Amplitude selectivity versus constant delay in minimum-phase lossless filters

A classic filter problem is the construction of minimumphase transfer functions which simultaneously possess linear phase in the passband and steep amplitude selectivity. This paper presents a detailed quantitative treatment of this problem for lossless transducers, and it is shown that the bandwidth for linear phase is strictly constrained independent of the complexity of the transducer once the amplitude selectivity is prescribed. Prototype, but nonrational, physically realizable transfer functions are constructed which satisfy the bandwidth constraints, and polynomials which approximate the prototypes are then calculated to yield physical linear phase ladder networks. The resultant structures, though minimum phase, can have respectable selectivity and phase properties. For example, an 8-element ladder is synthesized which realizes flat delay to ± 2.5 percent over 80 percent of the passband, with 60 dB of loss at twice cutoff, and 3-dB passband tolerance. Finally, the paper shows how to further improve amplitude selectivity at the price of small amounts of nonlinear phase distortion, and quantitative measures of this improvement are presented.