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PERFORMANCE OF AN ADAPTIVE BLIND EQUALIZER FOR QAM SIGNALS Antoinette Beasley and Arlene Cole-Rhodes Morgan State University Baltimore, MD 21251 Email: beasley@eng.morgan.edu, acrhodes@ieee.org ABSTRACT This paper presents a performance study of a blind adaptive channel equalization scheme. This scheme is based on a single cost function, which is created by combining two well-defined cost functions, the constant modulus algorithm (CMA) and the alphabet-matched algorithm (AMA). The combined cost function considers both the amplitude and the phase of the equalizer output, which allows for more efficient equalization of QAM signals. We present results that compare equalization of a 16-QAM signal using the new cost function with that obtained using the CMA cost function alone. The performance measures used for the comparison are the average mean-squared error (MSE) and the average symbol error rate (SER). INTRODUCTION We consider the problem of recovering an unknown signal that has been transmitted over an unknown frequency-selective channel using QAM modulation. The received symbol has been affected by intersymbol interference (ISI), which causes harmful distortions and presents a major difficulty in the recovery process. Signal detection at the receiver can be improved by combating this ISI, and we consider the use of an adaptive blind channel equalizer for this purpose. The constant modulus algorithm (CMA) is the most popular adaptive blind equalizer used today because of its relative simplicity and its good global convergence properties. It is very effective in equalizing signals that are of constant modulus. However, for non-constant modulus signals, CMA equalization can suffer from a considerable amount of residual ISI, and produce nonminimal convergence. . Prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. In [1], Barbarossa and Scaglione propose an alphabet-matched algorithm (AMA), which is able to provide better equalization of non-constant modulus signals, and they introduce a scheme that uses CMA to initialize the alphabet-matched algorithm (AMA). The AMA cost function tries to force the equalizer output to belong to the signal constellation of interest, and it is shown in [1] that this method is able to lower the residual ISI and improve the convergence rate over that of CMA alone. However, the AMA equalizer works well only when the initialization is good. An issue here is how to automatically switch from the initialization step using CMA, to the AMA. Using too few iterations of CMA will lead to ineffective equalization overall, while using too many iterations increases the runtime without improving performance. In this paper we present a new adaptive channel equalization scheme based on combining the two welldefined cost functions of the constant modulus algorithm (CMA) and the alphabet-matched algorithm (AMA) into a single cost function. We show through simulation that the scheme based on this combined cost function is able to provide improved results when compared to one using CMA alone. SYSTEM MODEL We consider a communications environment, which is coherent and synchronous with single-tone signaling. Carrier timing and waveform recovery have also been achieved. The received data block is of length N symbols and the equalizer attempts to remove the distortions introduced by the channel. Both the channel and equalizer are constrained to be linear, time-invariant FIR filters. The time-invariance assumption on the channel is reasonable because we consider short blocks of data. Noise in the channel is modeled as zero-mean additive white Gaussian noise (AWGN). Figure 1 provides a block diagram for the communications system. Page 1 of 1 Authorized licensed use limited to: Morgan State University. Downloaded on February 10,2023 at 23:55:11 UTC from IEEE Xplore. Restrictions apply. Figure 1: Channel and Equalizer The channel is a single-input single-output (SISO) system with an input/output relationship characterized by T −1 x(n) = ∑ h(k ) s (n − k ) + v(n), n = 0,..., N (1) k =0 where T is the number of channel zeros and {h(0) …. h(T-1)} is the channel impulse response, s(n) is the transmitted symbol sequence, and v(n) is AWGN noise in a data block of length N. We assume the symbol sequence in the block of data is independently and identically distributed (i.i.d.) and is non-Gaussian. The equalizer output, y(n) at time n is defined as y ( n) = L/2 ∑ w x(n − l ) l =− L / 2 l (2) where {wl : -L/2 ≤ l ≤ L/2} are the equalizer weights. BLIND EQUALIZATION ALGORITHM I. Background The scheme that we propose combines the