Ripple-pass function

Abstract

The well-known relation for an all-pass function is generalized by the introduction of two parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_{a}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_{b}</tex> making <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(s)=\frac{EvP(s)-k_{a}OdP(s)}{EvP(s)+k_{b}OdP(s)}</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(s)</tex> is a Hurwitz polynomial, while <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">EvP(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">OdP(s)</tex> are its even and odd parts, respectively. It is shown that the amplitude, phase, and group delay of such a generalized all-pass function ripple, and that the ripples are dependent on the two introduced parameters and their ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K = k_{a}/k_{b}</tex> . Thus the name "ripple-pass function." Some interesting and important features of the discussed function have been considered here. The ripple-pass function is suitable for practical applications such as amplitude, phase, and/or delay equalization, or for design of narrow-bandpass or bandstop (notch) filters. The ripple-pass function can be easily realized by using simple passive and active networks.