Quintic fractional error function for designing cascade-type phase compensators

This paper investigates the design of the tenth-order allpass phase compensator with cascade structure. This cascade-type compensator consists of five biquadratic (biquad) allpass sections, and those biquad sections are cascade-connected. To design this cascade-type phase compensator, we first derive a phase error function called quintic fractional (QF) error function, and then employ this QF-error function as the cost function that is minimized for optimizing the coefficients of the tenth-order phase compensator. The compensator coefficients are optimized such that the compensator’s phase response fits a prescribed ideal phase response (phase specification) with the maximum of the QF-error function being minimized. The QF-error function is a rational function whose numerator and denominator are the quintics of the unknown compensator coefficients. Utilizing the QF-error function as a cost function in optimizing the compensator’s coefficients enables the nonlinear minimization to be carried out from a reasonably good starting point, which in turn leads to a convergent design solution. This is the key motivation for deriving the QF-error function and then employing it to design the tenth-order cascade-type phase compensator. Moreover, since the tenth-order compensator comprises exclusively the biquad allpass sections, and the stability of the biquad sections is considerably easy to check, one can confirm the stability of the designed tenth-order compensator by checking the stability of the five cascade-connected biquads. Two illustrative design examples are included for demonstrating the usefulness of the QF-error function in terms of starting the minimization from a good initial point and thus producing a convergent solution. The two examples also illustrate the simplicity of utilizing the cascade structure for the stability check.