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Signals and Spectra Chap. 2
Bandlimited Waveforms
DEFINITION.
DEFINITION.
A waveform w(t) is said to be (absolutely) bandlimited to B hertz if
W(f) = F[w(t)] = 0, for |f| Ú B
(2–156)
A waveform w(t) is (absolutely) time limited if
w(t) = 0, for |t| 7 T
(2–157)
THEOREM. An absolutely bandlimited waveform cannot be absolutely time limited,
and vice versa.
This theorem is illustrated by the spectrum for the rectangular pulse waveform of Example
2–6. A relatively simple proof of the theorem can be obtained by contradiction [Wozencraft
and Jacobs, 1965].
The theorem raises an engineering paradox. We know that a bandlimited waveform cannot
be time limited. However, we believe that a physical waveform is time limited because the
device that generates the waveform was built at some finite past time and the device will
decay at some future time (thus producing a time-limited waveform). This paradox is resolved
by realizing that we are modeling a physical process with a mathematical model and perhaps
the assumptions in the model are not satisfied—although we believe them to be satisfied. That
is, there is some uncertainty as to what the actual time waveform and corresponding spectrum
look like—especially at extreme values of time and frequency—owing to the inaccuracies of
our measuring devices. This relationship between the physical process and our mathematical
model is discussed in an interesting paper by David Slepian [1976]. Although the signal may
not be absolutely bandlimited, it may be bandlimited for all practical purposes in the sense
that the amplitude spectrum has a negligible level above a certain frequency.
Another interesting theorem states that if w(t) is absolutely bandlimited, it is an analytic
function. An analytic function is a function that possesses finite-valued derivatives when they
are evaluated for any finite value of t. This theorem may be proved by using a Taylor series
expansion [Wozencraft and Jacobs, 1965].
Sampling Theorem
The sampling theorem is one of the most useful theorems, since it applies to digital communication
systems. The sampling theorem is another application of an orthogonal series expansion.
SAMPLING THEOREM.
-q 6 t 6qby
where
w(t) =
Any physical waveform may be represented over the interval
n=q
q
a n = f s w(t) sin {pf s[t - (n/f s )]}
dt
L pf s [t - (n/f s )]
-q
a
n=-q
a sin {pf s[t - (n/f s )]}
n
pf s [t - (n/f s )]
(2–158)
(2–159)