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Derivations and Relations of Various Cost Functions for Allpass Phase-Equalizing Filter Design

This paper derives a new error function for the design of an allpass phase-equalizing filter (PEF). The new error function is nonlinear with respect to the unknown PEF coefficients, and minimizing the error function can obtain the unknown coefficients. Then, the relationship between the existing phase-error function in the literature that is derived by using the Taylor series expansion and this new error function is revealed. Furthermore, the relationships among the new error function and two existing frequency-response error functions are derived. Therefore, this paper has two objectives. One is to develop a new error function for designing an allpass PEF, and the other is to reveal the relationships among the newly developed error function and various existing error functions, including a phase-error function and two frequency-response error functions. Those error functions can be used as cost functions to be minimized in developing new optimization techniques for the design of an allpass PEF.