - Cannel Matrix Tis matrix is known as te Pseudo inverse for a general m x n matrix were H H H =,,,,,,,,,,,, =,,, Note tat te off diagonal elements in te matrix H H H are not zero, because te off diagonal elements are non zero in values. Zero forcing equalizer tries to null out te interfering terms wen performing te equalization, i.e. wen solving for x te interference from x is tried to be nulled and vice versa. Wile doing so, tere can be amplification of noise. Hence Zero forcing equalizer is not te best possible equalizer.(jinag et al., 0) However, it is simple and reasonably easy to implement. For BPSK Modulation in Rayleig fading cannel, te BER is defined as P b = ( E b / No ( E / ) b No Were P b - Bit Error Rate E b / N o - Signal to noise Ratio 3. MMSE Equalizer Matematics A Minimum Mean Square Error (MMSE) estimator describes te approac wic minimizes te mean square error (MSE), wic is a common measure of estimator quality. Te main feature of MMSE equalizer, is tat it does not usually eliminate ISI completely but, minimizes te total power of te noise and ISI components in te output (Satis Kumar, et al., 0; Jinag et al., 0). Let x be an unknown random variable, and let y be a known random variable. An estimator x^ (y) is any function of te measurement y, and its mean square error is given by.6.7 MSE = E {(X^ -X )}.8 Were te expectation is taken over bot x and y. Te MMSE estimator is ten defined as te estimator acieving minimal MSE. In many cases, it is not possible to determine a closed form for te MMSE estimator. In tese cases, one possibility is to seek te tecnique minimizing te MSE witin a particular class, suc as te class of linear estimators (Co et al., 00). Te linear MMSE estimator is te estimator acieving minimum Mean square error among all estimators of te form AY + b. If te measurement Y is a random vector, A is a matrix and b is a vector. Let us now try to understand te matematics for extracting te two symbols wic interfered wit eac oter.
N. S. Kumar & K. R. S. Kumar / Songklanakarin J. Sci. Tecnol. 33 (3), 335-340, 0 337 In te first time slot, te received signal on te first receive antenna y = x +, x + n = [, ] x + n x.9 Te received signal on te second receive antenna is, y =, x +, x + n = [,, ] x + n x.0 Were y, y are te received symbol on te first and second antenna respectively, y, y are te received symbol on te first and second antenna respectively, is te cannel from st transmit antenna to st, is te cannel from nd transmit antenna to st, is te cannel from st transmit antenna to nd, is te cannel from nd transmit antenna to nd x, x are te transmitted symbols and n, n are te noise on st and nd receive antennas. Te equation can be represented in matrix notation as follows: y y, = x,, x Equivalently, n + n. Y = HX + N. Were, Y = Received Symbol Matrix. H = Cannel matrix. X = Transmitted symbol Matrix. N = Noise Matrix. Te Minimum Mean Square Error (MMSE) approac tries to find a coefficient W wic minimizes te criterion, E {[W y-x ] [W y-x ] H } Were W - Equalization Matrix H - Cannel Matrix and n - Cannel noise y - Received signal. To solve for x, we need to find a matrix W wic satisfies WH =I. Te Minimum Mean Square Error (MMSE) detector for meeting tis constraint is given by, W = [H H H+ N o I) - H H.3 Tis matrix is known as te pseudo inverse for a general m x n matrix Were H H H =,,,,,,,,,,,, =,,, Wen comparing te eq.(.3) to te eq.(.5) in Zero Forcing equalizer, apart from N o I te term bot te equations are comparable. In fact, wen te noise term is zero, te MMSE equalizer reduces to Zero Forcing equalizer. 4. Simulation Results and Discussions Simulation Analysis : Te simulations were carried out at RF signal processing lab Now let us study te simulation results of ZF equalizer receiver BER performance caracteristics. Let us vary te receiver antenna keeping transmitter antenna constant for equalizer based receiver at a particular e b /N 0 value using BPSK modulation metod. Figure (a-c) and table sow, as te no of receivers (n) is increased keeping te no of transmitters (m) as constant for a zero forcing receiver. Figure (a) sows m is fixed wit and n is varied. Similarly figure (b) and (c ) also sows te m is fixed wit, and 3 and n is varied respectively. Figure (d) sows te consolidated result in te form of bar cart comparison. It is evident tat te Bit Error Rate (BER) decreases as te number of transmitter increases, in Zero Forcing Equalizer. Te following observations are made. Te Zero Forcing Equalizer removes all ISI and is ideal only wen te cannel is noiseless. Wen te cannel is noisy, te Zero Forcing Equalizer will amplify te noise greatly at frequencies f were te cannel response H (jðf) as a small magnitude (i.e. near zeroes of te cannel) in te attempt to invert te cannel completely. Te ZF equalizer tus neglects te effect of noise altogeter, and is not often for wireless links. However, it performs well for static cannels wit ig SNR. Simulation Analysis : Figure (a-c) and table sows tat as te no of receivers (n) is increased keeping te no of transmitters (m=, 3 and 4) as constant.it is observed tat te Bit Error Rate (BER) decreases in MMSE equalizer. Figure (d) sows te consolidated result in te form of bar cart comparison. From te above following observations are made.a more balanced linear equalizer is te Minimum Mean Square Error Equalizer, wic is not eliminate ISI completely but instead minimizes te total power of te noise and ISI com-.4
