Multimodulus Blind Equalization Algorithm for Cross QAM Signal Constellations

Transcription

1 Multimodulus Blind Equalization Algorithm for Cross QAM Signal Constellations Improved Jenq-Tay Yuan, Lin-Wei Chang, and Kun-Da Tsai Department of Electronic Engineering, Fu Jen Catholic University Taipei 5, Taiwan, P.R.C. vuani,ee.u.edu.tw Abstract-This work proposes a gain control (GC) compensation scheme that improves performance of a multimodulus blind equalization algorithm for cross quadrature amplitude modulation (QAM) signal constellations. The improved multimodulus algorithm (MMA) for cross QAM signal constellations may improve carrier-phase recovery, which in turn generates a lower symbol-error rate (SER) than original MMA for cross QAM signal constellations. I. equalizer output, respectively; RR and RI are given INTRODUCTION JMMA,IJ(n) EL(yR(n) R1) JM,MAQ(n) EL(YI (n)- R JMMAQ(n) in which sr(n), if lyi(n)l <85 EL(YI )] (n) -R) IYR(n)l <85 if, YR(n) if () >85 where is a constant that is a function of signal constellation under consideration. Notably, two different constants, R1 and R, are used in (). As an example, Fig. depicts cross constellation for a 3-QAM input and threshold 8 This constellation is obtained from a square constellation of 6 x 6 36 points, by removing one outer point in each corner; R1 and R are given by - can be used to recover transmitted data symbols, s(n), where h(n) c(n) * f(n) denotes impulse response of combined channel-equalizer system. The constant modulus algorithm (CMA) [1] for blind equalization is well known to require a separate carrier recovery system (for example, a phase-locked loop) for phase recovery, because CMA cost function is invariant under a phase rotation in constellation. Oh and Chin [6], and Yang, Werner and Dumont [8] proposed a modified CMA called multimodulus algorithm (MMA) to solve problem of arbitrary phase rotation inherent in CMA. Their modified algorithm has following cost function R Y(1 1±+3) ) 8. and R11 (1±+3±+5 ) 6 (y. respectively. As stated in [8], a single modulus can also be used, but doing so would increase probability of converging to so-called "wrong solution". However, as will be demonstrated herein, use of MMA cost function given by () for cross constellations may obstruct carrier-phase recovery because two different sets of statistics are used along each dimension, causing a high symbol-error rate (SER), unless a suitable gain control (GC) compensation scheme is provided. The use of two sets of statistics along each dimension in () causes tap-weight vector of MMA for cross JR (n) + JI (n) E{ [y(n) RR} + E{ [yi (n) -RI]}' () where yr(n) and y (n) are real and imaginary parts of /6/$.(6 IEEE. E{S (n) I, JMMA,I (n) EL(yR(n) -R)], >85 h(i)s(n i)+e(i)w (n i) JMMAL and R,I and si (n) denote real and imaginary parts of s(n), respectively. Yuan and Tsai [9] analyzed MMA for square quadrature amplitude modulation (QAM) signal constellations. Their analysis indicated that MMA alone may be able to remove ISI and simultaneously correct phase error, because it implicitly incorporates a phase-tracking loop, which automatically recovers carrier phase. Yang, Werner and Dumont [8] modified MMA in (1) to take advantage of statistics of symbols used in cross constellations when transmitted symbol statistics are of QAM cross constellations for which number of bits per symbol is odd. The in-phase (or real) and quadrature (or imaginary) cost functions for square constellations in (1) are modified as follows for cross constellations with two different sets of statistics along each dimension: Adaptive channel equalization without a training sequence is known as blind equalization [1] - [7]. The major advantage of such a technique is that no training sequence is required to start or restart system when communication unexpectedly breaks down. Figure 