Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers 2 The Wiener or Minimum Mean Square Error (MMSE) solution is the most often used, but it is e

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1 Quasi-Minimum-BER Linear Combiner Equalizers Jose Tellado and John M. Cio Information Systems Laboratory, Durand 112, Stanford University, Stanford, CA Phone: (415) Fax: (415) Technical Area: Communication Theory or Signal Processing Abstract It is ell knon that Minimum Mean Square Error (MMSE) and Minimum Peak Distortion (MPD) criteria do not Minimize Bit Error Rate (MinBER), and for some channels they can have a large degradation compared to the MinBER solution. Based on the strenghts and eaknesses of the MMSE and MPD criteria, e have dened to ne solutions, hich e call Minimum Noise Limited Peak Distortion (MNLPD) and Min-. The rst solution is based on MPD but avoids the problem of noise enhancement. The second solution nds the optimal of a family of equalizers derived from the MMSE and MPD criteria. We have shon by numerical calculation that by using the MNLPD or the Min- criteria e can achieve almost MinBER performance. The proposed methods have larger complexity, but can be solved using ell knon techniques since they can be described as simple convex optimization problems. 1 Introduction Linear equalizers (LE) and Decision Feedback Equalizers (DFE) are to commonly used techniques to combat intersymbol interference (ISI) distortion. The equalizers considered in this ork use only linear combinations of the channel observations and the past decisions (DFE only). There are other structures that use nonlinear combinations but are more complex.

2 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers 2 The Wiener or Minimum Mean Square Error (MMSE) solution is the most often used, but it is ell knon that it does not correspond to the Minimum Bit Error Rate (MinBER) solution [1], hich is our most common performance goal. Besides the MMSE solution, there is the Minimum Peak Distortion (MPD) criterion that as rst described in [2], hich is not optimal in MinBER sense either, but can outperform the MMSE solution in some cases. Since neither the MMSE solution nor the MPD solution achieve the MinBER, there is still room to reduce the BER by using a linear-combiner LE or DFE structure. There has recently been several other approaches to this problem (e.g. [3],[4]), but all methods increase performance at the expense of extra computation or longer convergence time. The proposed solution is a superset of MMSE and MPD linear combiners and attempts to use the advantages of both methods to achive MinBER. This paper is organized as follos. Section 2 describes the problem and introduces the notation. Section 3 briey describes the most commonly used minimization criteria, namely, BER, MSE and PD, and introduces a ne criterion, hich e call Noise Limited Peak Distortion (NLPD). Section 4 describes a ne family of equalizer coecients based on the above criteria. Finally, Section 5 discusses simulation results to sho the utility of this ne family of equalizers. 2 Problem Formulation pk y k k z k x^ k-d n k b k Channel Receiver Figure 1: Baseband model for digital transmission system

3 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers 3 We consider a standard model of a linear Pulse Amplitude Modulated (PAM) system depicted in Fig. 1. The inputs are independent and equiprobable binary (1) random data. This study could be extended to the more general complex-valued and multilevel cases but the computations ould be more complex. The system has an overall nite impulse response p k of length + 1 and additive zero-mean hite Gaussian noise n k, hich is independent ith and is of poer spectral density 2. The channel output y k is given by: y k = X m=0 p m?m + n k : (1) The channel is folloed by a forard equalizer of lenght N, given by the vector = [ 0 : : : N?1] T and a feedback equalizer b = [b 1 : : : b M ] T of length M. The case M = 0 corresponds to the Linear Equalizer (LE), otherise e have a Decision Feedback Equalizer (DFE). To simplify the discussion, e ill only describe the M = 0 case, although e ill sho numerical results for both cases. If e rite the input to the equalizer in vector form yk = [y k : : : y k?n +1] T, the equalizer output can be expressed as the inner product T yk. From Eq. 1, the vector yk can be ritten in matrix form as: y k. y k?n = p 0 p 1 : : : p 0 0 : : : 0 0 p 0 p 1 : : : p 0 : : : : : : 0 p 0 p 1 : : : p 1 p ?N? n k. n k?n (2) hich can be ritten more compactly as yk = P xk + nk (3) Care should be taken not to mistake vectors (e.g. xk) ith their components (e.g. ). The M 0 case follos if e dene a matrix P ~ = [P; 0 I 0] and vectors ~yk = [y; xk?d] and ~ = [; b] hich generalize P, yk and in the folloing equations to include the feedback terms from b. For details, see e.g. [7]. In general, for the equalizer to perform optimally, it ill not be possible to get an estimate of (represented as ^ ) but rather a delayed version ^?D here D accounts for the delay of the cascade of the channel p and the equalizer. The probability of error P err, for this model is given by

