There are various way to obtain the polynomials referring to a passband different from [-1 1]. The simplest one is to evaluate the polynomials for [-1 1] passband using standard methods and then apply a frequency transformation. Let be s the domain where the polynomials are determined and s’ the new domain where the passband is [-1 -alfa]. A suitable frequency transformation is s’=a*s+b, where the constants a and b can be determined by imposing the correspondence of the passband frequencies in the teo domains: -j à -j, jà -j*alfa. It is found a=(1-alfa)/2, j*b=(1+alfa)/2. In a similar way can be found the transformation from [-1 1] to [+beta 1].

You can now evaluate the polynomials P, E, F referring to the passband [-1 1] (if transmission zeros are imposed they must be transformed from s’ to s using inverting the above linear equation). Then you can obtain F’ and E’ by transforming the roots of F and E (all these polynomials are monic). Transforming the roots of P gives the transmission zeros in s’ domain; for obtaining P’ you need the highest degree coefficient of P’ which can be obtained by imposing the request return loss level at one of the passband edges.

Regards

Giuseppe Macchiarella

Dipartimento di Elettronica e Informazione - Politecnico di Milano

Piazza Leonardo Da Vinci n. 32 - 20133 Milano - Italy