Page 710 - The Mechatronics Handbook

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It is observed that X s ( f ) consists of a scaled and periodically repeated version of X( f ) with period F s .
As shown in Fig. 23.9(c), it is obvious that when F s − F M ≥ F M , there is no overlapping of the spectra and
the signal x(t) can be recovered completely from x s (t). However, if F s − F M < F M the replicas of X(f ) will
overlap, resulting in a distorted spectrum and as such, x(t) can no longer be recovered from its sampled
version. Consequently, in order to recover x(t) from its samples, the sampling frequency should be such
that

F s – F M ≥ F M

that is,

F s ≥ 2F M

This is called the Nyquist sampling theorem. The minimum sampling frequency F s = 2F M is called the
Nyquist frequency. Sampling a signal at less than the Nyquist frequency results in a spectral distortion
termed aliasing. Furthermore, sampling a signal at a frequency of at least Nyquist frequency implies that
an ideal low-pass filter (LPF) with a gain of 1/F s and cutoff frequency F c can be used to recover its original
spectrum, where F M ≤ F c ≤ F s – F M .
Suppose we want to reconstruct x(t) from its samples. Assume that X( f ) is the spectrum of x(nT s ),
with no aliasing, as shown in Fig. 23.9(c). Thus,

1
 ----Xf(), f ≤ F s
----
X a f() =  F s 2 (23.36)

 f > F s
 0, ----
2
Note that

∞
X a f() = ∑ xnT s )e – j2πnfT s (23.37)
(
n=∞
–
so that its inverse Fourier transform is given by

∞ F /2 j2πft−nT )
(
1
(
x a t() = ---- ∑ xnT s ) ∫ s e s df
s
F s n=∞– – F /2
(23.38)
∞ sin [ π tnT s )/T s ]
(
–
= ∑ xnT s )---------------------------------------------
(
(
–
n=∞ π tnT s )/T s
–
This is the formula for the reconstruction of x(t) from its samples. That is, x(t) is generated by
multiplying the appropriately shifted function g(t) = sinc(tF s ) by the corresponding samples of x(nT s ).
Practical Sampling 8–10
The above discussion on sampling is based on the idealized models of periodic impulse sampling and
bandlimited interpolation. In practice, CT signals are not precisely bandlimited just as impulse signals
and ideal low-pass filters do not exist physically. Figure 23.10 represents the block diagram for the
conversion of continuous signals into their discrete forms. The continuous-time signal is prefiltered,
sampled, quantized, and finally encoded into finite-length words of, say, b bits. The prefilter, which is also
called anti-aliasing filter (AAF), is a low-pass filter that is needed to limit the input signal bandwidth to
F s /2 prior to sampling to avoid aliasing. In practice this filter will possess non-ideal characteristics, hence

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