Signals and Systems - Electrical Engineering

7.2 Uniform Sampling 429

Bandlimited or Not?

The following, taken from David Slepian’s paper “On Bandwidth” [66], clearly describes the uncertainty about bandlimited

signals:

The Dilemma—Are signals really bandlimited? They seem to be, and yet they seem not to be.

On the one hand, a pair of solid copper wires will not propagate electromagnetic waves at optical frequencies and

so the signals I receive over such a pair must be bandlimited. In fact, it makes little physical sense to talk of energy

received over wires at frequencies higher than some finite cutoffW, say 10^20 Hz. It would seem, then, that signals

must be bandlimited.

On the other hand, however, signals of limited bandwithWare finite Fourier transforms,

s(t)=

∫W

−W

e^2 πiftS(f)df

and irrefutable mathematical arguments show them to be extremely smooth. They possess derivatives of all orders.

Indeed, such integrals are entire functions oft, completely predictable from any little piece, and they cannot vanish

on anytinterval unless they vanish everywhere. Such signals cannot start or stop, but must go on forever. Surely

real signalsstart and stop, and they cannot be bandlimited!

Thus we have a dilemma: to assume that real signals must go on forever in time (a consequence of bandlimit-

edness) seems just as unreasonable as to assume that real signals have energy at arbitrary high frequencies (no

bandlimitation). Yet one of these alternatives must hold if we are to avoid mathematical contradiction, for either

signals are bandlimited or they are not: there is no other choice. Which do you think they are?

Remarks

n In practice, the exact recovery of the original signal may not be possible for several reasons. One could be
that the continuous-time signal is not exactly band limited, so that it is not possible to obtain a maximum
frequency causing frequency aliasing in the sampling. Second, the sampling is not done exactly at uniform
times—random variation of the sampling times may occur. Third, the filter required for the exact recovery
is an ideal low-pass filter, which in practice cannot be realized; only an approximation is possible. Although
this indicates the limitations of sampling, in most cases where: (1) the signal is band limited or approx-
imately band limited, (2) the Nyquist sampling rate condition is satisfied in the sampling, and (3) the
reconstruction filter approximates well the ideal low-pass filter, the recovered signal closely approximates
the original signal.
n For signals that do not satisfy the band-limitedness condition, one can obtain an approximate signal that
satisfies that condition. This is done by passing the non-band-limited signal through an ideal low-pass
filter. The filter output is guaranteed to have as maximum frequency the cut-off frequency of the filter
(see Figure 7.4). Because of the low-pass filtering, the filtered signal is a smoothed version of the original
signal—high frequencies of the signal have been removed. The low-pass filter is called anantialiasing
filter, since it makes the approximate signal band limited, thus avoiding aliasing in the frequency domain.
n In applications, the cut-off frequency of the antialiasing filter is set according to prior knowledge. For
instance, when sampling speech, it is known that speech has frequencies ranging from about 100 Hz to