1 illustrates an equivalent baseband model with a channel impulse response of c(n). The channel input, additive white Gaussian noise, and equalizer input are denoted by s(n), w(n), and u(n), respectively. The data symbols transmitted, s(n), is assumed to consist of stationary independently and identically distributed (i.i.d.), real or complex non-gaussian random variables belonging to a finite alphabet A. The channel is possibly a nonminimum phase linear time-invariant filter. The equalizer input, u(n) s(n) * c(n) + w(n) is n sent to a tap-delay-line blind equalizer intended to equalize distortion caused by inter-symbol interference (ISI) without a training signal, where * denotes linear convolution. The of blind output equalizer y(n) u(n) * f(n) s(n) * h(n) + w(n) * f(n) - E{fs)(n)) by R,R 98

2 constellations to be updated according to values of real and imaginary parts of equalizer output, y(n), as follows; f (n + 1) f (n) - *VJMMA f (n) _ A *MMA f(n)- peu(n) u(n) (3) where e(n) er (n) + j * ei (n) in which II. er(n) YR(n) (YR (n) _Ri), and ej (n) y1(n) (y7(n) -R1), < 8 and YR(n)l < 8 er(n) YR(n) (YR (n) Ri), ande (n) y1 (n)* (yj(n) R), < 8 and YR(n)l > 8 er (n) yn(n)(yrn) R), and e (n) yi (n) (Y(n)-R1), > 8 and YR(n)l < 8 er(n) yr(n)(yrn) R), and ej (n) y1 (n)*(yj(n) -?), > 8 and YR(n)l > 8 LOCAL MINIMA OF MMA FOR QAM CROSS CONSTELLATIONS A. MMA Cost Functionfor QAM Cross Constellations Using unit step function U(n), defined as be removed. Accordingly, MMA for QAM cross constellations, like its square counterpart, may remove phase jitter without use of a separate carrier tracking loop. This situation is in contrast to that of CMA, whose cost function is insensitive to phase of equalizer output, such that CMA alone cannot achieve phase recovery, causing an additional carrier recovery system (such as a phase-locked loop) to be required for phase recovery. B. Local Minima ofmma for Cross Constellations The term, u(iy_(n)l -S5) + u(iyr(n)l -S5), takes three possible values, which are, 1, and. These three values result in following three MMA cost functions at each iteration, n, denoted respectively, by JMAt (n), JMMA 1 (n), and JMAA (n), according to values of real and imaginary parts of equalizer output, y(n): JMVAO(n) A -Rl.KZh(i) ±Rl JMMAl(n) A - (R + R)S JMA(n) A-R.QLh(i) +.R where A Re{E{s(n)} h(i)} h(i)+(r+r) (5) U(n) O<, MMA cost function for cross I,n>O constellations [which is a weighted sum of in-phase and quadrature cost functions shown in ()], can be expressed as, JMAL(n) rre E{s (n)}zh(i)} jxk55eh(i) +5 jh(i)j h(l) j) -R1 ok h(/))1r1] ±K(RI R)KoZh(i) +(R jrh) [U(IYy(n) -') + U( YR (n) 8)] where 7s E[ls(n), ks E[(n) a p(k) h(k) hr(k) + jh1(k) ejo(k) is kth position of yielding combined channel-equalizer impulse response ve,ctor E {S (n)}.r3(k) cos.(k) + 3. ksr3 (k) h [...,h(- 1),h(O),h(1),...] and ChR(k) + h;i (k) tan -1 h1 (k) As in [9], MMA for QAM c,ross hr (k) constellations is analyzed for a complex i.i.d., zero-rrnean source and a complex baseband channel, by excluding additive channel noise. The assumptions of each membe,r of symbol alphabet being equiprobable in source sequ ence and of equalizer being eir doubly-infinite in length o)r of finite-length fractionally spaced remain valid. Notably, first term in (), ReLE{s(n)}zh( which contains phase information of blind equallizer output, still exists in cross constellation case. As mentic)ned in [9], this phase information enables a possible phase errc r to ks h(i) + Zh(i) ) Notably, JMMA (n) is obtained when jy1(n) <8 and YR(n)j <, which corresponds to x 16 inner square constellation points depicted in Fig., whereas JMMA 1(n) is obtained when eir jy1(n) >8 and yr(n)j jyj(n) < or <8 and yr(n)j > which corresponds to 16 outer constellation points depicted in Fig.. JMMA (n) is obtained when Iy (n)l > 8 and YR (n)l > 8, which