UploadFile_6417

142 Chapter 5 THE DISCRETE FOURIER TRANSFORM
the z-domain. We then extend the DFS to finite-duration sequences, which
leads to a new transform, called the discrete Fourier transform (DFT).
The DFT avoids the two problems mentioned and is a numerically computable
transform that is suitable for computer implementation. We study
its properties and its use in system analysis in detail. The numerical computation
of the DFT for long sequences is prohibitively time-consuming.
Therefore several algorithms have been developed to efficiently compute
the DFT. These are collectively called fast Fourier transform (or FFT)
algorithms. We will study two such algorithms in detail.
5.1 THE DISCRETE FOURIER SERIES
In Chapter 2 we defined the periodic sequence by ˜x(n), satisfying the
condition
˜x(n) =˜x(n + kN), ∀n, k (5.1)
where N is the fundamental period of the sequence. From Fourier analysis
we know that the periodic functions can be synthesized as a linear combination
of complex exponentials whose frequencies are multiples (or harmonics)
of the fundamental frequency (which in our case is 2π/N). From
the frequency-domain periodicity of the discrete-time Fourier transform,
we conclude that there are a finite number of harmonics; the frequencies
are { 2π N k, k =0, 1,...,N − 1}. Therefore a periodic sequence ˜x(n) can
be expressed as
˜x(n) = 1 N
N−1
∑
k=0
˜X(k)e j 2π N kn , n =0, ±1,..., (5.2)
where { ˜X(k), k =0, ±1,...,} are called the discrete Fourier series coefficients,
which are given by
˜X(k) =
N−1
∑
n=0
˜x(n)e −j 2π N nk , k =0, ±1,..., (5.3)
Note that ˜X(k) isitself a (complex-valued) periodic sequence with fundamental
period equal to N, that is,
˜X(k + N) = ˜X(k) (5.4)
The pair of equations (5.3) and (5.2), taken together, is called the discrete
Fourier series representation of periodic sequences. Using W N
△
= e
−j 2π N to
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.