Signals and Systems - Electrical Engineering

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

we have that

Px=

1

8

∫^4

0

cos(πt+π/ 2 )dt+

1

8

∫^4

0

dt= 0 +0.5=0.5

The first integral is the area of the sinusoid over two of its periods, thus zero. So we have thatx(t)

is a finite-power but infinite-energy signal, whiley(t)andz(t)are finite-power and finite-energy

signals. n

nExample 1.12

Consider an aperiodic signalx(t)=e−at,a>0, fort≥0 and zero otherwise. Find the energy and

the power of this signal and determine whether the signal is finite energy, finite power, or both.

Solution

The energy ofx(t)is given by

Ex=

∫∞

0

e−^2 atdt=

1

2 a

<∞

for any value ofa>0. The power ofx(t)is then zero. Thus,x(t)is a finite-energy and finite-power

signal. n

nExample 1.13

Consider the following analog signal, which we call acausalsinusoid because it is zero fort<0:

x(t)=

{

2 cos( 4 t−π/ 4 ) t≥ 0

0 otherwise

This is the kind of signal that you would get from a signal generator that is started at a certain

initial time (in this case 0) and that continues until the signal generator is switched off (in this

case possibly infinity). Determine if this signal is finite energy, finite power or both.

Solution

Clearly, the analog signalx(t)has infinite energy:

Ex=

∫∞

−∞

x^2 (t)dt

=

∫∞

0

4 cos^2 ( 4 t−π/ 4 )dt→∞