High-order allpass cascade-connected phase-system design utilizing fractional polynomial errors

Abstract

ABSTRACT This paper proposes a nonlinear optimization method for designing a high-order recursive allpass digital phase system that comprises several cascade-connected second-order (SO) allpass sections. This method first splits a given phase design specification (approximation target) into a set of equal sub-specifications, and then the equal sub-specification acts as the ideal phase for each of the SO sections. Since an SO section can be designed via employing a linear programming to minimize a linear fractional error function, this splitting process generates an excellent starting point (good initial values for the coefficients of the cascaded SO sections) for further minimizing the overall approximation error. This paper first derives a wide variety of fractional polynomial error functions for designing cascade-connected allpass phase systems of different high orders, and then shows that a high-order cascade-connected allpass phase system can be designed by utilizing a nonlinear solver to minimize the peak deviation of such a fractional polynomial error. Using the derived fractional polynomial error functions in the system design enables the system designer to find an excellent starting point for further minimizing the approximation error and thus reach a convergent design solution. This paper first derives a set of fractional polynomial errors, and then adopts the eighth-order cascade-connected phase system design to exemplify the effectiveness of the fractional-polynomial-error-based design tactic. Furthermore, the stability of the cascade-connected high-order allpass system can be readily checked via confirming that all of the cascaded SO sections have their poles inside the unit circle in the complex plane.