Minimax Design of Low-Complexity Allpass Variable Fractional-Delay Digital Filters

This paper proposes noniterative and iterative linear programming (LP) procedures for designing low-complexity allpass variable fractional-delay (VFD) digital filters in the minimax sense. Expressing each coefficient of an allpass VFD filter as a polynomial in the VFD parameter p, we show that the frequency response error of an allpass VFD filter can be written as a pure imaginary part divided by its denominator. Thus, the minimax design can be approximately formulated as an LP problem through neglecting the denominator, which leads to a noniterative minimax design. To improve the minimax design accuracy, we propose an iterative LP procedure that utilizes the denominator from the preceding iteration as a known. The iterative LP minimization is repeated until it converges to the minimax solution. Moreover, we also present a two-stage algorithm for optimizing the optimal variable range p ∈ [p_Min, p_Max] of the VFD parameter p and successively reducing the subfilter orders. Design examples are given to show that both noniterative and iterative LP methods can achieve much better minimax designs (smaller peak errors) than the existing iterative weighted-least-squares (WLS) approaches, which aim to minimize the peak errors of VFD response and variable phase response, respectively. Moreover, the resulting allpass VFD filters have lower complexities than those from the iterative WLS approaches.