On optimal filters with maximum number of constraints on amplitude and phase characteristics

Abstract

For optimal filters specified amplitude and phase (or group delay) characteristics, it is required that all the free parameters of the transfer function be used for the approximation. To achieve this requirement, the number of constraints on the amlitude and phase characteristics is defined first on general interpolation bases. Constant or arbitrarily prescribed lowpass or bandpass group delay and amplitude characteristics are approximated in the maximally flat, ripple or mixed sense for lumped filters, whereas highpass and band rejection characteristics are also considered for distributed or sampled data filters. The relationship between the number of free parameters and the number of amplitude and phase constraints for a given degree is derived for non-reciprocal as well as for reciprocal lossy and reciprocal reactant cases. Conditions are derived for the distribution of free parameters between the passband and stopband amplitude and phase characteristics for transfer functions with Hurwitzian denominator and also for monotonic amplitude characteristics in the transition band. Some published separate and simultaneous approximation methods are evaluated and compared on the above bases. It is pointed out that some of these methods do not satisfy the above requirements, although the optimal solution would exhibit higher performances.