Abstract
The Fourier transform of a discrete sequence yields a continuous complex function, the so-called continuous spectrum. This chapter describes how to express a periodic sequence using sampled spectral sequences in accordance with the sampling theorem and discrete Fourier transformation. The pair of time and spectral sequences forms a discrete Fourier transform (DFT) pair. The sampling theorem gives conditions and formulation for sampling a continuous function as a discrete sequence from which the original continuous function can be reconstructed without any deformation. The interpretation of the theorem is given in terms of the DFT and the partial sum of the Fourier series expansion for a continuous periodic function. In understanding the DFT schemes, it is informative seeing what happens if the sampling theorem is violated. Modifications of sequences through interpolation and decimation are displayed using numerical examples.
The original version of this chapter was revised.
An erratum to this chapter can be found at DOI 10.1007/978-4-431-54424-1_10
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Tohyama, M. (2015). Sampling Theorem and Discrete Fourier Transform. In: Waveform Analysis of Sound. Mathematics for Industry, vol 3. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54424-1_7
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DOI: https://doi.org/10.1007/978-4-431-54424-1_7
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