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Linear recursive operator's response using the discrete Fourier transform

A classic problem in signal processing is that of analysing empirical data in order to extract information contained within that data. The primary goal of this article is to employ the discrete Fourier transform (DFT) techniques for approximating, to a prescribed accuracy, the response of a shift-invariant recursive linear operator to a finite-length excitation. In this development, the required properties of the Fourier transform (FT) are first reviewed with particular attention directed toward the stable implementation of shift-invariant recursive linear operators. This is found to entail the decomposition of such operators into their causal and anticausal component operators. Subsequently, relevant issues related to the approximation of the FT by the DFT are examined. This includes the important properties of the non-uniqueness of mapping between a sequence and a given set of DFT coefficients. In the unit-impulse response approximation, DFT is shown to provide a useful means for approximating the unit-impulse response of a linear recursive operator. This includes making a partial fraction expansion of the operator's frequency-response. The error incurred in using the DFT for effecting the unit-impulse response approximation is then treated. This error analysis involves the introduction of one-sided exponential sequences and their truncated mappings that arise in a natural fashion when employing the DFT. These concepts form the central theme of the article.