Consideration on the optimum interpolation and design of linear phase filterbanks with high attenuation in stop bands

In the literatures[5] and [10], a systematic discussion is presented with respect to the optimum interpolation of multi-dimensional signals. However, the measures of error in these literatures are defined only in each limited block separately. Further, in these literatures, most of the discussion is limited to theoretical treatment and, for example, realization of higher order linear phase FIR filter bank is not considered. In this paper, we will present the optimum interpolation functions minimizing various measures of approximation error simultaneously. Firstly, we outline necessary formulation for the time-limited interpolation functions psi(m)(t) (m = 0, 1,..., M-I) realizing the optimum approximation in each limited block separately, where m are the index numbers for analysis filters. Secondly, under some assumptions, we will present analytic or piece-wise analytic interpolation functions phi(m)(t) minimizing various measures of approximation error defined at discrete time samples n = 0, +/-1,+/-2,.... In this discussion, phi(m)(n) are equal to psi(m),(n) n = 0, +/-1, +/-2,.... Since psi(m)(t) are time-limited, phi(m)(n) vanish outside of finite set of n. Hence, in designing discrete filter bank, one can use FIR filters if one wants to realize discrete synthesis filters which impulse responses are phi(m)(n). Finally, we will present one-dimensional linear phase M channel FIR filter bank with high attenuation characteristic in each stop band. In this design, we adopt the cosine-sine modulation initially, and then, use the iterative approximation based on the reciprocal property.