In the literature [9], the optimum discrete interpolation of one-dimensional signals is presented which minimizes various measures of approximation error simultaneously. In the discussion, the ratio λ of the weighted norm of the approximation error and that of the corresponding input signal plays an essential role to determine the structure of the set of signals. However, only the upper bound of λ is provided in [9]. In this paper, we will present more exact and systematic discussion of the optimum discrete interpolation of one-dimensional signals which minimizes various measures of approximation error at the same time. In this discussion, we will prove that the exact value of λ is identical with the upper limit, for ω (|ω| π), of the largest eigen value of a matrix including the weighting function W(ω) and the Fourier transforms of the optimum interpolation functions. Further, we will give a sufficient condition for W(ω) under which the ratio λ is equal to one, where the approximation error, if it is interpolated by sinc, is included in the set of band-limited signals defined by W(ω). Finally, as application of the presented approximation, we will propose a direction to interactive signal processing on Internet and a transmultiplexer system included in it. The transmultiplexer system included in this discussion can realize flexible arrangement of sub-bands which is inevitable in realizing the above proposal on interactive signal processing.

In the main part of this paper, we present a systematic discussion for the optimum interpolation approximation in a shift-invariant wavelet and/or scaling subspace. In this paper, we suppose that signals are expressed as linear combinations of a large number of base functions having unknown coefficients. Under this assumption, we consider a problem of approximating these linear combinations of higher degree by using a smaller number of sample values. Hence, error of approximation happens in most cases. The presented approximation minimizes various worst-case measures of approximation error at the same time among all the linear and the nonlinear approximations under the same conditions. The presented approximation is quite flexible in choosing the sampling interval. The presented approximation uses a finite number of sample values and satisfies two conditions for the optimum approximation presented in this paper. The optimum approximation presented in this paper uses sample values of signal directly. Hence, the presented result is independent from the so-called initial problem in wavelet theory.

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 ψm(t) (m=0,1,. . . ,M-1) 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 φm(t) minimizing various measures of approximation error defined at discrete time samples n=0, 1, 2,. . . . In this discussion, φm(n) are equal to ψm(n) n=0, 1, 2,. . . . Since ψm(t) are time-limited, φm(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 φm(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.

In the literature, the optimum discrete interpolation approximation is presented. However, this approximation is the optimum for the union of the set of band-limited signals and the set of the corresponding approximation errors. In this paper, under several assumptions, we present two optimum extended discrete interpolation approximations such that the set of the corresponding approximation errors is included in the set of signals if we ignore some negligible component of error. In this paper, we assume that the decimated sampling interval T satisfies T M, where M is the number of paths of the filter bank. The maximally distinct or under sampled filter banks treated in this paper have FIR analysis filters with or without a continuous pre-filter and FIR synthesis filters with a band-limited continuous D/A filter. The first discussion is useful for designing a kind of down-converter which transforms HDTV signal with wide band-width to SDTV signal with narrow band-width, for example. In this discussion, we assume that the SDTV signal is limited in |ω|π/T and the Fourier spectrum of the HDTV signal has wider band but is approximately included in the corresponding narrow band of the SDTV signal. In the well-known scalable coding of signals, if the spectrum of a signal with higher resolution is not included approximately in the spectrum of the corresponding signal with lower resolution, the quality of the quantized output signal with lower resolution will become quite low practically. As shown in [3], however, scalable coding has received much attention recently in the fields of HDTV/SDTV compatibility. Therefore, it will be natural to consider that the spectrum of HDTV signal with higher resolution is similar to and is included approximately in the corresponding spectrum of SDTV signal with lower resolution. The analysis filters are FIR filters with a continuous pre-filter approximately band-limited in |ω|π/T. To improve the quality of the SDTV signal, the whole spectrum component of the HDTV signal is used in the presented down-converter. Another discussion is a general theory of approximation for filter banks using the prescribed analysis filters. In this discussion, although some modification for the band-width is introduced in the process of analysis, the final band-width of the receiver is limited in |ω| π. The FIR analysis filters do not have pre-filter. The condition imposing on the set of signals is more general than the corresponding condition in the first optimum approximation theory. Finally, we present the optimum transmultiplexer TR. In general, under the condition that the receiver filters are prescribed, a transmultiplexer has approximation error between the original signal and the transferred signal. However, the presented TR minimizes approximately the supreme value of arbitrary positive measures of approximation error that can be defined, totally or separately, with respect to all the channels. Note that, in the presented discussion, we can prescribe the degree of FIR filters used in TR, strictly.