Signals and Systems - Electrical Engineering

426 CHAPTER 7: Sampling Theory

0 1 2 3

− 2

− 1

0

1

2

x(

t),

x(0.2

n)

(a)

0 1 2 3

− 2

− 1

0

1

2

x(

t),

x(0.4

n)

(b)

0 1 2 3

− 2

− 1

0

1

2

t

x(

t),

x(0.5

n)

(c)

0 1 2 3

− 2

− 1

0

1

2

t

x(

t),

x(

n)

(d)

FIGURE 7.3

Sampling ofx(t)=2 cos( 2 πt+π/ 4 )with sampling periods (a)Ts=0.2, (b)Ts=0.4, (c)Ts=0.5, and (d)Ts= 1

sec/sample.

In this case it can be shown that the sampled signal repeats periodically every two samples, since

x((n+ 2 )Ts)=x(nTs), which can be easily checked. According to the Nyquist sampling rate condi-

tion, this is the minimum number of samples per period allowed before we start having aliasing.

In fact, if we lets=max= 2 πcorresponding to the sampling periodTs=1, the samples in

Equation (7.13) are

x(nTs)=2 cos( 2 πn+π/ 4 )=2 cos(π/ 4 )=

√

2

and the sampled signal is

√

2 δTs(t). WithTs=1, the sampled signal cannot be possibly con-

verted back into an analog sinusoid. Thus, we have lost the information provided by the sinusoid.

Undersampling (getting too few samples per unit time) has changed the nature of the original

signal.

We use MATLAB to plot the continuous signal and four sampled signals (see Figure 7.3) for differ-

ent values ofTs. Clearly, whenTs=1 sec/sample there is no similarity between the analog and the

discrete signals due to frequency aliasing. n