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Application of the Mittag-Leffler expansion to sampling discontinuous signals

机译：Mittag-Leffler扩展在不连续信号采样中的应用

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摘要

In applying Shannon's sampling theorem, evaluation of the sampled signal Fourier spectrum is based on the fact that sampling the continuous-time signal is the result of multiplying the signal by distributions. If the signal has discontinuities, a multiplication of distributions - an undefined operation - is encountered. Such undefined operation has led to errors in the literature which to date accompany the formulation of sampling of signals containing discontinuities. This paper presents an approach to evaluating the product of distributions as a means of sampling discontinuous signals, eliminating such errors. It is shown that the value of the product of distributions may be found by invoking the Mittag-Leffler expansion. As an illustration of errors that have existed for decades and still exist in the digital signal processing literature whenever discontinuous signals are sampled the approach of impulse invariance provides a case in point. It was already noted that this approach has an inherent error. Yet, impulse invariance is still considered as one of the two main approaches for converting analogue to digital filters. In this study, the true spectra of sampled discontinuous signals are evaluated, and a new approach to the transformations between continuous-time and discrete-time systems eliminating the error, is proposed.
机译：在应用香农采样定理时，对采样信号傅立叶频谱的评估基于以下事实：对连续时间信号进行采样是信号乘以分布的结果。如果信号具有不连续性，则会遇到分布的乘法运算-未定义的运算。这种不确定的操作导致了迄今为止在文献中出现的错误，这些错误伴随着包含不连续信号采样的制定。本文提出了一种评估分布乘积的方法，作为一种对不连续信号进行采样的方法，可以消除此类误差。结果表明，可以通过调用Mittag-Leffler展开来找到分布乘积的值。作为对已经存在数十年并在数字信号处理文献中仍然存在的错误的说明，每当对不连续信号进行采样时，脉冲不变性就是一个很好的例子。已经注意到，这种方法具有固有的错误。然而，脉冲不变性仍被视为将模拟滤波器转换为数字滤波器的两种主要方法之一。在这项研究中，评估了采样的不连续信号的真实频谱，并提出了一种消除连续时间和离散时间系统之间误差的新方法。

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  • 来源
    《Signal Processing, IET》 |2013年第9期|863-878|共16页
  • 作者

    Corinthios M.J.;

  • 作者单位

    ??cole Polytechnique de Montr??al, Montr??al, Qu??bec, Canada|c|;

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