conventional CMA with a constellation matched error (CME) term, similar to the modified CMA (MCMA) scheme presented by He et. al. in [2]. The authors show that for QAM signals, this new scheme improves the convergence rate of the adaptive equalizer and is more robust than the conventional CMA for low signal-tonoise ratios (SNRs). The cost function presented is a weighted sum of two individual cost functions, one is CMA and the other is a constellation matched error (CME) term. The authors state that the CME term should satisfy certain desirable properties namely uniformity, symmetry, and zero/maximum penalties at the zero/maximum distance from the QAM constellation points. Uniformity implies that the CME function does not favor any one data symbol in the alphabet over another, while symmetry implies that the function is symmetric around each constellation symbol. It is also desirable that the CME assigns the zero/maximum penalty at the zero/maximum distance from a constellation point. A zero penalty occurs when an equalized symbol lies directly on a constellation point and the maximum penalty occurs midway between two adjacent constellation points. In this work we propose the use of AMA, presented by Barbarossa and Scaglione [1] as the CME term. AMA satisfies both the uniformity and symmetry properties while satisfying a minimum/maximum penalty assignment at the zero/maximum deviation points. For both the CMA and the AMA, the cost function J (w) to be minimized is a function of the T difference between the equalizer output, y (n) = w x n , and the known constellation points. The cost function for CMA is given by ( ) 2 2 J CMA ( w) = E y (n) − R2 , (3) 4 E c(i ) , and c(i ), i = 1,..., M are the where R2 = 2 E c(i ) { { } } known constellation points, M is the number of constellation points. The notation E{⋅} denotes the expected value, taken over the entire block of equalizer output symbols. This cost function attempts to restore the shape of the constellation by taking into account the distance between the equalizer output symbols and the computed constant modulus of the known constellation symbols. CMA does not take the phase of the constellation points into consideration. CMA is well studied and is the most popular algorithm used for blind adaptive equalization. The AMA cost function, proposed by Babarossa and Scaglione [1] is given by J AMA ( w ) = E 1 − M ∑e i =1 − y ( n )−c (i ) 2σ 2 2 (4) The AMA cost function uses the same notation as above and a parameter, σ, is used to control the width of the nulls that the function places around each constellation point. Its value is chosen so that these nulls do not overlap for adjacent constellation points. The AMA cost function attempts to restore the shape of the constellation by considering the distance between the equalizer output symbol and each of the known constellation symbols. It assigns an appropriate penalty based on this distance. When an equalized symbol is sufficiently close to a constellation point the cost function is minimized. The opposite is true when an equalized symbol is far from the constellation points. AMA is known to depend on good initialization for optimization of the cost function. In our equalization scheme we combine the two cost functions described above into a single cost function similar to that proposed in [2], but modified as given below: Page 2 of 2 Authorized licensed use limited to: Morgan State University. Downloaded on February 10,2023 at 23:55:11 UTC from IEEE Xplore. Restrictions apply. J (w) = JCMA(w) + β J AMA(w) . (5) This combined cost function attempts to restore the constellation shape by taking into account both the amplitude and phase of the equalized symbol. The parameter β is a fixed scaling factor between the CMA and AMA terms. Making the substitution of the CMA and AMA expressions into equation (5), we obtain M − y(n)−c(i) 2 2 2 J (w) = E ( y(n) − R2 ) + β E1 − ∑e 2σ i=1 { } 2 (6) The update for the equalizer weights in this proposed scheme is based on the following stochastic block gradient descent rule: w n +1 = w n − µ ∇ w J ( w ) (7) where µ is the algorithm step-size. The equalizer output T is given as y (n) = w x n , for equalizer weight vector, w and a received data vector, x n , which is a function of time n. The gradient of the cost function is given in equation (10) as the linear combination of the gradients of the CMA and AMA cost functions shown in equations (8) and (9) respectively: ∇ w J CMA ( w) = E 4 ( 2 y ( n) − R2 ) y ( n) * x (8) 2 2 M 1 * − y( n)−c(i ) / 2σ ∇wJ AMA(w) = E∑ 2 xn ( y(n) −c(i))e i=1 σ (9) Note that the * operator denotes the complex conjugate. Thus the