SUMMARYThe applications of the least-mean-square-approximation technique to filter synthesis are discussed, and explicit expressions for the characteristic function of all-pole lowpass filters are derived using a weighted least mean-square error norm. The weight function depends on one variable parameter which controls the shape of the magnitude response both in the passband and in the stopband. It is shown that most of the filter functions in common use are special cases of this approximation procedure, including Legendre monotonic passband filters as a limiting, degenerate case. Also, a useful generalization of Legendre filter functions is proposed.
SUMMARYThe explicit formula for the characteristic function of all-pole lowpass filters with monotonic passband magnitude response is derived. It is shown that the characteristic function of Legendre, Halpern, LSM and other known classes of monotonic magnitude all-pole filters that are synthesized from the magnitude squared function, can be obtained by assigning different values to a variable parameter in the general expression obtained. The approximation method consists of using a weighted least-mean-square norm to minimize the area under the first derivative of the characteristic function.
SUMMARYA new approach has been presented to find characteristic functions of a certain class of monotonic low-pass filters of non-maximally flat type. The well-known Legendre filters, and the more recently introduced class H filters, both belonging to this class, are derived as special cases of the application of least-squares approximation technique by optimizing the asymptotic loss. It is shown that class H filters provide higher stopband attenuation than the Legendre filters but have less favourable passband magnitude response. On the other hand, if the minimization of the passband loss is at a premium then the unconstrained least-squares monotonic approximants, referred to as LSM filters, yield the best results. Therefore, among all filters with monotonic magnitude response, the Legendre filters have neither optimum passband, nor optimum stopband performance, though they are suitable for some applications.
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