Improved hybrid variable and fixed step size least mean square adaptive filter algorithm with application to time varying system identification

In this paper a new simplified adaptive filter algorithm is introduced which is based on the hybrid operation of variable step-size and fixed step-size least mean square adaptive algorithm. In this proposed algorithm the variable step-size is used in the first stage, the algorithm adopts the fixed step size least mean square (LMS) whenever an acceptable mean square error threshold is reached that ensures the required steady state error and stability. The simulation results obtained show that the new algorithm outperforms the standard least mean square (LMS) in the desired transient-response, and outperforms the normalized least mean square (NLMS) algorithm in the desired transient and the steady-state response. It is shown that this new algorithm is able to track time-varying systems with better performance response. Also, the computational-complexity for this algorithm is reduced as compared with the ordinary least mean square (LMS).

1 Introduction

It is well known that the step-size parameter is responsible for adjusting the performance of the LMS algorithm. The speed of convergence of the LMS algorithm is proportional to its step-size parameter ; a larger step-size results in a lesser transient-time. At the same time, achieving a small misadjustment requires the use of a small step-size parameter [1]–[3]. These conflicting requirements initiate and motivate the use of the Variable Step-Size LMS algorithm (VSSLMS), which is considered as a compromise solution to be used in order to solve these two conflicting requirements. The VSSLMS algorithm is a self educating technique derived from the LMS algorithm that is used to improve performance [1], [2]. Every tap of the adaptive filter is given a specific time-varying step-size parameter . The LMS recursion formula is written as in (1).

$$\begin{align*}\begin{split} w_{i}(n+1)=w_{i}(n)+ & 2u_{i}(n)e(n)x(n-i),\\ &\quad i=0, 1, \ldots N-1 \end{split}\tag{1} \end{align*}$$ where, is the ith-element of the tap-weight vector and is its step-size parameter at iteration (n). The update of the step-size is related to the corresponding stochastic gradient term gi(n)=e(n)x(n-i), which is observed over a number of successive iterations of the algorithm, is increased if the gi(n) value indicates a positive (or negative) value consistently. This case is occurring when the adaptive filter tap weights are far away from convergence region. As the adaptive filter converges to near the optimum-values, the averages of the gi(n) terms come near zero and they oscillate more rapidly between positive and negative values. This is sensed by the algorithm and the resultant step size is gradually reduced to some pre-specified limits. If the case is changed and the system starts to follow a new optimum value, the gradient terms gi(n) will again show a consistent values (positive/negative), which will lead the algorithm to increase the step-size again, this situation continues till the averages of the gi(n) terms come near zero, and so on the algorithm detects the gi(n) terms in order to specify whether to increase or decrease the step-size [1]. An upper limit to the step-size parameters should be specified to prevent the system from instability. Also, a lower limit should be specified to prevent the system from slow response to sudden changes [1]. Specifying the limit of the step-size parameters that provide the stable operation of the VSSLMS is not easy as for the fixed step-size case [1] which is given by: $$\begin{equation*} 0 < u < \frac{1}{3tr[R]} \tag{2} \end{equation*}$$