4 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers 4 P err = P r f?d 6= ^?Dg = P r f T Pxk + T nk < 0j?D = 1g (4) P err = E " Q!#?D T Pxk X! = Q?D T Pxk kk kk (5) There are several criteria for optimizing the lter coecients f k g. Since the most meaningful measure of performance for a digital communication system is the average probability of error, e should choose the coecients to minimize this criteria. Unfortunately, the expression for the average probablity of error is a highly nonlinear function of the lter coecients. Thus more practical criteria have been used ith the most idespread being the peak distortion criterion and the mean square error criterion. The next section ill summarize the most important features of currently used equalizers and introduce our ne design criterion. 3 Equalizer Coecient Design 3.1 Minimum Bit Error Rate Criterion The Minimum Bit Error Rate (MinBER) criterion can be simply stated as X? MinBER = arg min Q?D T Pxk kk! (6) Since this function is nonlinear, there can be several minima and therefore, there is no simple solution to this problem in general. Hoever, for a certain class of channels, [4] states that a gradient based algorithm such as: k+1 = k? r k P r f?d 6= ^?Dg (7) ill converge to the MinBER. For convergence of a more general class of channels, [4] proposes a suboptimal algorithm based on the same criterion. A problem ith these gradient based algorithms is that although they are globably convergent for some cases, the rate of convergence is hard to evaluate.

5 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers Mean Square Error Criterion Mimimun Mean Square Error (MMSE) Equalizers choose the tap eight coecients to minimize the mean square value of?d? z k. The problem formulation is therefore h i? MSE = arg min E j?d? T Pxkj 2 (8) The solution is closed form and ell documented in the literature. For more details see e.g. [1],[7]. 3.3 Peak Distortion Criterion The Peak Distortion (PD) is dened as the orst case intersymbol interference at the output of the equalizer. If e call pj the j-th column vector of P, then the equalized value can be expresed as z k = X j6=d( T pj)?j + ( T pd)?d (9) here the rst term in the sumation is the interference and the second term is the desired value. The orst case interference occurs hen?j = signf T pjg, resulting in the value of the peak distortion given by P D() = X j6=d j T pjj (10) Therefore the problem formulation is min X j6=d j T pjj; (11) subject to : T pd = 1 (12) here the constraint is introduced to ensure e obtain, of the lters that produce the same desired signal poer, the lter ith the minimum distortion. The above problem is convex as as rst noticed by [2]. The draback ith the above solution is that it ignores the noise, hich can suer the problem called noise enhancement. Noting that minimizing distortion is equivalent to maximizing the eye diagram opening leads to the folloing equivalent formulation of the problem in Eq :

6 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers 6 max min ;?D=1 T Pxk (13) subject to : T pd = 1 (14) With a simple change of variable [6], e can rerite Eq as: max t (15) subject to : T Pxk t; 8xk;?D = 1 (16) T pd = 1 (17) For a to-level transmission, the constrant in Eq. 17 is only a normalizing constant and could be set to any other positive value ithout changing the probablity of error. On the other hand, if e change that constraint to a norm constraint on the lter coecients, e obtain the folloing convex optimization problem: max t (18) subject to : T Pxk t; 8xk;?D = 1 (19) T 1 (20) Intuitively, this optimization problem nds the equalizer lter coecients that maximize the eye diagram opening, ithout amplifying the noise poer at the output of the equalizer given by 2 kk 2. We ill call this criterion the Noise Limited Peak Distortion (NLPD) criterion and denote the optimal solution as NLP?. The D performance of the NLPD solution is discussed in Section 5, and is shon to be far more superior to the PD solution, especially for DFE. 4 Weighted Combination of MSE and PD Criteria The solution to Eq. 8,? MMSE, minimizes the average total distortion (ISI plus noise) at the output of the equalizer. On the other hand, the solution to Eqs ,? MP D, minimizes the orst case distortion of the ISI alone. To natural questions arise as to hich solution performs better for our desired criterion, BER, and ho far these solutions are from the MinBER. As ill be shon in Section 5, depending on the channel,? MMSE or

7 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers 7? MP D could be the better choice. The reason for this can be seen by looking more closely at the P err expression in Eq. 5. Since the Q(x) function decreases very rapidly as its argument x increases, it turns out that the P err is usually dominated by a fe terms in the summation. The PD criterion minimizes the biggest term in the sum, but does not try to decrease the other terms. On the other hand, the MSE criterion minimizes the set, but its quadratic cost function does not penalize enough some values that ill be very signicant in the P err computation. No, consider the family of solutions =? MSE + (1? )? P D; 0 1 (21) This superset of equalizer coecients includes the? MSE and? P D as special cases. Moreover, it includes intermediate solutions that have a loer PD than the MSE solution, and hich exhibit better behavior than the PD solution for the non-orst-case-terms. Calling? the solution that minimizes BER over, the folloing relationships hold: X min Q?D T Pxk kk! X Q min " X xk! x? k?dt P k? k Q!?D MSE? T Pxk X!# k MSE? k ; Q?D PD? T Pxk x k? k PD k (22) (23) Since? ill outperform both the MMSE and MPD solutions, the remaining question is ho far is it from the MinBER solution. For all the cases tested, the dierence is negligible. The folloing section shos some numerical values. 5 Numerical Results The highly nonlinear nature of Eqs makes it extremly hard to compute bounds that are suciently tight to be useful. To demonstrate the near optimal behaviour of? in the MinBER sense, e ill no present our simulation results. All BER quantities are calculated from the exact formula for P err based on Eq. 5.

