A New Time-Varying Convergence Parameter for the LMS Adaptive Filtering Algorithm

A New Time-Varying Convergence Parameter for the LMS Adaptive Filtering Algorithm

Siddappaji Dr. K L Sudha

Associate Professor, ECE Professor, ECE

BMS College of Engg, Bangalore-19 DSCE, Bangalore-78

Abstract – A number of variable step size LMS algorithms are used to improve the performance of conventional LMS algorithm. This paper proposes a new Time varying LMS (NTVLMS) algorithm by making use of a nonlinear function to get the variable step size parameter µ. The proposed algorithm is applied to adaptive noise cancellation system.The simulation results show that the performance is better compared to conventional LMS and TVLMS algorithms.

Keywords – Adaptive filter, Convergence rate, LMS algorithm, New Time- varying LMS algorithm (NTVLMS), Time- varying LMS algorithm (TVLMS), Steady state misadjustment.

1. Introduction

   The least-mean-square (LMS) algorithm was developed by Widrow[1][2]. It is still used in adaptive signal processing for its simplicity, less computation, ease of implementation and good convergence property. The LMS algorithm is described by the equation:

   e(n) = d(n) XT(n)W(n) —————–(1)

   w(n+1) = w(n) + 2µ(n)e(n)X(n)———(2)

   Where w(k) is filter coefficient at time k, µ(k) is step size, e(k) is adaptation error and x(k) are the filter input respectively. Equation (1) shows that LMS algorithm uses an adaptation error and a variable step size to update the filter coefficients. In this algorithm, the step size parameter µ(n)is constant. The choice of this parameter µ is very important to the convergence and stability. As in [2], the stability of the convergence of LMS algorithm requires the step size parameter µ satisfy the condition

   gives faster convergence but a large SSM so, suitably select µ based on eqn(3). But it is still far from optimum trade-off between SSM and convergence rate. Then TVLMS algorithm is proposed to solve this problem. The TVLMS algorithm uses a suitable step size in the initial stage to speed up the convergence and the step size is adjusted to a smaller value gradually during the convergence.

   In section 2, the TVLMS is discussed and their performance is introduced. In section 3, the new TVLMS algorithm is proposed. In section 4, the proposed algorithm is applied to noise cancellation system and simulation results are presented. Finally draws the conclusion.

2. TVLMS algorithm

   Many variable step size LMS algorithms are used to achieve the optimum convergence rate and SSM performance. In TVLMS algorithm [4]-

   [5] the step size parameter is found out by using the equation as

   µ(n) = (n) × µo ————————-(4)

   whereµois the value of step size parameter in conventional LMS algorithm. This µois used to update µ(n) in this algorithm.

   (n) is a decaying factor. We will consider the following decaying law:

   (n) = 1/(1+ )———————-(5)

   Where C, a, b are positive constants that will determine the magnitude and the rate of decrease for(n). According to the above law, C has to be a positive number larger than 1. When C = 1, (n) will be equal to 1 and the TVLMS algorithm will be the same as that of conventional LMS algorithm. As (n) decreases with respect to time, the

   0 µ

   ()

   .———–(3)

   convergence parameter µ(n) decreases and rate of convergence increases compared to LMS

   Where tr(R) is the trace of the autocorrelation matrix of input x and max is the maximum eigenvalue of R.

   In general a smaller step size leads to a small steady state misadjustment (SSM) but a slower convergence rate. While a larger step size

   algorithm, but still convergence rate is high and the computational complexity of this algorithm is large.

3. New TV LMS algorithm

   Based on the performance of LMS and TVLMS algorithm, a New TVLMS algorithm is proposed. This algorithm can improve the convergence rate with respect to LMS and TVLMS algorithms and computational complexity is largecompared to LMS algorithm and less compared to TVLMS algorithm. The proposed algorithm updates the step size by

   µ(n) = d + µo tan-1 ( f n +g ) ————-(6)

   whered, f and g are positive constants select suitably.µois the value of step size parameter in conventional LMS algorithm. This µois used to update µ(n) in this algorithm. The proposed algorithm gives good result for additive white Gaussian noise with variable input power as well as variable noise power.