gradient expression for the combined scheme is: ∇wJ(w) = ∇wJCMA(w) + β ∇wJAMA(w) (10) and it should be noted that the averaged block gradient is computed at each iteration. The step-size, µ used in equation (7) is adaptive and is similar to that proposed in [3] T µ =α xn w n ∇ w J (w )x n T . (11) The cost function given in (5) combines the previously described CMA and AMA equalizers into a single equalizer. Note that if β is zero, we have the pure CMA equalizer and if β is very large, (5) is essentially controlled by the AMA equalizer. For a fixed β, it should also be noted that when the equalizer is far from convergence, the AMA cost function remains fairly flat, and the evolution of the equalizer is controlled by CMA. Once CMA has converged, the evolution can then be controlled by the AMA. Thus the combined cost function eliminates the need to monitor CMA for convergence, which was an issue discussed in [1]. Further, it provides some additional gains in the region where CMA has not yet converged, and the AMA initialization is poor. II. Experimental Setup The effectiveness of the combined scheme of equation (5) is critically dependent on the scaling parameter, β between the two terms of the cost function. Thus it is of importance to determine the optimal value of β. The goal of this section is to find an acceptable initial range of β values for simulation purposes, and we note that these initial values may be sub-optimal. The following experimental setup was used for testing purposes. Consider a channel with impulse response ho = [1, 0, 0.5]. This channel is considered to be a good communications channel because of its fairly flat magnitude response, and it was previously used for testing purposes in [3] and [5]. We transmit a data sequence of length, N=500 symbols using 16-QAM modulation and use an equalizer of length L=15. Figure 2 provides plots of the average MSE versus SNR (dB) for different values of the scaling parameter, β. The MSE is computed as the error between the composite channel impulse and the impulse, which corresponds to perfect equalization. For these tests each algorithm was run for 100 iterations, and then averaged over a total of 500 Monte Carlo runs. The dashed curve presents the results of using CMA equalization alone with phase recovery and it is shown for comparison purposes. Here, CMA does not equalize this QAM data sequence even at an SNR of 30 dB (see [5]). Figure 2 shows that the value of the scaling parameter, β is indeed a determining factor in algorithm performance. At lower SNR values, specifically 5 to 15 dB, equalization is not achieved but in the cases of higher SNR where it has been achieved, the β = 30 curve produced the minimal values of the average MSE. This β value is likely to be sub-optimal since it was selected to be the best value among a limited range of tested values. We note that the curves for β = 100 and Page 3 of 3 Authorized licensed use limited to: Morgan State University. Downloaded on February 10,2023 at 23:55:11 UTC from IEEE Xplore. Restrictions apply. l000 lie above CMA. This is consistent with the performance of AMA under poor initialization. Figure 2: Scaling factor curves for Channel ho It is expected that the optimizing value of β will change with SNR, so for future simulations a stochastic approximation (SA) technique will be used offline to automatically determine the optimal values of β, at different levels of SNR for a specific channel. An iterative stochastic gradient method will be used to adjust the parameter, β to minimize a specified mathematical loss function. In particular, we require that value of the scaling parameter β that minimizes the MSE at a fixed SNR. The SA algorithm allows us to update β iteratively as follows, while averaging over noisy measurements of the loss function: β k +1 = β k − a k gˆ k ( β k ) . previous section and results are shown for 100 iterations of each algorithm. The two different performance measures used for evaluation are (i) the average MSE taken between the composite channel (equalizer and channel) impulse and the impulse corresponding to perfect equalization, and (ii) the average SER taken between the channel input and equalizer output sequences. In this case the MSE and SER are estimated by averaging a total of 1500 Monte Carlo runs. Figure 3 compares three different schemes via the average MSE, while Figure 4 provides a comparison using the average SER. The CMA only curve is produced using 100 iterations of the CMA equalizer alone. The curve corresponding to the original CMA/AMA equalizer uses CMA to initialize AMA (see [1]). In this scheme, 100 iterations of CMA were