Tracking analysis of the sign algorithm without the Gaussian constraint

This paper is concerned with the analysis of the sign algorithm (SA) when used to adapt a finite impulse response (FIR) filter with randomly time-varying target weights. The analysis is done under the assumption that positive and negative polarities of the noise are equally probable and that the noise probability density function at the origin exists and is strictly positive, This assumption fits many noise distributions encountered in applications. Expressions of the excess mean square error ξ and the mean square weight misalignment η are derived. It is found that both ξ and η are independent of the type of distribution of the filter input. Both ξ and η are proportional to the reciprocal of the noise probability density function at the origin. The step sizes that minimize ξ and η are found to be independent of both the variance and the type of distribution of the noise. Given the sum of the mean square target weight fluctuations, it is found that ξ (resp. η) is independent (resp. dependent) on both the mean squares of individual target weight fluctuations and the mutual correlation among them. The tracking properties of the SA are found to be strongly related to the ones of the LMS algorithm. It is shown that the charts of ξ and η versus the step size of the SA can be obtained from the corresponding ones of the LMS algorithm via a simple linear transformation that depends only on the noise distribution. The above results hold for both continuous and discrete distributions of the input of the filter