Systems, Sampling and Quantization
3
sampling
sampling interval
spectrum
images
Sampling Theorem
In order to obtain the signal coming out from a linear system it is sufficient
to apply the convolution operator between the input signal and the impulse
response.
1.2
The Sampling Theorem
In order to perform any form of processing by digital computers, the signals
must be reduced to discrete samples of a discrete-time domain. The operation
that transforms a signal from the continuous time to the discrete time is called
sampling, and it is performed by picking up the values of the continuous-time
signal at time instants that are multiple of a quantity T , called the sampling
interval. The quantity F
s
= 1/T is called the sampling rate.
The presentation of a detailed theory of sampling would take too much space
and it would become easily boring for the readership of this book. For a more
extensive treatment there are many excellent books readily available, from the
more rigorous [66, 65] to the more practical . Luckily, the kernel of the theory
can be summarized in a few rules that can be easily understood in terms of the
frequency-domain interpretation of signals and systems.
The first rule is related to the frequency representation of discrete-time vari-
ables by means of the Fourier transform, defined in appendix A.8.3 as a special-
ization of the Z transform:
Rule 1.1 The Fourier transform of a function of discrete variable is a function
of the continuous variable , periodic
3
with period 2.
The second rule allows to treat the sampled signals as functions of discrete
variable:
Rule 1.2 Sampling a continuous-time signal x(t) with sampling interval T pro-
duces a function ^
x(n) = x(nT ) of the discrete variable n.
If we call spectrum of a signal its Fourier-transformed counterpart, the fun-
damental rule of sampling is the following:
Rule 1.3 Sampling a continuous-time signal with sampling rate F
s
produces a
discrete-time signal whose frequency spectrum is a periodic replication of the
spectrum of the original signal, and the replication period is F
s
. The Fourier
variable for functions of discrete variable is converted into the frequency vari-
able f (in Hz) by means of
= 2f T =
2f
F
s
.
(7)
Fig. 1 shows an example of frequency spectrum of a signal sampled with
sampling rate F
s
. In the example, the continuous-time signal had all and only
the frequency components between -F
b
and F
b
. The replicas of the original
spectrum are sometimes called images.
Given the simple rules that we have just introduced, it is easy to understand
the following Sampling Theorem, introduced by Nyquist in the twenties and
popularized by Shannon in the forties:
3
This periodicity is due to the periodicity of the complex exponential of the Fourier trans-
form.
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