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92
Signals and Spectra Chap. 2
f s /2
= 1
(2–165)
(f s ) 2 e -j2m(n-m)(f/fs) df = 1 d
L-f s /2
f nm
s
Thus, the w n (t), as given by (2–161), are orthogonal functions with K n = 1/f s . Using Eq.
(2–84), we see that Eq. (2–159) follows. Furthermore, we will show that Eq. (2–160) follows
for the case of w(t) being absolutely bandlimited to B hertz with f s Ú 2B. Using Eq. (2–84)
and Parseval’s theorem, Eq. (2–40), we get
q
a n = f s w(t)w n
*(t) dt
L
-q
q
= f s W(f)£ n
*(f) dt
L
-q
(2–166)
Substituting (2–164) yields
f s /2
a n = W(f)e +j2pf(n/f s)
df
L
-f s /2
(2–167)
But because W(f) is zero for |f| 7 B, where B … f s /2, the limits on the integral may be
extended to (-q, q) without changing the value of the integral. This integral with infinite
limits is just the inverse Fourier transform of W(f) evaluated at t = n/f s . Consequently,
a n = w(n/f s ), which is (2–160).
From Eq. (2–167), it is obvious that the minimum sampling rate allowed to reconstruct
a bandlimited waveform without error is given by
(2–168)
This is called the Nyquist frequency.
Now we examine the problem of reproducing a bandlimited waveform by using N sample
values. Suppose that we are only interested in reproducing the waveform over a T 0 -s
interval as shown in Fig. 2–17a. Then we can truncate the sampling function series of
Eq. (2–158) so that we include only N of the wn(t) functions that have their peaks within the
T 0 interval of interest. That is, the waveform can be approximately reconstructed by using N
samples. The equation is
w(t) L
(f s ) min = 2B
n=n 1 +N
a
n=n 1
a n w n (t)
(2–169)
where the { w n (t)} are described by Eq. (2–161). Figure 2–17b shows the reconstructed waveform
(solid line), which is obtained by the weighted sum of time-delayed (sin x)/x waveforms
(dashed lines), where the weights are the sample values a n = w(n/f s ) denoted by the dots.
The waveform is bandlimited to B hertz with the sampling frequency f s Ú 2B. The sample
values may be saved, for example, in the memory of a digital computer, so that the waveform
may be reconstructed at a later time or the values may be transmitted over a communication
system for waveform reconstruction at the receiving end. In either case, the waveform may be