A Novel Blind Equalizer Based on Dual-Mode MCMA and DD Algorithm

Transcription

1 A Novel Blid Equalizer Based o Dual-Mode MCMA ad DD Algorithm Seokho Yoo, Sag Wo Choi 2,JumiLee 2, Hyougmoo Kwo 2, ad Iickho Sog 2, School of Iformatio ad Commuicatio Egieerig, Sugkyukwa Uiversity, 3 Cheocheo Dog, Jaga Gu, Suwo , Korea syoo@ece.skku.ac.kr 2 Departmet of Electrical Egieerig ad Computer Sciece, Korea Advaced Istitute of Sciece ad Techology, 373- Guseog Dog, Yuseog Gu, Daejeo 35-7, Korea {swchoi, jmlee, kwo}@sejog.kaist.ac.kr, i.sog@ieee.org Abstract. We address a ew blid equalizer icorporatig both the good iitial covergece characteristic of the dual-mode modified costat modulus algorithm (MCMA) ad the low residual error characteristic after covergece of the decisio-directed (DD) algorithm. I the proposed scheme, a covergece detector is employed to help switchig from the dual-mode MCMA to the DD algorithm. We have observed that the proposed scheme exhibits a good overall performace i compariso with the CMA, MCMA, ad dual-mode MCMA. Itroductio I may moder commuicatio systems icludig digital mobile ad digital TV systems, data is ofte trasmitted through ukow chaels ad is thus subject to itersymbol iterferece (ISI), mostly due to the chael dispersio characterized by the o-ideal ature of the chael. The ISI is a primary cause degradig the performace of digital commuicatio systems. Thus miimizig ISI i digital commuicatio chael is crucial for the improvemet of system performace at high speed trasmissio rate. Beig a efficiet tool to extract the trasmitted symbol sequece by couteractig the effects of ISI, a equalizer icreases the probability of correct symbol detectio. Data-aided algorithms iitialize ad adjust equalizer coefficiets with a kow traiig sequece from the trasmitter before iformatio-bearig data This research was supported by the Miistry of Sciece ad Techology (MOST) uder the Natioal Research Laboratory (NRL) Program of Korea Sciece ad Egieerig Foudatio (KOSEF), for which the authors would like to express their thaks. Correspodig author. Y.-S. Ho ad H.J. Kim (Eds.): PCM 25, Part II, LNCS 3768, pp , 25. c Spriger-Verlag Berli Heidelberg 25

2 72 S. Yoo et al. trasmissio. Use of a traiig sequece, however, reduces the badwidth efficiecy ad may become impractical whe updatig of the coefficiets should be performed frequetly at the receiver ed. It is therefore desirable to equalize a chael without the aid of a traiig sequece, resultig i the self-recoverig, blid, or o-data aided equalizatio [] [3]. Amog the major advatages of blid equalizatio techiques is that o traiig sequece is ecessary to start-up or restart the equalizatio system whe the commuicatio breaks dow upredictably. Blid equalizatio methods also offer potetial improvemet i system capacity by elimiatig the traiig overhead. Numerous studies o blid equalizatio ca be foud i the literature. For example, ormalized slidig-widow costat-modulus ad decisio-directed (DD) algorithms have bee proposed i [4], establishig a lik betwee blid equalizatio ad classical adaptive filterig. A miimum-disturbace techique was proposed i [5] to avoid the gradiet oise amplificatio problem ad achieve improved stability ad robustess with low computatioal complexity. The multimodulus algorithm (MMA) itroduced i [6] takes advatage of the symbol statistics of such sigal costellatios as osquare ad very dese costellatios. Amog the various adaptive blid equalizatio algorithms, the Godard algorithm [] is oe of the best kow ad simplest adaptive blid equalizatio algorithms. This algorithm was also developed idepedetly ad exteded as the costat modulus algorithm (CMA) i [2]. Sice the CMA is phase-blid, the equalizer output has a arbitrary phase rotatio after covergece. Some performace improvemet of DD algorithm has also bee achieved i [7] by cotrollig the step size parameter accordig to the regios i which the equalized output lies i a costellatio. A particular problem of the CMA ad modified CMA (MCMA) is that the residual mea square error (MSE) i the steady state is sometimes ot sufficietly small for the system to exhibit adequate performace. Miimizig the residual MSE ad/or speedig up the covergece rate, the dual-mode algorithms such as those cosidered i [8] are possibly a plausible solutio to improvig the overall performace. The dual-mode algorithms possess a faster covergece rate ad lower residual MSE tha the CMA ad MCMA at a small additioal complexity to detect the covergece ad/or a ope eye patter. The cocurret CMA ad soft DD adaptatio proposed i [9] has lower computatioal requiremets tha the cocurret CMA ad DD algorithm. I this paper, we propose a ew blid equalizatio algorithm employig the dual-mode MCMA with modified parameters i the blid mode to improve the covergece rate ad the DD algorithm i the steady state mode to reduce the residual MSE. 2 The Chael ad Equalizer Models Assume that the trasmitted data {a } are a idepedet ad idetically distributed (i.i.d.) zero-mea sequece with idepedet real ad imagiary parts

