We present a systematic theory for the optimum sub-band interpolation using a given analysis or synthesis filter bank with the prescribed coefficient bit length. Recently, similar treatment is presented by Kida and quantization for decimated sample values is contained partly in this discussion [13]. However, in his previous treatment, measures of error are defined abstractly and no discussion for concrete functional forms of measures of error is provided. Further, in the previous discussion, quantization is neglected in the proof of the reciprocal theorem. In this paper, linear quantization for decimated sample values is included also and, under some conditions, we will present concrete functional forms of worst case measures of error or a pair of upper bound and lower limit of those measures of error in the variable domain. These measures of error are defined in Rn, although the measure of error in the literature [13] is more general but must be defined in each limited block separately. Based on a concrete expression of measure of error, we will present similar reciprocal theorem for a filter bank nevertheless the quantization for the decimated sample values is contained in the discussion. Examples are given for QMF banks and cosine-modulated FIR filter banks. It will be shown that favorable linear phase FIR filter banks are easily realized from cosine-modulated FIR filter banks by using reciprocal relation and new transformation called cosine-sine modulation in the design of filter banks.

It is known that an anticausal IIR filter can be realized in real time by using the time reversed section technique. When combined with a casual IIR filter, the overall transfer function can yield exact linear phase characteristic in theory. This paper presents a new method for designing complex IIR digital filters with exact linear phase. The design problem of IIR filters with exact linear phase can be reduced to magnitude-only filter design. The proposed procedure is based on the formulation of an eigenvalue problem by using Remez exchange algorithm. By solving the eigenvalue problem to compute the real maximum eigenvalue, the solution of the rational interpolation problem can be achieved. Therefore, the optimal filter coefficients are easily obtained through a few iterations. The proposed design algorithm not only retains the speed inherent in Remez exchange algorithm, but also Simplifies the interpolation step because is has been reduced to the computation of the real maximum eigenvalue. Several examples are presented to demonstrate the effectiveness of the proposed method.

A novel method is presented for designing discrete coeffcient FIR linear phase filters using Hopfield neural networks. The proposed method is based on the minimization of the energy function of Hopfield neural networks. In the proposed method, the optimal solution for each filter gain factor is first searched for, then the optimal filter gain factor is selected. Therefore, a good solution in the specified criterion can be obtained. The feature of the proposed method is that it can be used to design FIR linear phase filters with different criterions simultaneously. A design example is presented to demonstrate The effectiveness of the proposed method.