EDITOR: B. SOMANATHAN NAIR
1. INTRODUCTION
In analog-to-digital (A/D) conversion, analog signals
are first sampled, then held, and finally converted into digital signals. Rate
of sampling is computed based on the Nyquist
sampling theorem, which states that:
An analog signal can
be recovered from its samples completely if and only if sampling is done under
the condition that sampling frequency used is equal to, or greater than, twice
the maximum frequency of that signal.
In mathematical form, the
sampling theorem may be stated as
fs ³ 2fm (1)
where fs
is the sampling frequency and fm
is the maximum frequency of input analog signal.
To
illustrate the sampling theorem, consider the sampling of analog audio signals
in the range of 100 Hz to 4 kHz. In this case, the maximum audio frequency fm = 4 kHz. Therefore the
sampling frequency required is
fs ³ 8 kHz
2. PROOF OF SAMPLING
THEOREM
We find that sampling is a convolution operation.
This operation may be expressed mathematically as
y(t) = x(t)
p(t) (2)

where x(t) is the input analog signal, p(t)
is the sampling pulses, and y(t) is the sampled output. The star symbol represents the
convolution operation. Convolution in time domain can be seen to be equal to
multiplication in frequency domain. In the Fourier-transform domain, we express
the convolution related to sampling as
Y(w) = X(w)P(w) (3)
where Y(w) is the Fourier-transform y(t),
X(w) is the Fourier-transform of x(t), and P(w) is the Fourier-transform
of p(t). Figure 1 shows an
example of a time-domain input signal x(t), and Fig. 2 shows the frequency
components X(w) in that signal, obtained
by Fourier transformation of x(t). It may be noted here that x(t)
is the waveform that we observe on an oscilloscope and X(w) is the pattern that we observe on a spectrum analyzer. From Fig. 2, we
observe that the frequency components are discrete signals with each signal
component having finite amplitude. We have drawn a blue-colored dotted line as
the envelope covering the samples in the figure.
As
stated above, for sampling, we use an impulse- (delta-) function train p(t), defined as
p(t) = d(t-nT) (4)
where n =
0, 1, 2, … , and T is any unit time period.
where 2π/Ts = ws is the amplitude of the sampled pulses. From Eq. (5), we
find that Fourier transform of a time-domain impulse train is a
frequency-domain impulse train. Substituting Eq. (5) into Eq. (3) yields
Equation (6) may be expanded as
Y(w) = ws [X(w)d (w) + X(w)d(w -ws) + X(w)d(w -2ws)
X(w)d(w -nws) +

X(w)d(w +ws) + X(w)d( w +2ws)
X(w)d(w +nws)] (7)

From Eq. (7), we find that sampling operation has
produced a sample of order zero and an infinite
number of side-band samples of orders 1, 2, 3, …, n. We also notice that the zeroth-order sample is located at w = 0, the first-order
samples at w = ±
ws, the second-order samples
at w = ±
2ws, and so on. The amplitudes
of all these samples are equal to ws [X(w)d(w)]. Figures 3, 4, and 5
give the pictorial representations of the sampling operation, where we have
used a compressed version of the
envelope (with a maximum frequency of wm) of X(w) for convenience.
In Fig. 3, as stated above,
we find that the product function ws X(w) is located at w = ws, ±
2ws, ±
3ws, and so on. The shapes of envelope,
shown in Fig. 3, as stated above, represent the product term wsX(w), and are arbitrarily chosen here [the actual shape of these is dependent
on the actual shape of the term X(w)]. We also notice that the
bandwidth of each of the samples is equal to 2wm, where wm is the maximum frequency of the sampled signal.
From Fig. 3, we find that the
samples remain just touching each
other, if ws = 2wm. In Fig. 4, we notice that
the samples are isolated from each other and are not touching each other. This occurs
for ws > 2wm. under these two
conditions, the original signal may be recovered from the samples by using a
low-pass filter with cut-off frequency ws = 2wm. it may be noticed that forws > 2wm, there exists a guard band between each of the samples;
this helps in designing the low-pass filter because cut-off frequency is not
sharp. In the figures, the sample corresponding to the original signal is shown
in red lines. It is this sample that we have to recover. The rest of the
samples to be neglected are shown in blue lines.
Figure 5 shows the condition
when ws < 2wm. Under this condition, the
samples overlap each other and separating them becomes difficult. This situation
is known as aliasing.
From the explanations given
above, we conclude that original signal can be recovered from its samples if we
sample the signal at the rate of ws ≥ 2wm. But this is the statement of the sampling theorem.
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