3 A Novel Blid Equalizer Based o Dual-Mode MCMA ad DD Algorithm 73 derived from a quadrature amplitude modulatio (QAM) costellatio. Let the causal ad liear time-ivariat chael has coefficiets {h(),h(),,h(l )} with L the legth of the chael impulse respose, chael memory, or chael order. The, the received sigal at time idex is L x = h (k)a k + v, () k= where v is a i.i.d. additive white Gaussia oise (AWGN) ad deotes complex cojugate. To recover {a }, the received sigals {x } are passed through a equalizer modelled as a N-tap FIR filter with coefficiets {w(),w(),,w(n )}. The output is the y = W H X, (2) where X =[x,x,,x N+ ] T is the vector of the received sigals, W = [w(),w(),,w(n )] T is the vector of the equalizer tap weights (coefficiets), ad the superscript H deotes the complex cojugate traspose. 3 A Novel Scheme for Blid Equalizatio 3. Dual-Mode MCMA Let the error sigal of the dual-mode MCMA be e = γ e MCMA where γ ad β are adaptive parameters, ad + β e DD, (3) e MCMA = y,r (y 2,R R 2,R)+iy,I (y 2,I R 2,I), (4) e DD = y â (5) with i =. Here, y,r ad y,i are the real ad imagiary parts of the equalizer output y, respectively, ad the hard decisio output â of y is a estimate of a. The real quatities R 2,R ad R 2,I i (4) are obtaied as ad R 2,R = E[a4,R ] E[a 2,R ] (6) R 2,I = E[a4,I ] E[a 2,I ] (7)

4 74 S. Yoo et al. by settig the derivative of a o-covex cost fuctio with respect to the equalizer tap weights to be zero to miimize the cost fuctio. Usig a stochastic gradiet algorithm as the updatig rule, the vector W is adapted by W + = W µ e X, (8) where µ is the step size. The dual-mode MCMA is kow to have good performace i terms of the covergece rate ad residual MSE whe it uses a sigmoid fuctio i the costructio of the relatio betwee γ ad β. Note that the sigmoid fuctio () is the cumulative distributio fuctio (cdf) of the logistic pdf [], oe of the well-kow heavy-tailed pdf s. To illustrate simply ad clearly a drawback of the dual-mode MCMA, let us cosider the adaptive parameters ad β sigmoid γ sigmoid = emcma e DD of the dual-mode MCMA with the sigmoid fuctio = g( e DD ) (9) { g( e DD )} () g(x) =, a >. () +e a(x.5) From (3), (9), ad (), it is obvious that the compoet γ sigmoid e MCMA of the error sigal (3) will ot be zero i the steady state sice γ sigmoid is ot zero eve whe the chael is perfectly equalized. This results i a large output error level (relative to the case where e = β sigmoid e DD ) i the steady state after the equalizer has coverged completely. I additio, the parameter a i () has some restrictio o its rage due to a tradeoff relatioship that a large value of a icreases the covergece rate but results i a large error sigal i the steady state, ad vice versa. 3.2 The Proposed Algorithm for Blid Equalizatio To overcome the drawback of the dual-mode MCMA, we propose a method i which oly the DD algorithm operates i the steady state thereby improvig the residual MSE performace of the dual-mode MCMA i the steady state. The proposed equalizatio algorithm cosists of the dual-mode MCMA, DD algorithm, ad a covergece detector as show i Figure. The combiatio proposed i Figure basically attempts to utilize the advatages of the dualmode MCMA ad DD algorithm, thereby improvig both the covergece rate ad the residual MSE i the steady state.

