Closed-form solutions for discrete-time elliptic transfer functions

Presents closed-form solutions in explicit form for the direct derivation of discrete-time elliptic transfer functions having denormalized lowpass (LP), highpass (HP), bandpass (BP), or bandstop (BS) loss-frequency characteristics. The proposed method is based on the derivation of a suitable discrete-time z-domain normalized lowpass elliptic prototype reference transfer function and on the transformation of that transfer function into the desired discrete-time z¯-domain denormalized LP, HP, BP, or BS elliptic transfer function using a recently advanced z-to z¯ frequency transformation technique. The required discrete-time z-domain normalized lowpass prototype reference transfer function is derived by establishing the fact that if it is related to a continuous-time s-domain normalized lowpass elliptic prototype reference transfer function through the conventional s-to-z bilinear frequency transformation. The parameters of the former as expressed in terms of Jacobian elliptic functions, are related to those of the latter through the ascending Landen transformation