Behavior of the least mean square algorithm with a periodically time‐varying input power

Abstract

The paper analyzes the transient and steady-state performances of a least mean square algorithm in the rarely-studied situation of a time-varying input power. A scenario of periodic pulsed variation of the input power is considered. The analysis is carried out in the context of tracking a Markov plant with a white Gaussian input. It is shown that the mean square deviation (MSD) converges to a periodic sequence having the same period as that of the variation of the input power. Expressions are derived for the convergence time and the steady-state peak MSD. Surprisingly, it is found that neither the transient performance nor the steady-state performance degrades with rapid variation of the input power. On the other hand, slow input power variation causes degradation in both the transient and steady-state performances for given amplitude of variation of the input power. In the case of a time-invariant plant, neither rapid nor slow variation of the input power causes degradation in the steady-state performance. On the other hand, there is degradation in the transient performance for slow variation of the input power.

Stochastic analysis of the diffusion least mean square and normalized least mean square algorithms for cyclostationary white Gaussian and non‐Gaussian inputs

“…+ four statistically independent cross terms which will average to zero. (16) We need to average both sides of ( 16). Now,…”

“…Diffusion algorithms [1][2][3][4][5][6][7][8][9][10][11][12] and cyclostationary inputs [13][14][15][16][17][18][19][20][21][22][23] are two separate important subjects in the theory of adaptive filtering. The purpose of this article is to combine the two subjects and to analyze the stochastic behavior of diffusion algorithms for cyclostationary inputs.…”