4. Application to the adaptive noise cancellation system

   The basic adaptive noise cancelling system is shown in fig (1) below

   fig(1) Block diagram of adaptive noise cancellation system

   A signal is transmitted over a channel is received by the receiver with uncorrelated noise x0(n). The signal s(n) and noise x0(n) combined to form the desired signal d(n) = s(n) + x0(n). A second signal input to the adaptive filter is noise x(n), which is uncorrelated with the signal but correlated in some unknown way with noise x0(n). The noise x(n) applied as an input to the adaptive filter to produce an output y(n) is close enough to the replica of x0(n).

   In this system the signal x(n) is processed by

   the filter that automatically adjust its weights through the above mentioned algorithms with respect to the error signal e(n).

   i=0

   i=0

   y(n)= N1 w i x(n i)= WT(n)X(n)——-(7)

   X(n) = [x(n), x(n-1), ——– x(n-N+1)]T

   W(n) = [w0(n), w1(n), ——– wN-1(n)]T

   Where N is the order of the filter, wiis the filter coefficient.

   To implement LMS based adaptive noise cancellation system, it is very important to choose the following parameters,filter order, initial values of the filter coefficients, and value of the step size parameter, etc.

5. Simulation Results

   The simulation results of the three algorithms are given as follows.

   The condition of simulation is:

   1. The order of the adaptive filter is 2

   2. The input signal is continuous wave of sin signal.

   3. The noise is a white Gaussian noise

   4. The amplitude of the sin signal is 0.01 and 0.2, corresponding to SNR -43 dB and -17dB.

   5. The algorithm was implemented for 200iterations and 2000 iterations for each case.

      1. Dynamic range comparison with LMS and TVLMS algorithm

         From fig (2), the proposed algorithm has faster convergence rate in case of both high and low signal-to-noise ratio (SNR).

         Signal amplitude of 0.2

         Here,

         Signal amplitude of 0.01

         Fig(2) The variable step size curve

      2. Performance comparison with LMS and TVLMS algorithm

         From fig (3) and fig (4),

         1. The proposed algorithm has faster convergence rate

         2. The steady error of the proposed algorithm is less compared to LMS and TVLMS algorithms in case of both low signal-to- noise ratios (SNR) and high signal-to- noise ratios (SNR).

      Output waveform

      Fig(3) sine signal of amplitude 0.2

      Fig(4) sine signal of amplitude 0.01

6. Conclusions

The proposed New TVLMS algorithmshows improved performance over the conventional LMS and TVLMS algorithms under white Gaussian noise environment. This algorithm is optimum for specified values of the constant parameters d, f, g and µo. The concept of this paper is useful to the implementation of Radar communication system for the cancellation of jamming signal, interference or noise cancellation in case of Digial communication and satellite communication system and it is useful to other automation fields.

References

1. Widrow B, et al. Stationary and nonstationary learning characteristics of the LMS adaptive filter. Proc.IEEE, 64:1151-1162,1976.

2. Widrow B, Stearn S.D. Adaptive Signal processing.New York:Prentice-Hall,1985.

3. S. Haykin, Adaptive Filter Theory,Third Edition, New York: Prentice-Hall,2002.

4. YS. Lau, Z. M. Hussain, and R. Harris, A time- varying convergence parameter for the LMS algorithm in the presence of white Gaussian noise, Submitted to the Australian Telecommunications, Networks and Applications Conference (ATNAC), Melbourne, 2003.

5. Yuu-Seng Lau+, Zahir M. Hussian++ and Richard Harris, Performance of Adaptive Filtering Algorithms: A Comparative Study Centre for Advanced Technology in Telecommunications

   (CATT), RMIT University, Melbourne, Victoria, Australia.

6. Hongbing Li and HailinTian A new VSS-LMS Adaptive Filtering Algorithm and its Application in Adaptive Noise Jamming Cancellation system, 2009 IEEE.

7. SanjeevDhull ,SandeepArya& O.P. Sahu, Performance Evaluation of Adaptive Filters Structures for Acoustic Echo Cancellation, International Journal of Engineering (IJE), Volume (5) : Issue (2), pp. 208 – 216 : 2011