5 A Novel Blid Equalizer Based o Dual-Mode MCMA ad DD Algorithm 75 a Chael x Equalizer W y â + - v Dual-Mode MCMA DD Algorithm Covergece Detector Fig.. A block diagram of the proposed system Blid Mode: At the begiig of the equalizatio process, the blid mode error e DD will ted to be large. Thus, based o the simplificatio +exp( x) exp( x) forx sufficietly large i (9) ad (), we propose to use ad γ prop = e α η (2) = emcma e DD β prop e α2 η, (3) where α ad α 2 are positive umbers, ad η = η η, (4) η =( ψ)η + ψ e DD 2, <ψ< (5) with η =. Whe the preset output error level (represeted approximately by η )hasa larger value with respect to the previous output error level (represeted approximately by η ), γ prop has a large value while β prop is small, ad vice versa. Sice we cosider oly the covergece rate but ot the residual MSE i the blid mode, we have more flexibility at the expese of some more complexity for the tuig of the parameters α, ψ, adα 2. Because of this advatage, we ca use larger values of both γ prop ad β prop i the blid mode. Cosequetly, the covergece is expected to be faster i the proposed method tha i the dual-mode MCMA usig (9) ad (). Steady State Mode: I our cotext, the steady state meas a state after a iitial covergece has bee attaied durig the traiig period with the dual mode MCMA. Oce the proposed equalizer begis to coverge, the dualmode MCMA is switched ito the DD algorithm by a covergece detector. The covergece rate after the shift from the dual-mode MCMA to the DD algorithm

6 76 S. Yoo et al. gets lower ad lower while the residual MSE gets smaller ad smaller. Sice oly the DD algorithm is employed i the steady state mode, the residual MSE is expected to be reduced more compared with the dual-mode MCMA. Let the error sigal of the proposed algorithm i the steady state mode be where the adaptive gai λ is give by e prop = λ (y â ), (6) λ = e MCMA. (7) Note that e prop = e MCMA e DD. Normally a larger λ results i a faster covergece at the cost of less residual MSE reductio, ad vice versa; the ubiquitous tradeoff betwee the covergece rate ad residual MSE reductio. We are to select λ cosiderig this tradeoff betwee the covergece rate ad residual MSE reductio. The choice (7) of λ essetially allows us to chage smoothly from the dual-mode MCMA to the DD algorithm ad has bee foud to cope effectively with variat SNR ad chaels i simulatios also. Covergece Detector: If the DD algorithm is triggered too early before the proposed scheme coverges, the covergece will be slow, ad if it starts too late after the covergece, the equalizer may coverge to a differet state. It is therefore highly importat to adequately determie the istat of the switchig to the DD algorithm. To derive a measure for the detectio of covergece, let us cosider Figure 2, where S t deotes a time iterval of legth C ad d t = e DD e DD C S t S t = C ( ) e DD C (t )+l C edd C (t 2)+l, t =2, 3,. (8) l= We assume d = ad the superscripts before ad after refer to before ad after the covergece, respectively. Sice the time iterval St before would be located o the steep slope ad St after o the covergece floor, we ca fid the (approximate) time istat of the covergece by computig the value of d t ad comparig it with a referece value. It is clear that d t is small also at the begiig of the blid mode where the output error level is large. If we judge the covergece solely o the basis of the value of d t, we might therefore ed up with a udesirable result. I [8], by otig that a ope eye coditio ca be expressed as e DD = y â < D 2, a detectio scheme edd < D 4 is proposed, where D is the miimum distace betwee the symbols i the costellatio. To detect the covergece more correctly by smoothig the effects of the fluctuatio of the errors { e DD }

