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Sec. 2–8 Discrete Fourier Transform 99
1.0
w(t)
|W(f) |
Delta function
1
with weight –– 2
0 – f s
0
f s
t
f
(a) Waveform and Its Spectrum
w w (t)
T
–– 2
|W w (f) |
T
0
t
– f s f s
f
(b) Windowed Waveform and Its Spectrum
w sw (t)
T
–––
2t
|W sw (f) |
t
t
– f s
1
1
– –– fs
––2 f
f s
2
s
f
(c) Sampled Windowed Waveform and Its Spectrum ( f s 1/t )
w psw (t)
T
–––
2t
|W sw (f) | or t | X(n) |
t
T 0
– f s
1
– –– f 1 0
f fs
––2 f s
2
s
(d) Periodic Sampled Windowed Waveform and Its Spectrum (f 0 1/T)
Figure 2–20
Comparison of CFT and DFT spectra.
1 2 3 4 5 6 7 8
f
n
where f = n/T and ¢t = T/N. The sample values used in the DFT computation are
x(k) = w(k ¢t), as shown in the left part of Fig. 2–20c. Also, because e -j(2p/N)nk of Eq. (2–176)
is periodic in n—in other words, the same values will be repeated for n = N, N + 1, ...
as were obtained for n = 0, 1, ...—it follows that X(n) is periodic (although only the first N values
are returned by DFT computer programs, since the others are just repetitions). Another way
of seeing that the DFT (and the IDFT) are periodic is to recognize that because samples are used,
the discrete transform is an example of impulse sampling, and consequently, the spectrum must
be periodic about the sampling frequency, f s = 1/¢t = N/T (as illustrated in Fig. 2–18 and