Minimax l∞-Optimal at Time-Domain Linear Filtering

Abstract

The article is devoted to one approach to optimal filtration. We consider a Wiener filter scheme. The proposed statement has two differences from the classical one. The first difference is that the input influences (useful signal and interference) are limited to the maximum absolute values, and not variances. The second difference is that the quality criterion is also the maximum absolute value, and not the variance of the error. Thus, the quadratic criterion in Wiener’s formulation is replaced by a criterion in the form of the l∞-norm (Chebyshev-norm). Therefore, the proposed problem is called the l∞optimal filtering problem. An original way of selecting input filters for signals for this task is proposed. The method allows creating sets of signals with complex limitations of the absolute values of the signals and their derivatives. The calculation of the quality criterion reduces to Bulgakov's problem of the accumulation of perturbations. For a system with discrete time, the quality criterion is written in the form of a sum of an infinite series. Convergence conditions of the series are obtained. If the conditions of convergence are satisfied, an infinite series with any desired accuracy can be replaced by its partial sum. In this case, a quality criterion is obtained in the form of the l1-norm of the impulse response of the filter. It is proposed to numerically search for the impulse response of an optimal filter by the method of subgradient descent. An example of searching for a l∞-optimal filter is considered. The result is compared with classic bandpass filters. The possibility of reducing the phase delay of the filter in the passband is shown.