evidently represents wrong solutions since outer point in each corner does not exist. () The general form of all possible stationary points of MMA for cross constellations can be derived []. Without any loss of generality, only JM,,(n) in (5) is considered, such aj,vmajai(n) aj.,1(n) and that VJIM I(n) r I+ (k), + 66o7r(i) - (R1+ R)os (6) i#k 99 E{S (n)} r3 (k)[- sin (k)] (7) where r and are unit vectors in and (k) directions, respectively. Equation (7) is crucial as it restricts (k) to only eight different values. Equation (6) yields one result when O(k) E {,-, z, -} and anor when {T 3~ 5~ 7~ r ~, it has (k)e' ',. When O(k)E,, been shown in [] that R 7S E {s(n) }±+ 3ks s

3 (R+R )o7q,and Es(n)}+ 3k57s equalizer output y(n) yr(n) + jy1(n) is divided by.871 to yield compensated equalizer output + y'(n)y~(n YR (n) + + j1(n)[yr y (n) jy (n) (n) jy, (n)] wwhich is n o7q E{s(nY'}+3kscsi correspond to local minima of JmAj o(n), JMMA,l (n) and JMMAA(n), respectively, for steady-state mode operations (M.871 applied to update tap-weight vector of MMA: f(n+1) f(n)-u.e* (n).u(n) where e(n) er(n)+ j *ei(n) 1). III. GAIN CONTROL COMPENSATION PROPOSED in FOR (hl,r(k)) R 1.393, E{s(n)} ± 3ksUs.871 and E{s(n)}±+3ks u7 ' E{s(n) ±3k er(n) YR (n) (YR (n) -R) IV. COMPUTER SIMULATIONS The results of computer simulation are presented to compare performance of MMA in terms of SER for cross QAM constellations, obtained with incorporation of a GC with that obtained without. The GC is employed to change magnitude of equalizer output such that magnitude of combined channel-equalizer impulse response of all local minima equals unity at each iteration, after magnitude of ISI has been equalized. This in turn compensates for tap-weight vector update of MMA for cross constellations thus allowing dynamic phase-tracking capability to remain after magnitude has been equalized. A typical voice-band complex communication channel with transfer function.66, respectively. These results s reveal that magnitude of equalizer output will be distorted three cost by functions, Jm,m(n), Jmm1(n) and JA(n), once "eye" of signal constellation is open. This distortion may obstruct carrier-phase recovery unless an appropriate compensation technique is incorporated, because one unique feature of MMA is that once it starts functioning, its magnitude equalization and phase-tracking capabilities remain simultaneously effective. However, this carrier-phase recovery takes much longer than does magnitude equalization. The latter may be accomplished in fewer than iterations whereas former may take 5 iterations or even more. Consequently, MMA may still yield a high SER, because equalizer output still exhibits a large phase rotation even once magnitude equalization is accomplished ( ISI < -db ). The use of a GC for each of three cost functions is refore proposed to compensate for magnitude of equalizer output, following magnitude equalization, which in turn is used to update tap-weight > 8 and <, vector. For instance, when ly,(n)l which and e1(n) yi (n). (y1(n) -R1) However, this gain control operation should not be executed until magnitude equalization has been accomplished. The problem now is to determine when to apply this gain control operation during blind equalization period. The following scheme is proposed. The number of times three MMA cost functions [i.e., JMvMA (n), JvmMA I(n), andaj.,(n)] used are counted, from very beginning of blind equalization (iteration n ) until iteration n, according to absolute values of real and imaginary parts of equalizer output, y(n). The of cumulative numbers times JMMAo (n), JMMA I (n), andjmm (n) used, are No (n), N1(n) and N(n), respectively, from n to n. Initially, No () N1 () N () is set. Then, gain control n at iteration is not executed operation N(n) <.1, since as long as unless No (n) + N1 (n) + N(n) probability