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 [67]. 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
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.