run to initialize AMA, which is then run for 100 iterations for a total of 200 iterations. The combined scheme is run for only 100 iterations using β= 30. We note that the combined cost function equalizer is comparable in complexity to that of the original CMA/AMA equalizer. At lower SNR values, specifically 5 to 15 dB, equalization performance suffers in all cases. But at higher SNR values the combined scheme shows considerable improvement over the CMA only case, and comparable performance to the original CMA/AMA case. This is observed in Figure 3 with the average MSE. Using the average SER, the results of Figure 4 show that at higher SNR the combination of CMA and AMA provides a significant improvement over CMA alone and allows for more efficient equalization. (12) β k is the parameter estimate, a k is the algorithm step-size and gˆ k ( β k ) is the gradient approximation all at iteration k. It has been determined experimentally that the optimal value of β for this channel lies in the range of 10 to 40 for varying SNRs. PERFORMANCE EVALUATION RESULTS In this section we present results of the initial evaluation of our combined scheme (based on β=30) and compare it to the original CMA-initialized AMA scheme [1], and also to CMA alone after phase recovery. The experimental setup is identical to that described in the Figure 3:Average MSE vs. SNR dB for Channel ho Page 4 of 4 Authorized licensed use limited to: Morgan State University. Downloaded on February 10,2023 at 23:55:11 UTC from IEEE Xplore. Restrictions apply. ACKNOWLEDGEMENTS The authors would like to thank Brian Sadler and Ananthram Swami of the Army Research Laboratory for their many helpful suggestions. REFERENCES [1] S. Barbarossa, A. Scaglione, “Blind Equalization using Cost Functions Matched to the Signal Constellation”, Proc. 31st Asilomar Conf. Sig. Sys. Comp., Pacific Grove (CA), vol. 1, pp. 550-54, Nov. 1997. Figure 4: Average SER vs. SNR dB for Channel ho CONCLUSIONS AND FUTURE WORK In this work, we have presented initial results of a new equalization scheme that combines the conventional CMA cost function with the alphabetmatched algorithm (AMA) in a weighted sum. This combined cost function performs equalization by taking into account both the amplitude and phase of the equalized output symbol. It also takes advantage of the global convergence properties of CMA, and combines this with the improved convergence rate of AMA near its optimum. We have compared the combined scheme with a CMA only scheme and the original CMA/AMA equalizer of [1] by measuring the MSE and the SER. The combined cost function approach has been shown to provide more efficient equalization of QAM signals than can be achieved using CMA alone and comparable performance to the original CMA/AMA equalizer. The initial results presented here show that this equalizer provides significant improvements over CMA alone for QAM signals, while adding minimal complexity. The value of the scaling factor β used for performance evaluation was most likely sub-optimal, and may require further optimization. But we are already able to observe a significant improvement in performance using this combined cost function scheme. In future work we will focus on fully optimizing the β parameter for improved performance of the combined single-cost function scheme. We will use a stochastic approximation algorithm to determine offline its optimal values for different channels at different SNR values, and further test and evaluate the equalizer presented here on different channels to determine its effectiveness and robustness for QAM constellations. [2] Lin He, M. G. Amin, C. Reed, Jr., R. C. Malkemes, “A Hybrid Adaptive Blind Equalization Algorithm for QAM Signals in Wireless Communications”, IEEE Trans. on Signal Proc., Vol. 52, No. 7, pp. 2058-2069, July 2004. [3] A. Swami, S. Barbarossa, B. M. Sadler, and G. Spadafora, “Classification of digital constellations under unknown multipath propagation conditions”, Proc. SPIE, Digital Wireless Comm. II, Orlando, FL, April 2000. [4] C.R. Johnson, Jr. et al, “Blind Equalization Using the Constant Modulus Criterion: A Review”, Proc. IEEE vol. 86, no. 10, pp. 1927-50, Feb 1995. [5] A. Beasley, A. Cole-Rhodes, B. Sadler, A. Swami, “Adaptive Blind Equalization using an AlphabetMatched Algorithm”, Proceedings of the Collaborative Technology Alliances Conference – Communications & Networks, pp. 429-433, April 2003. [6] J. C. Spall. Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. Hoboken, NJ, Wiley 2003. Page 5 of 5 Authorized licensed use limited to: Morgan State University. 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