Realization of two transfer functions with complex coefficients by a transfer function with hypercomplex coefficients

Abstract

AbstractThis paper shows that two complex coefficient transfer functions can be realized by a single hypercomplex coefficient network, utilizing all four outputs of the hypercomplex network. the realization is considered for the case where the two arbitrary complex coefficient transfer functions do and do not have a common denominator. It is shown that the transfer functions can be realized by the hypercomplex coefficient network of the same order in the former case and by the hypercomplex coefficient network with the half‐order in the latter case. the realization of the transfer functions with the common denominator is interesting, but the general method to derive the transfer functions is not known.From such a viewpoint, this paper proposes the discrete‐type Kautz approximation with the positive frequency as the domain of approximation, including the optimal pole determination algorithm. By this approach, two complex coefficient transfer functions with an n‐th‐order common denominator are derived from the two 2n‐th‐order real or complex coefficient transfer functions. the method can be considered as one that simultaneously approximates the amplitude and the phase. the proposed approximation method is an approximation only for the positive frequency domain, and the property of the analytic signal that the negative frequency component is zero can be utilized. As a result, the method proposed in this paper has the advantages that the degree of freedom in the filter design is enhanced and the sampling frequency can be halved. In addition, the network structure can be simplified.