This letter presents, as an extension of a previous work of the present authors, a design method of IIR digital filters, which is guaranteed in principle to be stable, by using the interpolation of the characteristic function of the filter for a set of prescribed values of the function and, in addition, its higher derivatives. As a numerical example, an IIR digital filter having the characteristics such that it has a maximal flatness in the passband and prescribed steepness in the transition band is designed to show the effectiveness of the proposed method.

An IIR digital low pass filter with flat monotonic passband, equiripple stopband and narrower transition bandwidth than that of Inverse Chebyshev digital filters of the same order is designed. The requisite equiripple stopband is realized by designing the filter in Deczkeys' w-plane. The characteristic functions are designed so as to have a root of multiplicity n at ω = 0 to ensure the n degree of flatness of the passband, and to have a pair of complex conjugate roots with coordinates constrained such that the magnitude response of the passband attenuates monotonically. The freedom in the coordinate of the complex conjugate roots is exploited to minimize the transition bandwidth. The equations are derived that give the minimum transition bandwidth of the proposed filter, which is considerably narrower than that of Inverse Chebyshev filters. It is showen through practical numerical examples that the order of the proposed filter is as low as half that of the Inverse Chebyshev filter satisfying the same specification.

This letter presents a method of optimal design of IIR digital filters with the minimum number of interpolation points in magnitude response. Those filters are classified into four types and each of them is designed by a simple formula. Maximal flatness, ripple property, filter order, and steepness in transition band are discussed.

The authors recently proposed a design method of stable IIR digital filters based on the interpolation by rational characteristic functions of filters, for a set of values of these characteristic function and, in addition, their higher derivatives prescribed at a number of frequency. This method can be further extended so that, despite usage of a less number of interpolation points, almost the same filter characteristics as one obtained by the former method can be realized. This paper presents an improved design method for making the transfer function meet strict magnitude specifications. The method proposed in this paper is especially efficient for designing a filter whose characteristics is specified not only in the passband but also in the transition band with relatively narrow bandwidth.

The approximation of the gain characteristics of linear phase FIR digital filters is reduced to the approximation by cosine polynomials. Therefore we can easily obtain an optimum solution under the LMS of Chebyshev error criterion. However the optimum solution does not always meet practical specifications, especially in the case where the gain is specified strictly at some angular frequencies. On the other hand in such a case, it is known that interpolation technique can be suitably applied for the approximation mentioned above. However, in this case, we encounter another difficulty in the approximation caused by interpolation. In order to overcome the above difficulty, this paper proposes a new method utilizing both of the interpolation and LMS techniques. Some parameters included in approximating functions are used to satisfy prescribed interpolating conditions and the other parameters are used to minimize the approximation error under the LMS criterion. In addition, interpolation technique is extended to include the case in which also higher derivatives are taken into interpolation conditions to make smooth interpolation. An example is shown to illustrate the effectiveness of the proposed method.