Advance in Electronic and Electric Engineering. ISSN 2231-1297, Volume 4, Number 6 (2014), pp. 587-592 Research India Publications http://www.ripublication.com/aeee.htm Performance Comparison of ZF, LMS and RLS Algorithms for Linear Adaptive Equalizer Prachi Sharma, Piush Gupta and Pradeep Kumar Singh Electronics & Communication, Jaypee University of Engineering & Technology, Guna, Madhya Pradesh, INDIA. Abstract For high speed data communication, extracting true data from the noisy transmitted data corrupted with inter-symbol interference (ISI), equalizers are a necessary component of the receiver architecture. Adaptive algorithms have been extensively used in communication signal processing. The recent digital communication systems need equalizers with a fast converging rate, that cannot be met by conventional adaptive filtering algorithms. ZF and LMS are widely used due to their simplicity and robustness, but fail to complete convergence criteria. RLS exhibit better performances, but is complex and unstable, and hence avoided for practical implementation. This paper analyses the performance of ZF, LMS and RLS algorithms for linear adaptive equalizer. Keywords: Adaptive algorithm, ZF, LMS, RLS, BER, ISI. 1. Introduction One of the most important advantages of the digital transmission systems is higher reliability in noise environment in comparison to their analog counterparts. But often the digital information, i.e. the transmitted pulses is smeared out so that pulses corresponding to different symbols are not separable, and this phenomenon is known as inter-symbol interference (ISI). [1]The need for equalizers arises from the fact that the channel has amplitude and phase dispersion which results in the interference of the transmitted signals with one another. So, in order to solve this problem equalizers are designed. Equalizer is meant to work in such a way that Bit Error Rate (BER) should be low and Signal-to-Noise Ratio (SNR) should be high. Equalizer gives the inverse of channel to the Received signal and combination of channel and equalizer gives a flat
588 Prachi Sharma et al frequency response and linear phase. The noise performance of static equalizer is not very good. Most of the times the transmission system s transfer functions are not known. Also, the channel s impulse response may vary with time. The result of this is difficulty in the equalizer designing. So, mostly preferred scheme is adaptive equalizers. An adaptive equalizer is an equalization filter that automatically adapts to time-varying properties of the communication channel. It is a filter that self-adjusts its transfer function according to an optimizing algorithm. Adaptive algorithms have been extensively used in the fields of biomedical, image processing, communication signal processing and many more. [2]Fig.1 shows the block diagram of an adaptive equalizer, where Random Noise Generator (1) provides the input signal & Random Noise Generator (2) provides additive white noise to corrupt the channel output. The adaptive equalizer performs the task of correcting the distortions produced by channel in presence of additive white noise. Fig. 1: Block diagram of Adaptive Equalizer. 2. Equalization Algorithms 2.1 Zero Forcing (ZF) The ZF-equalizer is designed using the peak-distortion criterion. Ideally eliminates all ISI. The names zero forcing corresponds to bringing down to inter-symbol interference to zero in a noise free case. This will be useful when ISI is significant compared to noise. Zero-forcing equalizers ignore the additive noise and may significantly amplify noise for channels with spectral nulls. The limitations of ZFE are: [3]1. Even though the channel impulse response has finite length, the impulse response of the equalizer needs to be infinitely long. [3]2. At some frequencies the received signal may be weak. To compensate the magnitude of the zero forcing equalizer grows very large. As a consequence any noise added after the channel gets boosted by a large factor and destroys the overall SNR. Furthermore the channel may have zeros in its frequency response that cannot be inverted at all.