338 N. S. Kumar & K. R. S. Kumar / Songklanakarin J. Sci. Tecnol. 33 (3), 335-340, 0 (a) (b) (c) (d) Figure. Comparison of BER analysis for m x n Antenna configurations of ZF Equalizer. Table. Bit Error Rate values for mxn antenna configurations of ZF Equalizer Sl.No Bit Error Rate value for = E b / N o - 0dB m xn Value mxn Value mxn Value x 0.5 3x 0.367 4x 0.434 x3 0.907 3x3 0.57 4x3 0.960 3 x4 0.46 3x4 0.99 4x4 0.65 4 x5 0.07 3x5 0.3 4x5 0.605 5 - - 3x6 0.08556 4x6 0.099
N. S. Kumar & K. R. S. Kumar / Songklanakarin J. Sci. Tecnol. 33 (3), 335-340, 0 339 (a) (b) (c) (d) Figure. Comparison of BER analysis for m x n Antenna configurations MMSE Equalizer Table. Bit Error Rate values for mxn antenna configurations of MMSE Equalizer Sl.No Bit Error Rate value for = E b / N o - 0dB m xn Value mxn Value mxn Value x 0.33 3x 0.633 4x 0.75 x3 0.77 3x3 0.883 4x3 0.93 3 x4 0.385 3x4 0.448 4x4 0.5 4 x5 0.099 3x5 0.077 4x5 0. 5 - - 3x6 0.079 4x6 0.087
340 N. S. Kumar & K. R. S. Kumar / Songklanakarin J. Sci. Tecnol. 33 (3), 335-340, 0 ponents in te output. Hence from te above graps it is evident tat te BER decreases as te number of receiving antenna increases wit respect to number of transmitting antenna in MMSE equalizer based MIMO receiver. Conclusion Tis paper presents a simulation study on te performance comparison of ZR and MMSE Equalizer based MIMO receiver.tis paper compares te matematics and simulation results of two equalizers based MIMO receivers. Te simulation results sow tat te BER caracteristics for te two different types of equalizers namely ZF, and MMSE.Two types of simulation analysis are carried at RF signal processing lab. Te Simulation analysis discuss about m x n ZF Equalizer analysis.tis is by varying te receiver antenna configuration and keeping transmitter antenna constant for a particular type of E b /N 0 value using BPSK modulation metod. Te Zero Forcing Equalizer removes all ISI and is ideal only wen te cannel is noiseless. Wen te cannel is noisy, te Zero Forcing Equalizer will amplify te noise greatly at frequencies f were te cannel response H (jf) as a small magnitude (i.e. near zeroes of te cannel) in te attempt to invert te cannel completely. From te simulation results its is summarized tat Zero forcing equalization fails in te most of application due to te following. Already te cannel impulse response as finite lengt but te impulse response of te equalizer need to be infinitely long. Te cannel may consist of zeros in its frequency response but tat cannot be inverted. Some frequencies may be small, upon compensation it grows large. Addition of noise also gets boosted up and tus spoils te over all signal to noise ratio. Hence it is considered to a good receiver under noise free conditions. Simulation analysis discusses about MMSE Equalizer based MIMO receiver.based on te matematical model and simulation results te it is inferred tat MMSE equalizer based receiver is removes a marginal noise but does not eliminate completely also doesn t amplify as te case of zero forcing.hence MMSE equalizer based is a best coice tan Zero forcing equalizer based receiver. Acknowledgement Te autors express teir sincere tanks to, Te Management, Te Director Academics (SNR Caritable Trust),Te Principal, Sri Ramakrisna Engineering College, for teir constant support and encouragement given to us. Te Autors also extend teir eartfelt tanks to te DC members Dr.R.Rangarajan Te Dean Dr.Maalingam engineering college and Dr.Sankar Narayanan Te Dean EASA college of engineering for teir tecnical support and guidance to complete tis researc work. References Co, K and Yoon, D. 00 On te general BER expression of one and two dimensional amplitude modulations, IEEE Transactions on Communications, vol. 50, pp. 074-080. David Gesbert, Mansoor Safi, Dan-san Siv, Smit, P.J., et al. 003. From teory to practice : An Over View of MIMO space time coded wireless system, IEEE Journal on selected areas in Communication, (3), 8-30. Jiang, Yi, Varanasi, M.K., Jian Li.0. Performance Analysis of ZF and MMSE Equalizers for MIMO Systems: An In-Dept Study of te Hig SNR Regime, IEEE Transactions on Information Teory. 57(4), 008-06. Jindal, N. 005. Hig SNR analysis of MIMO broadcast cannels, Proc. IEEE Int. Symp. Information Teory, Adelaide, Australia, 30-34. Satis Kumar, N. and Sankar Kumar, K.R. 0. Performance Analysis of m x n Equalizer Based Minimum Mean Square Error (MMSE) Receiver fo r MIMO Wireless Cannel. International Journal of Computer Applications 6(7), 47 50, doi: 0.50/0-76. Tse, D. and Viswanat, P. 005. Fundamentals of Wireless Communications. Cambridge University Cambridge Press. Zang, H., Dai, Q., Zou and Huges, B. L. 006 On te diversity-multiplexing tradeo for ordered SIC receivers over MIMO cannels, IEEE International Conference on Communications (ICC), Istanbul, Turkey. Zang, X. and Kung, S.003 Capacity analysis for parallel and sequential MIMO equalizers, IEEE Transactions on Signal Processing, vol. 5, pp. 989-300.