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[Keyword] the optimum approximation(2hit)

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  • The Optimum Approximate Restoration of Multi-Dimensional Signals Using the Prescribed Analysis or Synthesis Filter Bank

    Takuro KIDA  Yi ZHOU  

     
    PAPER-Digital Signal Processing

      Vol:
    E79-A No:6
      Page(s):
    845-863

    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.

  • The Optimum Approximation of Multi-Dimensional Signals Based on the Quantized Sample Values of Transformed Signals

    Takuro KIDA  

     
    PAPER-Digital Signal Processing

      Vol:
    E78-A No:2
      Page(s):
    208-234

    A systematic theory of the optimum multi-path interpolation using parallel filter banks is presented with respect to a family of n-dimensional signals which are not necessarily band-limited. In the first phase, we present the optimum spacelimited interpolation functions minimizing simultaneously the wide variety of measures of error defined independently in each separate range in the space variable domain, such as 8 8 pixels, for example. Although the quantization of the decimated sample values in each path is contained in this discussion, the resultant interpolation functions possess the optimum property stated above. In the second phase, we will consider the optimum approximation such that no restriction is imposed on the supports of interpolation functions. The Fourier transforms of the interpolation functions can be obtained as the solutions of the finite number of linear equations. For a family of signals not being band-limited, in general, this approximation satisfies beautiful orthogonal relation and minimizes various measures of error simultaneously including many types of measures of error defined in the frequency domain. These results can be extended to the discrete signal processing. In this case, when the rate of the decimation is in the state of critical-sampling or over-sampling and the analysis filters satisfy the condition of paraunitary, the results in the first phase are classified as follows: (1) If the supports of the interpolation functions are narrow and the approximation error necessarily exists, the presented interpolation functions realize the optimum approximation in the first phase. (2) If these supports become wide, in due course, the presented approximation satisfies perfect reconstruction at the given discrete points and realizes the optimum approximation given in the first phase at the intermediate points of the initial discrete points. (3) If the supports become wider, the statements in (2) are still valid but the measure of the approximation error in the first phase at the intermediate points becomes smaller. (4) Finally, those interpolation functions approach to the results in the second phase without destroying the property of perfect reconstruction at the initial discrete points.