Digital transmission of information and digital signal
processing all require signals to first be "acquired" by a
computer. One of the most amazing and useful results in
electrical engineering is that signals can be converted from a
function of time into a sequence of numbers without
error: We can convert the numbers back into the
signal with (theoretically) no
error. Harold Nyquist, a Bell Laboratories engineer, first derived
this result, known as the Sampling Theorem, in the
1920s. It found no real application back then.
Claude
Shannon,
also at Bell Laboratories, revived the result once computers
were made public after World War II.
The sampled version of the analog signal
st
s
t
is
sn
T
s
s
n
T
s
,
with
T
s
T
s
known as the sampling interval. Clearly, the
value of the original signal at the sampling times is
preserved; the issue is how the signal values
between the samples can be
reconstructed since they are lost in the sampling
process. To characterize sampling, we approximate it as
the product
xt=st
P
T
s
t
x
t
s
t
P
T
s
t
,
with
P
T
s
t
P
T
s
t
being the periodic pulse signal. The resulting signal, as
shown in Figure 1, has nonzero
values only during the time intervals
n
T
s
−Δ2
n
T
s
+Δ2
n
T
s
Δ
2
n
T
s
Δ
2
,
n∈…101…
n
…
1
0
1
…
.
For our purposes here, we center the periodic pulse signal
about the origin so that its Fourier series coefficients are
real (the signal is even).
p
T
s
t=∑k=−∞∞
c
k
ei2πkt
T
s
p
T
s
t
k
c
k
2
k
t
T
s
(1)
c
k
=sinπkΔ
T
s
πk
c
k
k
Δ
T
s
k
(2)
If the properties of
st
s
t
and the periodic pulse signal are chosen properly, we can
recover
st
s
t
from
xt
x
t
by filtering.
To understand how signal values between the samples can be
"filled" in, we need to calculate the sampled signal's
spectrum. Using the Fourier series representation of the
periodic sampling signal,
xt=∑
k
=−∞∞
c
k
ei2πkt
T
s
st
x
t
k
c
k
2
k
t
T
s
s
t
(3)
Considering each term in the sum separately, we need to know
the spectrum of the product of the complex exponential and the
signal. Evaluating this transform directly is quite easy.
∫−∞∞stei2πkt
T
s
e−(i2πft)d
t
=∫−∞∞ste−(i2π(f−k
T
s
)t)d
t
=S(f−k
T
s
)
t
s
t
2
k
t
T
s
2
f
t
t
s
t
2
f
k
T
s
t
S
f
k
T
s
(4)
Thus, the spectrum of the sampled signal consists of weighted
(by the coefficients
c
k
c
k
)
and delayed versions of the signal's spectrum
(
Figure 2).
Xf=∑
k
=−∞∞
c
k
S(f−k
T
s
)
X
f
k
c
k
S
f
k
T
s
(5)
In general, the terms in this sum overlap each other in the
frequency domain, rendering recovery of the original signal
impossible. This unpleasant phenomenon is known as
aliasing.
If, however, we satisfy two conditions:

The signal
st
s
t
is bandlimited—has power in a
restricted frequency range—to
W
W
Hz, and

the sampling interval
T
s
T
s
is small enough so that the individual components in the
sum do not overlap—
T
s
<1/2W
T
s
12
W
,
aliasing will not occur. In this delightful case, we can
recover the original signal by lowpass filtering
xt
x
t
with a filter having a cutoff frequency equal to
W
W Hz.
These two conditions ensure the ability to recover a
bandlimited signal from its sampled version: We thus have the
Sampling Theorem.
The Sampling Theorem (as stated) does not mention the
pulse width
Δ
Δ.
What is the effect of this parameter on our ability to
recover a signal from its samples (assuming the Sampling
Theorem's two conditions are met)?
The only effect of pulse duration is to unequally weight
the spectral repetitions. Because we are only concerned
with the repetition centered about the origin, the pulse
duration has no significant effect on recovering a signal
from its samples.
The frequency
12
T
s
1
2
T
s
,
known today as the Nyquist frequency and the
Shannon sampling frequency, corresponds to the
highest frequency at which a signal can contain energy and
remain compatible with the Sampling Theorem. Highquality
sampling systems ensure that no aliasing occurs by
unceremoniously lowpass filtering the signal (cutoff frequency
being slightly lower than the Nyquist frequency) before
sampling. Such systems therefore vary the
antialiasing filter's cutoff frequency
as the sampling rate varies. Because such quality features
cost money, many sound cards do not have
antialiasing filters or, for that matter, postsampling
filters. They sample at high frequencies, 44.1 kHz for
example, and hope the signal contains no frequencies above the
Nyquist frequency (22.05 kHz in our example). If, however, the
signal contains frequencies beyond the sound card's Nyquist
frequency, the resulting aliasing can be impossible to remove.
To gain a better appreciation of aliasing, sketch the
spectrum of a sampled square wave. For simplicity
consider only the spectral repetitions centered at
−1
T
s
1
T
s
,
0
0,
1
T
s
1
T
s
.
Let the sampling interval
T
s
T
s
be 1; consider two values for the square wave's period:
3.5 and 4. Note in particular where the spectral lines go
as the period decreases; some will move to the left and
some to the right. What property characterizes the ones
going the same direction?
The square wave's spectrum is shown by the bolder set of
lines centered about the origin. The dashed lines
correspond to the frequencies about which the spectral
repetitions (due to sampling with
T
s
=1
T
s
1
)
occur. As the square wave's period decreases, the negative
frequency lines move to the left and the positive
frequency ones to the right.
If we satisfy the Sampling Theorem's conditions, the signal
will change only slightly during each pulse. As we narrow the
pulse, making
Δ
Δ
smaller and smaller, the nonzero values of the signal
st
p
T
s
t
s
t
p
T
s
t
will simply be
sn
T
s
s
n
T
s
,
the signal's samples. If indeed the Nyquist
frequency equals the signal's highest frequency, at least two
samples will occur within the period of the signal's highest
frequency sinusoid. In these ways, the sampling signal
captures the sampled signal's temporal variations in a way
that leaves all the original signal's structure intact.
What is the simplest bandlimited signal? Using this
signal, convince yourself that less than two
samples/period will not suffice to specify it. If the
sampling rate
1
T
s
1
T
s
is not high enough, what signal would your resulting
undersampled signal become?
The simplest bandlimited signal is the sine wave. At the
Nyquist frequency, exactly two samples/period would
occur. Reducing the sampling rate would result in fewer
samples/period, and these samples would appear to have
arisen from a lower frequency sinusoid.
"Electrical Engineering Digital Processing Systems in Braille."