Signals and Systems - Electrical Engineering

1.3 Continuous-Time Signals 83

Although this signal has infinite energy, it has finite power. LettingT=NT 0 whereT 0 is the period

of 2 cos( 4 t−π/ 4 )(orT 0 = 2 π/4), then its power is

Px=lim

T→∞

1

2 T

∫T

−T

x^2 (t)dt=lim

T→∞

1

2 T

∫T

0

x^2 (t)dt

= lim

N→∞

N

2 NT 0

∫T^0

0

x^2 (t)dt=

1

2 T 0

∫T^0

0

4 cos^2 ( 4 t−π/ 4 )dt

which is a finite value and therefore the signal has finite power but infinite energy. n

As we will see later in the Fourier series representation, any periodic signal is representable as a pos-
sibly infinite sum of sinusoids of frequencies multiples of the fundamental frequency of the periodic
signal being represented. These frequencies are said to beharmonically related, and for this case the
power of the signal is shown to be the sum of the power of each of the sinusoidal components—that
is, there is superposition of the power. This superposition is still possible when a sum of sinusoids
creates a nonperiodic signal. This is illustrated in Example 1.14.

nExample 1.14

Consider the signals x(t)=cos( 2 πt)+cos( 4 πt)and y(t)=cos( 2 πt)+cos( 2 t), −∞<t<∞.

Determine if these signals are periodic, and if so, find their periods. Compute the power of these

signals.

Solution

The sinusoids cos( 2 πt)and cos( 4 πt)periodsT 1 =1 andT 2 = 1 /2, sox(t)is periodic sinceT 1 /T 2 =

2 with periodT 1 = 2 T 2 =1. The two frequencies are harmonically related. The sinusoid cos( 2 t)

has as periodT 3 =π. Therefore, the ratio of the periods of the sinusoidal components ofy(t)is

T 1 /T 3 = 1 /π, which is not rational, and soy(t)is not periodic and the frequencies 2πand 2 are

not harmonically related.

Using the trigonometric identities

cos^2 (θ)=

1

2

( 1 +cos( 2 θ))

cos(α)cos(β)=

1

2

(cos(α+β)+cos(α−β))

we have that

x^2 (t)=cos^2 ( 2 πt)+cos^2 ( 4 πt)+2 cos( 2 πt)cos( 4 πt)

= 1 +

1

2

cos( 4 πt)+

1

2

cos( 8 πt)+cos( 6 πt)+cos( 2 πt)