7 A Novel Blid Equalizer Based o Dual-Mode MCMA ad DD Algorithm 77.9 before S t.8 S t before.7 Error level d t before after S t S t after.2 d t after Iteratio Fig. 2. A graph for the detectio algorithm ad cosequetly makig the detector less depedet o the problem, we propose to use the average E t = e DD (9) C S t of the output errors i S t i additio to d t i the decisio of the covergece. I summary, the steps of the detectio algorithm are as follows: i) Let t =2. ii) Check if d t <dad E t < D 4,whered is a positive costat. iii) If the results i ii) are both positive, the covergece detector switches dualmode MCMA ito DD algorithm at 5 t +. Otherwise, we repeat ii) ad iii) with t =3, 4,. The DD algorithm is kow [4], [8] to coverge surely whe a iitial covergece has already bee obtaied durig a traiig period, which occur for example whe the eye patter of the sigal is iitially ope. By usig (9) with the threshold guarateeig the eye to ope (i.e., D 4 < D 2 ) except for a severe distortio case, the covergece detectorcabeusedtoswitchfromthedual MCMA to DD algorithm. Step Size of the Proposed Algorithm: Usig the methods similar to the ormalized least mea square (NLMS) algorithm [], we ca derive the stability criterio for the proposed algorithm. The NLMS adjusts the step size µ such that the updated filter coefficiets would produce zero error with the curret data vector.

8 78 S. Yoo et al. First, let us adjust the step size µ i (8) such that the updated coefficiets would achieve the desired modulus whe applied to X. That is, we select µ MCMA such that From (8), we have W H + X = R 2,R + i R 2,I. (2) W+X H = W H X µ MCMA e MCMA X 2 (2) after some algebraic maipulatios. Substitutig (2) ad (2) ito (2) gives y,r + iy,i µ MCMA The solutio to the equatio (22) ca be show to be µ MCMA,R = e MCMA X 2 = R 2,R + i R 2,I. (22) y,r (y,r + R 2,R ) X 2 (23) ad µ MCMA,I = y,i (y,i + (24) R 2,I ) X 2 for the real ad imagiary parts of µ MCMA, respectively. Similarly, the step size for the DD algorithm is obtaied to be from ad µ DD = X 2 (25) W H +X =â (26) y,r + iy,i µ DD e DD X 2 =â. (27) Fially, let us derive the step size of the proposed algorithm usig the results (23) (25). Multiplyig both sides of (22) by γ prop µ DD ad both sides of (27) by β prop µ MCMA, ad the addig the results, we get y,r + iy,i = γprop µ DD γ prop µ MCMA + βprop µ DD e prop X 2 ( R 2,R + i R 2,I )+β prop γ prop µ DD + βprop µ MCMA µ MCMA â (28) after some algebraic maipulatios usig (3). Comparig (22), (27), ad (28), it looks reasoable at the first glace to choose the step size as µ = γ prop µ MCMA + βprop µ DD. (29)

9 A Novel Blid Equalizer Based o Dual-Mode MCMA ad DD Algorithm 79 Ufortuately, we have observed i prelimiary simulatios that the step size (29) is sometimes too large ad the algorithm may fail to coverge whe X 2 is too small. Oe simple solutio is clearly to use the modified step size µ prop = σ + γprop µ MCMA + βprop µ DD, (3) where σ is a small positive umber. The proposed algorithm coverges i the mea-square sese if the step size µ satisfies the coditio <µ<mi(µ prop ). (3) 4 Simulatio Results I all the simulatios herei, we have assumed that the clock of the received sigal is perfectly recovered ad a carrier phase offset does ot affect the equalizer. A simple 6-QAM costellatio has bee chose ad the sigal to oise ratio Imagiary Imagiary (a) Real (b) Real Imagiary Imagiary (c) Real (d) Real Fig. 3. The costellatios of the equalized 6-QAM sigals whe SNR=3dB. (a) CMA, (b) MCMA, (c) dual-mode MCMA, (d) proposed algorithm.

