Tracking analysis of the sign-sign algorithm for nonstationary adaptive filtering with Gaussian data

This correspondence is concerned with the analysis of the sign-sign algorithm (SSA) when used to track a plant with randomly time-varying parameters. The input of the plant and the plant noise are assumed stationary and Gaussian. The paper derives expressions of the steady-state excess mean square error ξ, the steady-state mean square weight misalignment η, and the step sizes that minimize each one of them. The paper also presents a comparison among the tracking properties of the SSA, the sign algorithm (SA), and the signed regressor algorithm (SRA). It is found that the three algorithms share the features that (1) ξ does not depend on the eigenvalue spread of the input covariance matrix, (2) η depends on the eigenvalue spread, and (3) the step size that minimizes ξ is different from the one that minimizes η. The minimum values of ξ attained by the SA and the SRA are equal to each other, and they are 1 dB less than the one attained by the SSA. The ratio of the minimum value of η of the SSA to the one of the SA is found to be dependent on the input eigenvalue spread; for equal eigenvalues, this ratio is equal to 1 dB. The minimum value of η of the SSA is found to be 1 dB higher than the one of the SRA independently of the input eigenvalue spread. It is found that an advantage of the SSA with respect to both the SA and the SRA is that the two optimum step sizes of the SSA are independent of the mean square plant input and the mean square plant noise