that equalizer output is wrong is less than or equal to.1, magnitude of ISI can be assumed to have been equalized such that M 1 (or in its steady-state mode of operation). However, at this point, carrier-phase recovery is still under way. Therefore, gain control compensation may improve performance of MMA in terms of phase recovery of equalizer output. This scheme will also be valid for or QAM cross constellations (such as 18-point signal constellation) or even for generalized MMA (GMMA), in which complex plane of in-phase and quadrature output samples of equalizer is divided into disjoint regions, all of which have ir own cost functions and moduli. In cross constellation case, such as for a 3-QAM input source, dispersion constants R1 and R are determined from 6 x 6 36 outer square constellation points and x 16 inner square constellation points, respectively, as displayed in Fig.. Hence, R1 and R are determined using different statistics of 3-QAM constellation whereas denominator of of all local minima, (E{s(n)} + 3kSS), is computed using overall statistics of 3-QAM constellation. Therefore, numerator and denominator of of all local minima of JMAO(n), J,, (n) and JMA(n) cannot be cancelled out when M 1, as y can in square constellation case [9]. Consequently, magnitudes of none of local minima are unity, which is considered to be perfect equalization when M 1. This situation is in contrast to case of square signal constellation, considered in [9], in which RR R I R (such that only a single set of statistics applies along each dimension) is computed using overall statistics of such that QAM constellation, ) + (hll (k) 1 at all four local h1(k) minima when M 1. For a 3-QAM input to which () applies, magnitudes of all of local minima for JMAg (n), JMMA I (n) and JMMA (n), can be computed as, C(z-1) [(-.5 -.j) + (.9 +.j)z-1 +(-.-.j)z + (.85+.5j)z +( j)z- + (.9.7j)z_5 +( j)z-6] adopted from [11] introducing around phase rotation is used in computer simulations. As presented in Fig. 1, transmitted data symbols s(n) are an independent, identically distributed 3-QAM sequence, and input to equalizer is YR(n)l

4 (with p -' ) when CFO is set to Af 5x-6. Simulation results reveal that gain control starts functioning when n 53 iterations with GC. Figure 7 shows that ensemble-averaged symbol-error rates in steady state are 37%, %, 6%, and 19% when 11, 1, 31 tap weights (without GC), and 11 tap weights (with GC) are used with CFO, Af 5 x -6, respectively. M -1 u(n) c(i)s(n - i)ejn) + w(n), where f/(n) zznaf is a carrier phase error in which Af is a carrier frequency offset (CFO). The real and imaginary parts of complex-valued additive white Gaussian noise w(n) are assumed to be independent and have equal variance, such that signal noise ratio (SNR) is db. The simulation experiments described herein employ a complex equalizer of a transversal filter structure with one of following three different number of tap weights (i.e., 11, 1, and 31 tap weights) with 5,, and 15 units of time delay, respectively. All of tap weights were initialized by setting central tap weight to one and ors to zero. The following performance indexes are used to evaluate performance. (1) Ensemble-averaged inter-symbol interference (ISI) (an indicator of performance of magnitude equalization defined as alone), r ISI (db) lolog k h(k)l -max(jh(k)l) max(h(k)) over V. CONCLUSIONS The MMA for cross QAM signal constellations proposed by Yang, Werner and Dumont [8] uses two different sets of statistics along each dimension. Therefore, magnitude of combined channel-equalizer impulse response of all of local minima is not equal to unity once magnitude has been equalized. The resulting distortion of equalizer output tends to disturb carrier-phase recovery and consequently, MMA yields a high SER, because a large phase rotation remains in equalizer