10 72 S. Yoo et al. 5 5 CMA MSE (db) 5 MCMA Dual mode MCMA 2 25 proposed Sample Idex x 4 Fig. 4. The MSE of various schemes whe SNR=4dB E[ a (SNR) at the iput of the equalizer is defied as SNR = log h() 2 ] σ i 2 db, where σ 2 is the variace of the AWGN. As a measure of the performace, the residual MSE defied as MSE = log E[ y â 2 ] [db] (32) is used. All the simulated residual MSE values are averaged over 2 trials with the step size µ =.8. The proposed algorithm has four parameters, α, α 2, ψ, add which eed to be set oly at the iitial stage. We have chose the value of the parameters as α = 2, α 2 =.5, ψ =.9, D /4=.5 add =.8 ad used these values i all the followig simulatios to make the residual MSE smaller tha db: the values of the parameters ca be chose appropriately i other chaels also. For a 22-tap chael impulse respose adopted from [9], Figure 3 shows the costellatios of the equalized outputs, ad Figures 4 ad 5 depict the MSE trajectories whe SNR=3dB ad SNR=4dB, respectively. The CMA exhibits a slow covergece while the MCMA coverges faster with less residual MSE tha the CMA. It is evidet that the dual-mode MCMA achieves some performace improvemet i terms of the covergece rate ad residual MSE compared with CMA ad MCMA. The proposed algorithm presets the best performace (the covergece rate ad residual MSE) irrespective of the SNR. It is clearly observed that the proposed algorithm achieves the best residual MSE, although the proposed algorithm sometimes exhibits a slightly slower covergece tha the dual-mode MCMA. This is due to the iheret characteristics of the DD algorithm which, after the covergece, reduces the residual MSE

11 A Novel Blid Equalizer Based o Dual-Mode MCMA ad DD Algorithm CMA MSE (db) 5 MCMA 2 Dual mode MCMA 25 Proposed Sample Idex x 4 Fig. 5. The MSE of various schemes whe SNR=5dB. more while coverges slowly with a very small adaptive gai whe the SNR is high. The proposed algorithm always has a lower residual MSE tha the dualmode MCMA ad the differece of covergece rate betwee the two algorithms i the blid mode is egligible. 5 Coclusio I this paper, we have proposed a ew blid equalizer allowig reduced residual error level ad faster covergece compared with the CMA, MCMA, ad dual-mode MCMA. The proposed algorithm makes use of both the good iitial covergece characteristic of the dual-mode MCMA ad the low residual error characteristic of the DD algorithm after covergece. The proposed algorithm, i exchage for a slightly higher complexity whe compared to other covetioal algorithms, offers improved equalizatio performace. Simulatio results have supported the performace advatages of the proposed algorithm i geeral. Refereces. D. N. Godard, Self-Recoverig Equalizatio ad Carrier Trackig i Two- Dimesioal Data Commuicatio Systems, IEEE Tras. Comm., vol. 28, pp , Nov J. R. Treichler ad B. G. Agee, A New Approach to Multipath Correctio of Costat Modulus Sigals, IEEE Tras. Acoust., Speech, Sigal Process., vol. 3, pp , Apr. 983.

12 722 S. Yoo et al. 3. V. Weerackody, S. A. Kassam, ad K. R. Laker, Covergece Aalysis of a Algorithm for Blid Equalizatio, IEEE Tras. Comm., vol. 39, pp , Jue C. B. Papadias ad D. T. M. Slock, Normalized Slidig-Widow Costat- Modulus ad Decisio-Directed Algorithms: A Lik betwee Blid Equalizatio ad Classical Adaptive Filterig, IEEE Tras. Sigal Process., vol. 45, pp , Ja J. C. Li, Blid Equalisatio Techique Based o a Improved Costat Modulus Adaptive Algorithm, IEE Proc. Comm., vol. 49, pp. 45-5, Feb J. Yag, J. -J. Werer, ad G. A. Dumot, The Multimodulus Blid Equalizatio ad Its Geeralized Algorithms, IEEE Jour. Selected Areas Comm., vol. 2, pp , Jue F. J. Ross ad D. P. Taylor, A Ehacemet to Blid Equalizatio Algorithms, IEEE Tras. Comm., vol. 39, pp , May O. Macchi ad E. Eweda, Covergece Aalysis of Self-Adaptive Equalizers, IEEE Tras. Iform. Theory, vol. 3, pp. 6-76, Mar S. Che, Low Complexity Cocurret Costat Modulus Algorithm ad Soft Decisio Directed Scheme for Blid Equalisatio, IEE Proc. Visio, Image, Sigal Process., vol. 5, pp , Oct I. Sog, J. Bae, ad S. Y. Kim, Advaced Theory of Sigal Detectio, Spriger- Verlag, 22.. M. H. Hayes, Statistical Digital Sigal Processig ad Modelig, Joh Wiley ad Sos, G. Picchi ad G. Prati, Blid Equalizatio ad Carrier Recovery Usig a Stopad-Go Decisio-Directed Algorithm, IEEE Tras. Comm., vol. 35, pp , Sep. 987.