Signals and Systems - Electrical Engineering

114 C H A P T E R 1: Continuous-Time Signals

(a) Carefully plot the signaly(t)=dx(t)/dt, the derivative ofx(t).

(b)Can you compute

z(t)=

∫∞

−∞

[x(t)−0.5]dt

If so, what is it equal to? If not, explain why.

(c)Isx(t)a finite-energy signal? How abouty(t)?

1.18. Complex exponentials

For a complex exponential signalx(t)= 2 ej^2 πt:

(a) Determine its analog frequency 0 in rad/sec and its analog frequencyfin hertz. Then find the signal’s

period.

(b)Supposey(t)=ejπt. Would the sum of these signalsz(t)=x(t)+y(t)also be periodic? If so, what is

the period ofz(t)?

(c)Suppose we then generate a signalv(t)=x(t)y(t), with thex(t)andy(t)signals given before. Isv(t)

periodic? If so, what is its period?

1.19. Full-wave rectified signal—MATLAB

Consider the full-wave rectified signal

y(t)=|sin(πt)| −∞<t<∞

part of which is shown in Figure 1.21.

FIGURE 1.21

Problem 1.19.

− 2 − 1 0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

t

y(

t)

(a) As a periodic signal,y(t)does not have finite energy, but it has a finite powerPy. Find it.

(b)It is always useful to get a quick estimate of the power of a periodic signal by finding a bound for the

signal squared. Find a bound for|y(t)|^2 and show thatPy< 1.

(c)Use symbolic MATLAB to check if the full-wave rectified signal has finite power and if that value

coincides with thePyyou found above. Plot the signal and provide the script for the computation of

the power. How does it coincide with the analytical result?