Closed-Form Mixed Design of High-Accuracy All-Pass Variable Fractional-Delay Digital Filters

This paper presents a closed-form method for minimizing the weighted squared error of variable fractional-delay (VFD) of an all-pass VFD digital filter under an equality constraint on its normalized root-mean-squared (NRMS) error of variable frequency response (VFR). The main purpose is to reduce the squared VFD error as much as possible while keeping its NRMS VFR error exactly at a predetermined value. We first prove that the linearized VFR error of an all-pass VFD filter is almost the same as its linearized phase error, and then convert the equality-constrained weighted-least-squares (WLS) design into an unconstrained optimization problem through the minimization of a mixed error function that mixes the weighted squared VFD error and squared VFR error. To reduce the computational complexity, we derive a closed-form mixed error function by utilizing Taylor series expansions of trigonometric functions. Therefore, the error functions can be efficiently computed without discretizing the design parameters (frequency omega and VFD parameter p). The closed-form mixed error function not only reduces the computational complexity, but also speeds up the design process as well guarantees the optimality of the final solution. Furthermore, a two-point search (dichotomous search) scheme is proposed for finding the optimal range p is an element of [p(Min), p(Max)] of the VFD parameter p, and then the subfilter orders are optimized under a given filter complexity constraint (the number of all-pass VFD filter coefficients). This two-stage optimization process utilizes the NRMS VFD error as an error criterion. Design examples and comparisons are given to demonstrate that the closed-form mixed WLS method yields low-complexity all-pass VFD filters with a high-accuracy VFD response but without noticeably degrading its frequency response.