8 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers MinBERplot=, Min_alpha=*, MPDplot=x, MMSEplot=o Perr SNRmfb, p=[1.2,1.1, 0.2], LE(3), Delay=2 Figure 2: 5.1 Linear Equalization First, consider the channel used in [4] that is given by the vector p T = [1:2 1:1? 0:2]. In Fig. 2, e plot BER vs. Signal to Noise Ratio-Matched Filter Bound, SNR MF B = kpk 2 = 2. The delay chosen is D = 2 hich minimizes the MMSE. For this case, the MPD solution (x's) outperforms the MMSE solution (circles) at every noise value shon. Hoever, both solutions fall short of the MinBER (dashed). For example, at BER = 10?5, MinBER has a.5 db gain over MPD and over 6 db gain ith respect to MMSE. On the other hand, the Min- solution (*'s) is almost superimposed on the MinBER solution. Fig. 3 shos the eect of varying for three dierent SNR MF B. The cases = 0 and = 1 correspond to the MPD and MMSE solutions. As e can see, the best solution is somehere in beteen the MMSE and MPD solutions. For example, at SNR MF B = 30dB the best? = 0:3 As e increase the number of lter coeecients, the dierence beteen the MMSE solution and the MinBER solution decreases [4], as can be seen in Fig. 4, here the number of taps as increased to 5. This example supports the fact that MMSE solutions or MPD solutions can be the better choice depending on the channel parameters. For this example, at lo SNR MF B, the MMSE is better, but at high SNR MF B here the orst case distortion dominates the BER, MPD outperforms MMSE. In any case, the Min- solution is again indistinguishable from

9 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers SNR_mfb=30dB Perr 10 4 SNR_mfb=32.5dB 10 5 SNR_mfb=35dB alpha Figure 3: the MinBER solution. 5.2 Decision Feedback Equalization We ill no consider the DFE case, this time ith a channel of length 4, p T = [:35 :8 1 0:8]. In Fig. 5 e plot the BER for a DFE ith 2 feedforard taps and 3 feedback taps. The plots assume error free feedback path. As before, the Min- solution (*'s) is superimposed on the MinBER solution, outperforming the MMSE and MPD solutions by 2dB and 3dB respectively at BER = 10?6. In Fig. 5, the performance of the MNLPD solution (o's) is included and is shon to also overlap ith the MinBER solution. For all other examples tested, the Min- solution lies very close to the MinBER solution, but this is not alays the case for the MNLPD solution. 6 Conclusions For some channels designing LE and DFE using MSE or PD can lead to large degradation compared to the MinBER solution. Studying the exact BER formula for FIR channels, e have briey discussed the pros and cons of the MMSE and MPD criteria. We have shon several gures here the degradation ith respect to MinBER

10 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers MinBERplot=, Min_alpha=*, MPDplot=x, MMSEplot=o Perr SNRmfb, p=[1.2,1.1, 0.2], LE(5), Delay=4 Figure 4: can be over 6dB at the BER of interest. Based on the strenghts and eaknesses of the MMSE and MPD criteria, e have dened to ne solutions, hich e call Minimum Noise Limited Peak Distortion (MNLPD) and Min-. The rst solution is based on MPD but avoids the problem of noise enhancement. The second solution nds the optimal of a family of equalizers derived from the MMSE and MPD criteria. We have shon by numerical calculation that by using the MNLPD or the Min- criteria, e can achieve almost MinBER performance for cases here the MMSE suers degradations of over 6dB. The proposed methods have larger complexity, but can be solved using ell knon techniques since they can be described as simple convex optimization problems. References [1] J.G. Proakis. Digital Communications, McGra Hill, [2] R. W. Lucky. \Automatic Equalization for Digital Communications," Bell Syst. Tech. J.,, vol. 44 pp. 547{588, April 1965.

11 Tellado, J: Quasi-Minimum-BER Linear Combiner Equalizers MinBERplot=, Min_alpha=*, MPDplot=x, MNLPDplot=o, MMSEplot= Perr SNRmfb, p=[0.35,0.8,1,0.8], DFE(2,3), Delay=1 Figure 5: [3] S. Chen, E. Chng, B. Mulgre and G. Gibson, \Minimum-BER Linear-Combiner DFE," ICC'96, pp. 1173{1177, [4] Chen-Chu Yeh and J. Barry. \Approximate Minimum Bit-Error Rate Equalization for Binary Signaling," ICC'97, pp. 1095{1099, [5] R. W. Lucky, J. Salz and E.J. Weldon Jr. Principles of Data Communication, McGra-Hill, [6] S. Boyd and L. Vandenberghe. Convex Optimization, Lecture Notes for Introduction to Convex Optimization ith Engineering Applications, Stanford University, [7] J.M. Cio. Digital Data Transmission. Manuscript in preparation.