Performance Comparison of ZF, LMS and RLS Algorithms for Linear Adaptive 589 The ZF Equalizer belongs to the class of preset linear equalizers and it uses the Peak Distortion Criterion to evaluate the equalizer tap weights. Consider the communication system block diagram (with an equalizer) given in figure 2. Fig. 2. Block Diagram of ZF equalizer H (F) = H (f). e, f W 0, f W (1) H (f) = H (f) (2) H (f). H (f) = H (f) = H (f) (3) H (f), must compensate for the channel distortion. H (f) = ( ) = ( ) e ( ), f W (4) Inverse channel filter completely eliminates ISI caused by the channel Zero Forcing equalizer, ZF. 2.2 Least Mean Square (LMS) LMS algorithms acts as a desired filter by finding the filter coefficients that relate to producing the least mean squares of the error signal (difference between the desired and the actual signal). It is a stochastic gradient descent method that uses the gradient vector of the filter tap weights to converge on the optimal Weiner solution. In Figure.1 x(n) is the vector of tap inputs, d(n) is the desired response, y(n) is the output and e(n) is the estimated error. [4]From the method of steepest descent, the weight vector equation is given by: w(n + 1) = w(n) + μ[ (E{e (n)})] (5) Where µ is the step-size parameter and controls the convergence characteristics of the LMS algorithm, e 2 (n) is the mean square error between the beam former output y(n) and the Reference signal which is given by, e (n) = [d (n) w x(n)] (6) In order to achieve a fast initial convergence speed and to retain a fast tracking ability in the steady state, large value for step size is chosen. On the other hand, large step size will result in large steady maladjustment error. 2.3 Recursive Least Square (RLS) [5]RLS is an algorithm which recursively finds the filter coefficients that minimizes a weighted linear least squares cost function relating to the input signals. This is in contrast to LMS that aims to reduce the mean square error. In the derivation of the RLS, the input signals are considered deterministic, while for the LMS they are
590 Prachi Sharma et al considered stochastic. Compared to others, the RLS exhibits extremely fast convergence. However, this benefit comes at the cost of high computational complexity. As illustrated in Fig.2 the RLS algorithm has the same procedures as LMS, except that it provides a tracking rate sufficient for fast fading channel, moreover RLS is known to have stability issues due to covariance update formula p(n) which is used for automatics adjustment in accordance with the estimation error as follows: p(0) = δ I (7), where p is the inverse correlation matrix and δ is regularization parameter, positive constant for high SNR and negative constant for low SNR. For each instant of time n=1, 2, 3..., the filter output is calculated using the filter tap weights from previous iteration and current input vector, π(n) = p(n 1)x(n) (8) The intermediate time varying gain vector k(n) is calculated using k(n) = ( ) ( ) ( ) Where Λ is the forgetting factor lying in the range 0< Λ < 1. Then priori estimation error is given by ξ(n) = d(n) w (n 1)x(n) (10) The filter tap weight vector is updated according to w(n) = w(n 1) + k(n)ξ (n) (11) (9) Fig. 3: Block diagram of RLS filter. 3. Simulation Results Computer simulations were conducted to analyze the performance of ZF, LMS, and RLS algorithm. A channel equalization model in the training mode was used as shown in Fig.1. The equalizer input signal was given by Eq.(12). For ZF algorithm, 3000 symbols were generated as the QPSK signal as input and then bit error rates for different tap filter was computed as shown in fig 4. For LMS, 3000 QPSK symbols were generated as input with step size (μ)=0.01, as shown in Fig.5. [6]For RMS, simulation of BPSK, QPSK and 8-PSK was performed and results were obtained as shown in Fig.6. x(n) = h(k)u(n k) + v(n) (12)
Performance Comparison of ZF, LMS and RLS Algorithms for Linear Adaptive 591 Bit error probability curve for QPSK in ISI with ZF equalizer 10-1 sim-3tap sim-5tap sim-7tap sim-9tap Theoretical Bit Error Rate 10-2 10-3 0 1 2 3 4 5 6 7 8 9 10 Eb/No, db Fig. 4: BER curve for ZF algorithm. Bit error probability curve for QPSK in ISI with LMS equalizer LMS BER 10-1 Bit Error Rate 10-2 10-3 0 1 2 3 4 5 6 7 8 9 10 Eb/No, db Fig. 5: BER curve for LMS algorithm.
592 Prachi Sharma et al Fig. 6: BER curve for RLS algorithm. 4. Conclusion A comparison on the adaptive filtering algorithms (namely ZF, LMS and RLS) has been carried out based on their BER. In ZFE, on increasing the taps of the equalizer the error decreases. It was showed that the BER of LMS equalizer was less than that of 9-tap ZFE, hence LMS performs better as compared to ZFE. [6]In RLS, the BER decreased by 50.9%, 50.25%, 45.5% for BPSK, QPSK and 8PSK respectively. Thus, RLS meets channel equalization by reducing channel effects; however, its performance can be further improved by using neural network equalization. References [1] S.Haykins, Analog and Digital Communications, Prentice Hall, 1996. [2] Mahmood Farhan Mosleh, Combination of LMS and RLS Adaptive Equalizer for Selective Fading Channel, European Journal of Scientific Research, Vol.43 No.1 (2010). [3] Jon Mark and Weihua Zhuang (2003). "Ch. 4". Wireless Communications and Networking. Prentice Hall. p. 139. [4] Romuald Rocher, Daniel Menard, Olivier Sentieys, Accuracy evaluation of fixed-point based LMS algorithm, Volume 20, Issue 3, May 2010, Pages 640 652 [5] Simon Haykin, Adaptive Filter Theory, Prentice Hall, 2002. [6] Abhinand.J, C.M. Sujatha, Simulation of Adaptive Channel Equalization for BPSK, QPSK and 8-PSK, International Conference on Electronics & Communication Engg., ISBN: 978-93-81693-56-8.