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Sec. 2–7 Bandlimited Signals and Noise 91
and f s is a parameter that is assigned some convenient value greater than zero.
Furthermore, if w(t) is bandlimited to B hertz and f s Ú 2B, then Eq. (2–158) becomes
the sampling function representation, where
a n = w(n/f s )
(2–160)
That is, for f s Ú 2B, the orthogonal series coefficients are simply the values of the
waveform that are obtained when the waveform is sampled every 1/f s seconds.
The series given by Eqs. (2–158) and (2–160) is sometimes called the cardinal series. It
has been known by mathematicians since at least 1915 [Whittaker, 1915] and to engineers
since the pioneering work of Shannon, who connected the series with information theory
[Shannon, 1949]. An excellent tutorial paper on this topic has been published in the
Proceedings of the IEEE [Jerri, 1977].
Example 2–19 SAMPLING THEOREM FOR A RECTANGULAR PULSE
Using Eq. (2–158), evaluate the waveform generated by the Sampling Theorem for the case of
samples taken from a rectangular pulse. See Example2_19.m for the solution.
PROOF OF THE SAMPLING THEOREM.
We need to show that
w n (t) = sin {pf s[t - (n/f s )]}
(2–161)
pf s [t - (n/f s )]
form a set of the orthogonal functions. From Eq. (2–77), we must demonstrate that Eq.
(2–161) satisfies
L
q
-q
w n (t)w m
*(t)dt = K n d nm
(2–162)
Using Parseval’s theorem, Eq. (2–40), we see that the left side becomes
L
q
-q
q
w n (t)w m
*(t) dt = £ n (f)£ m
*(f) df
L
-q
(2–163)
where
£ n (f) = [w n (t)] = 1 f s
ßa f f s
be -j2p(nf/f s)
(2–164)
Hence, we have
L
q
-q
w n (t)w m
*(t) dt =
1
(f s ) 2 L
q
-q
cßa f 2
bd e -j2p(n-m)f/f s
df
f s