output in steady state. Simulation results indicate that proposed GC can compensate for magnitude distortion of equalizer output and n improves its carrier-phase recovery, resulting in a lower SER than that obtained without a GC. independent runs; () ensemble-averaged rotation angle (an indicator of a possible phase rotation of equalizer output), defined as phase angle of h(k) with maximum modulus over independent runs; (3) averaged symbol-error rate (SER) or average probability of symbol error, over independent runs; () mean-squared error (MSE), generated by ensemble-averaging squared error versus number of iterations n over independent learning curves. Figures 3-6 demonstrate ISI, angel of rotation of equalized constellation, MSE, and traces of SER for MMA with and without gain control (with step-size parameters,,, set to -5, 8x-6, and 8x-6, respectively, for 11, 1, and 31 tap weights) without CFO with three different number of tap weights with or without GC. Figure 3 shows that longer equalizer length without GC produces larger ISI. When using 11 tap weights, although MMA without GC yields a little lower ISI in steady state than that with GC, MMA without GC tends to obstruct carrier-phase recovery, as stated above. Consequently, equalized constellation of MMA without GC has an obvious average rotation angle, even in steady state, that clearly affects SER as depicted in Figs. and 6. As mentioned previously, as more tap weights are used in equalizer without GC, better phase recovery can be achieved. However, MMA with 11 tap weights with proposed GC provides smallest rotation angle, and refore lowest SER and MSE than MMA without GC. According to Foschini [7], a blind equalizer is switched to decision-directed equalization mode when an SER is around.1. Notably, Figs. 3 and show that magnitude is almost equalized before 15 iterations (when ISI < -db ) (see Fig. 3), while phase recovery continues even after n 15 (see Fig. ). The simulation results herein indicates that gain control begins when n 17 iterations when 11 tap weights are used with GC. Figure shows that angels of rotation of equalized constellation in steady state are, 7,., and.5 when 11, 1, 31 tap weights (without GC), and 11 tap weights (with GC) are used, respectively. Figure 6 shows that ensemble-averaged symbol-error rates in steady state are 18%, 11%, 6%, and.6% when 11, 1, 31 tap weights (without GC), and 11 tap weights (with GC) are used, respectively. Figure 7 presents traces of SER for MMA with and without gain control ACKNOWLEDGEMENT This work was supported by National Science Council under contract NSC R.O.C. (NSC), Taiwan, 9-13-E--. ls(n) y(n)1 REFERENCES [1] D. N. Godard, "Self-recovering equalization and carrier tracking in two-dimensional data communication system." IEEE Trans. Commun., vol. COM-8, pp , Nov [] J. R. Treichler and M. G. Larimore, "New Processing Techniques Based on Constant Modulus Algorithm," IEEE Trans. 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5 Transmitter s(n) Channe (n) Equalizer y(n) D (n), 1 tap weight without GC 8, Fig. 1 The simplified baseband model Cross Constellation for 3-QAM T- Ti a uj LU U 1 1 tap 1 tap,31 tap, T -5 L Fig. 5 Ensemble-averaged MSE performance without CFO. T Real Part -1r I- 6 E Fig. Cross constellation for a 3-QAM input 31 tap 1 tap 11 tap 1 tap weight without GC fr- _ LU uj 11 tap 1 tap 31 tap 1 tap weight without GC 3-QAM, SNR db m r 18 - / / l O Fig. 6 Averaged symbol-error rate performance without CFO. - - L Fig. 3 Ensemble-averaged ISI without CFO tap weight without GC 35 5 a) ) 1 1 tap weight wthout GO 1 tap weight without GC 3-QAM, SNR db 7 6 ry 5 Lu U 3-QAM, SNR db 11 tap 1 tap 31 tap cu ~ (15- P 1 tap 31 tap 5 v, x Fig. Ensemble-averaged rotation angle of equalized constellation without CFO Fig. 7 Averaged symbol-error rate performance with CFO Af 5x-6