Abstract
This short chapter introduces the discrete (finite) and fast Fourier transform. The numerical stability and conditioning of the Fourier matrix is mentioned. Applications using convolution and circulant matrices are considered, as is the periodogram or power spectral density. ◃
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Notes
- 1.
We will learn in Chap. 12 how to solve this equation.
- 2.
Because of the plethora or plenitude—perhaps even superfluity—of conventions and conventional notations for Fourier series and signal processing, the bookkeeping takes on more importance than usual. For example, what some people call the matrix \(\mathbf{F}\) actually corresponds to what other authors or software packages, for example, Matlab, mean by the inverse discrete Fourier transform. And then there is where to place the factor n, or n + 1 if you index from zero, or \(\sqrt{n + 1}\). You get the idea.
- 3.
Note that Matlab’s circulant matrix convention is the transpose convention to the above.
- 4.
Of course, it is only normwise well-conditioned. If you want a relatively small Y k to be accurate, you’ll have to work, and if there’s data error, you may be out of luck.
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Corless, R.M., Fillion, N. (2013). The Discrete Fourier Transform. In: A Graduate Introduction to Numerical Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8453-0_9
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