The Optimum Discrete Approximation of Band-Limited Signals without Necessity of Combining the Set of the Corresponding Approximation Errors

Yuichi KIDA  Takuro KIDA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E85-A   No.3   pp.610-639
Publication Date: 2002/03/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Digital Signal Processing
digital signal processing,  optimum interpolation,  filter bank,  transmultiplexer,  

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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.