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Signals and Spectra Chap. 2
MATLAB uses the DFT and IDFT definitions that are given by Eqs. (2–176) and
(2–177), except that the elements of the vector are indexed 1 through N instead of 0 through
N - 1. Thus, the MATLAB FFT algorithms are related to Eqs. (2–176) and (2–177) by
and
X = fft(x)
(2–178)
x = ifft(x)
(2–179)
where x is an N-element vector corresponding to samples of the waveform and X is the
N-element DFT vector. N is chosen to be a power of 2 (i.e., N = 2 m , where m is a positive
integer). If other FFT software is used, the user should be aware of the specific definitions
that are implemented so that the results can be interpreted properly.
Two important applications of the DFT will be studied. The first application uses the
DFT to approximate the spectrum W(f); that is, the DFT is employed to approximate the
continuous Fourier transform of w(t). The approximation is given by Eq. (2–184) and
illustrated by Example 2–20. The second application uses the DFT to evaluate the complex
Fourier series coefficients c n . The result is given by Eq. (2–187) and illustrated by
Example 2–21.
Using the DFT to Compute the Continuous Fourier Transform
The relationship between the DFT, as defined by Eq. (2–176), and the CFT will now be examined.
It involves three concepts: windowing, sampling, and periodic sample generation.
These are illustrated in Fig. 2–20, where the left side is time domain and the right side is the
corresponding frequency domain. Suppose that the CFT of a waveform w(t) is to be evaluated
by use of the DFT. The time waveform is first windowed (truncated) over the interval (0, T) so
that only a finite number of samples, N, are needed. The windowed waveform, denoted by the
subscript w, is
w w (t) = e w(t), 0 … t … T
- (T/2)
f = w(t) ß at b
0, t elsewhere T
The Fourier transform of the windowed waveform is
q
W w (f) = w w (t)e -j2pft T
dt = w(t)e -j2pft dt
L L
-q
0
(2–180)
(2–181)
Now we approximate the CFT by using a finite series to represent the integral, where
t = k ¢t, f = n/T, dt =¢t, and ¢t = T/N. Then
N-1
W w (f)| f=n/T L a w(k ¢t)e -j(2p/N)nk ¢t
k=0
(2–182)
Comparing this result with Eq. (2–176), we obtain the relation between the CFT and DFT; that is,
W w (f)| f=n/T L¢t X(n)
(2–183)