SUMMARYA filter-design oriented theory is presented for polynomials with prescribed phase properties. The phase function is specified by its values and/or higher derivative values at a set of given frequencies. The polynomials are generated by recurrence formulae whose coefficients are calculated by a recursive algorithm. Formulae are presented to calculate the higher derivatives of some composite functions of the phase which are utilized in the process of flat phase approximations.
Two different approaches are introduced for the design of non‐prototype ladder and lattice wave digital filters (WDFs) exhibiting arbitrary amplitude in the baseband (passband, transition band and stopband) and linear phase in the passband. The first approach is based on the phase correction of a minimum phase lattice (or ladder) WDF designed to satisfy the amplitude specifications in the three bands. In the second approach the amplitude and phase requirements are approximated simultaneously. It is devoted to the design of a lattice WDF that is constructed from the parallel arrangement of two allpass subfilters. The design procedure relies on preconstructing one of the subfilters to have exact linear phase at all frequencies and constructing the other to interpolate an arbitrary phase at a set of frequencies distributed all over the baseband. The hidden relationship between the amplitude and phase functions of the filter is utilized to approximate both of them.
The approximation problem is solved by applying an interpolation technique combined with the Remez exchange algorithm. Prototype filters with amplitude specifications in the passband and stopband and phase specifications in the passband are also considered as special cases. Design examples are presented to show the efficiency of the two methods.
An FIR filter design method with simultaneous amplitude and phase approximation is presented. Two components are proposed in the design method: simultaneous amplitude and phase interpolation in the passband and amplitude interpolation in the stopband. These interpolations are implemented by the Remez exchange algorithm.The polynomials for the interpolations are generated by recurrence formulae and the recurrence coefficients are calculated by a recursive method. The properties of the filters designed with the simultaneous approximation are investigated and compared with those of exact linear and minimum phase filters.
SUMMARYA simultaneous amplitude and phase approximation method is presented for reciprocal and non-reciprocal lumped or sampled filters realized by LC, CCD, active RC, cascaded SC, digital or wave digital IIR filters. The approximation is based on the fictive decomposition of a non-minimum phase filter into a minimum phase and an allpass network in such a way that some poles and zeros should coincide with each other and so the filter order should decrease. In this way the amplitude and phase approximations can be carried out alternately, though the minimum phase and the allpass networks do not occur in the resulting filter. Both approximations are based on linear interpolation of specified functions with a possible application of the Remez algorithm. The interpolation algorithms can be used in other arbitrary approximation problems as well. Illustrative examples are given for some lowpass and bandpass filters.
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