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Sec. 2–7 Bandlimited Signals and Noise 95
or
q
W s (f) = 1 (2–173)
T a W(f - nf s )
s n=-q
As is exemplified in Fig. 2–18b, the spectrum of the impulse sampled signal is the spectrum
of the unsampled signal that is repeated every f s Hz, where f s is the sampling frequency
(samples/sec). † This quite significant result is one of the basic principles of digital signal processing
(DSP).
Note that this technique of impulse sampling may be used to translate the spectrum of a
signal to another frequency band that is centered on some harmonic of the sampling frequency.
A more general circuit that can translate the spectrum to any desired frequency band
is called a mixer. Mixers are discussed in Sec. 4–11.
If f s Ú 2B, as illustrated in Fig. 2–18, the replicated spectra do not overlap, and the
original spectrum can be regenerated by chopping W s (f) off above f s /2. Thus, w(t) can be reproduced
from w s (t) simply by passing w s (t) through an ideal low-pass filter that has a cutoff
frequency of f c = f s /2, where f s Ú 2B.
If f s 6 2B (i.e., the waveform is undersampled), the spectrum of w s (t) will consist of
overlapped, replicated spectra of w(t), as illustrated in Fig. 2–19. ‡ The spectral overlap or tail inversion,
is called aliasing or spectral folding. § In this case, the low-pass filtered version of w s (t)
will not be exactly w(t). The recovered w(t) will be distorted because of the aliasing. This distortion
can be eliminated by prefiltering the original w(t) before sampling, so that the prefiltered
w(t) has no spectral components above |f| = f s /2. The prefiltering still produces distortion on
the recovered waveform because the prefilter chops off the spectrum of the original w(t) above
|f| = f s /2. However, from Fig. 2–19, it can be shown that if a prefilter is used, the recovered
waveform obtained from the low-pass version of the sample signal will have one-half of the error
energy compared to the error energy that would be obtained without using the presampling filter.
A physical waveform w(t) has finite energy. From Eqs. (2–42) and (2–43), it follows
that the magnitude spectrum of the waveform, |W(f)|, has to be negligible for |f| 7 B, where
B is an appropriately chosen positive number. Consequently, from a practical standpoint, the
physical waveform is essentially bandlimited to B Hz, where B is chosen to be large enough
so that the error energy is below some specified amount.
Dimensionality Theorem
The sampling theorem may be restated in a more general way called the dimensionality theorem
(which is illustrated in Fig. 2–17).
THEOREM. When BT 0 is large, a real waveform may be completely specified by
N = 2BT 0 (2–174)
† In Chapter 3, this result is generalized to that of instantaneous sampling with a pulse train consisting of
pulses with finite width and arbitrary shape (instead of impulses). This is called pulse amplitude modulation (PAM)
of the instantaneous sample type.
‡ For illustrative purposes, we assume that W(f) is real.
§ f s /2 is the folding frequency, where f s is the sampling frequency. For no aliasing, f s 7 2B is required, where
2B is the Nyquist frequency.