Signals and Systems - Electrical Engineering

7.2 Uniform Sampling 427

nExample 7.2

Consider the following signals:

(a) x 1 (t)=u(t+0.5)−u(t−0.5)

(b) x 2 (t)=e−tu(t)

Determine if they are band limited or not. If not, determine the frequency for which the energy

of the non-band-limited signal corresponds to 99% of its total energy and use this result to

approximate its maximum frequency.

Solution

(a) The signalx 1 (t)=u(t+0.5)−u(t−0.5)is a unit pulse signal. Clearly, this signal can be easily

sampled by choosing any value ofTs<<1. For instance,Ts=0.01 sec would be a good value,

giving a discrete-time signalx 1 (nTs)=1, for 0≤nTs=0.01n≤1 or 0≤n≤100. There seems

to be no problem in sampling this signal; however, we have that the Fourier transform ofx 1 (t),

X 1 ()=

ej0.5−e−j0.5

j

=

sin(0.5)

0.5

does not have a maximum frequency and sox 1 (t)is not band limited. Thus, any chosen value

ofTswill cause aliasing. Fortunately, the values of the sinc function go fast to zero, so that

one could compute an approximate maximum frequency that covers 99% of the energy of

the signal.

Using Parseval’s energy relation we have that the energy ofx 1 (t)(the area underx^21 (t)) is 1

and if we wish to find a valueM, such that 99% of this energy is in the frequency band

[−M,M], we need to look for the limits of the following integral so it equals 0.99:

0.99=

1

2 π

∫M

−M

[

sin(0.5)

0.5

] 2

d

Since this integral is difficult to find analytically, we use the following script in MATLAB to

approximate it.

%%%%%%%%%%%%%%%%%%%%%

% Example 7.2 --- Parseval’s relation and sampling

%%%%%%%%%%%%%%%%%%%%%

syms W

for k = 1:23;

E(k) = int((sin(0.5*W)/(0.5*W))ˆ2,0,k*pi)/pi

if E(k)>= 0.9900,

k

return

end

end