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CHAPTER 5
The Discrete
Fourier Transform
In Chapters 3 and 4 we studied transform-domain representations of discrete
signals. The discrete-time Fourier transform provided the frequencydomain
(ω) representation for absolutely summable sequences. The
z-transform provided a generalized frequency-domain (z) representation
for arbitrary sequences. These transforms have two features in common.
First, the transforms are defined for infinite-length sequences. Second,
and the most important, they are functions of continuous variables (ω
or z). From the numerical computation viewpoint (or from MATLAB’s
viewpoint), these two features are troublesome because one has to evaluate
infinite sums at uncountably infinite frequencies. To use MATLAB, we
have to truncate sequences and then evaluate the expressions at finitely
many points. This is what we did in many examples in the two previous
chapters. The evaluations were obviously approximations to the exact
calculations. In other words, the discrete-time Fourier transform and the
z-transform are not numerically computable transforms.
Therefore we turn our attention to a numerically computable transform.
It is obtained by sampling the discrete-time Fourier transform in the
frequency domain (or the z-transform on the unit circle). We develop this
transform by first analyzing periodic sequences. From Fourier analysis we
know that a periodic function (or sequence) can always be represented by
a linear combination of harmonically related complex exponentials (which
is a form of sampling). This gives us the discrete Fourier series (DFS) representation.
Since the sampling is in the frequency domain, we study the
effects of sampling in the time domain and the issue of reconstruction in
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