19-02-2013, 02:22 PM
Discrete Fourier Transform
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Introductory Remarks
There are many ways that the Discrete Fourier Transform (DFT) arises in practice but generally
one somehow arrives at a periodic sequence numbers. These numbers may arise, for example,
as a discretely sampled values of an analog function sampled over some period window and then
extended periodically. They may also arise as a discrete set of values from the measurements in
an experiment. Once again we would assume that they are extended periodically. In any case, the
DFT of the sequence is a new periodic sequence and is related to the original sequence via a DFT
inversion transform similar to the Inverse Fourier Transform. The DFT turns out to be very useful
in spectral analysis of both Fourier series and Fourier transforms.
Unfortunately, there are many different definitions of the DFT just as there are for the Fourier
transform. They are all related but the difference makes reading different books a bit of a chore. In
these notes we will adopt the definition used in the Matlab software since Matlab will use for the
most part in the numerical examples.