The selective coefficient update normalized least mean-square (SCU-NLMS) algorithm was proposed to reduce computational complexity while preserving close performance to the full-update NLMS algorithm, which brought it a lot of attention. In practical applications, the length of the unknown system impulse response is not known and, therefore, the length of the adaptive filter can be less than that of the unknown system particularly in situations when the unknown system impulse response is long. In all existing analysis of the SCU-NLMS algorithm, exact modeling of the unknown system is assumed, i.e., the length of the adaptive filter is equal to that of the unknown system impulse response. In this paper, we present mean-square performance analysis for the SCU-NLMS algorithm in an undermodeling situation and assuming independent and identically distributed (i.i.d.) input signals. The analysis model takes into account order statistics employed in the SCU-NLMS algorithm leading to accurate transient and steady state theoretical results. Analysis extends easily to the exact modeling case where expressions quantifying the algorithm mean-square performance are presented and shown to be more accurate than the ones reported in the literature. Simulation experiments validate the accuracy of the theoretical results in predicting the actual behavior of the algorithm.
The selective coefficient update normalized least mean-square (SCU-NLMS) algorithm belongs to the family of adaptive algorithms that reduces computational complexity by updating a portion of the adaptive filter coefficients at each iteration. The SCU-NLMS algorithm has been a viable choice in applications where long adaptive filters are used since it was shown to maintain the closest performance among partial update NLMS algorithms to the full-update NLMS algorithm [1], [2], [3], [4], [5], [6] and [7]. In all existing analysis of the SCU-NLMS algorithm [2], [3], [4] and [8], exact modeling of the unknown system is assumed, i.e., the length of the adaptive filter is equal to that of the unknown system impulse response. However, in practical applications, the length of the unknown system is not known, and in applications with long impulse responses, the system is normally undermodeled. Theoretical results of the SCU-NLMS algorithm with exact modeling do not necessarily apply in the undermodeling case and, therefore, there is a need to study the theoretical performance of the algorithm in an undermodeling situation.
The objective of this paper is to provide convergence analysis of the SCU-NLMS algorithm that describes its transient as well as its steady state mean and mean-square behavior in an undermodeling situation. Analysis assumes a zero-mean independent and identically distributed (i.i.d.) input signal, and will use the common independence assumption that the input signal vector is independent of the adaptive filter vector [9]. Assuming a more general input signal model, though desirable in the analysis of adaptive algorithms, makes providing explicit closed form expressions that clearly quantify the mean-square behavior of the algorithm a very difficult task due to order statistics employed by the algorithm [3], [4] and [8]. Our analysis model takes into account order statistics thus leading to results that predict very well the actual mean and mean-square behavior of the algorithm. In [4], order statistics were applied to the analysis of the selective partial update NLMS algorithm with exact modeling, which was shown to provide more accurate results than those in [3]. Our analysis will be extended to the exact modeling case, and will result in different equations for the algorithm MSE, final excess MSE, and stability bounds than those derived in [4]. This is because authors in [4] analyze the selective partial update NLMS algorithm presented in [3] that has a slightly different coefficient update equation than the one we are analyzing here and was proposed in [1] and [2]. Moreover, our analysis approach is different from that in [4] where authors also assume a certain model for the white input signal. In [8], mean-square analysis was performed for the SCU-NLMS algorithm, and the effect of order statistics appears in the analysis in a simplified manner. The analytical approach used here is different from that in [8]. Simulation experiments will be conducted to verify analysis, and also comparisons are made with theoretical results from [8] for the exact modeling case.
This paper presented mean and mean-square convergence analysis of the selective coefficient update NLMS (SCU-NLMS) algorithm in an undermodeling situation. Analysis yielded expressions for the algorithm mean coefficients error, coefficients error power, and MSE. Moreover, stability bounds, and a closed form expression for the algorithm steady state excess MSE were derived. The analysis were applied to the exact modeling case, where new expressions are derived for the algorithm MSE and stability bounds that predict the actual behavior of the algorithm more accurately than previously reported theoretical results in the literature. Simulation experiments conducted in undermodeling and exact modeling situations indicated the ability of the derived theoretical results to predict well the transient and steady state behavior of the algorithm.
