Signals and Systems - Electrical Engineering

1.3 Continuous-Time Signals 81

n An analog signal x(t)is said to beabsolutely integrableif x(t)satisfies the condition

∫∞

−∞

|x(t)|dt<∞ (1.14)

nExample 1.11

Find the energy and the power of the following:

(a) The periodic signalx(t)=cos(πt/ 2 +π/ 4 ).

(b) The complex signaly(t)=( 1 +j)ejπt/^2 , for 0≤t≤10 and zero otherwise.

(c) The pulsez(t)=1, for 0≤t≤10 and zero otherwise.

Determine whether these signals are finite energy, finite power, or both.

Solution

The energy in these signals is computed as follows:

Ex=

∫∞

−∞

cos^2 (πt/ 2 +π/ 4 )dt→∞

Ey=

∫^10

0

|( 1 +j)ejπt/^2 |^2 dt= 2

∫^10

0

dt= 20

Ez=

∫^10

0

dt= 10

where we used|( 1 +j)ejπt/^2 |^2 =| 1 +j|^2 |ejπt/^2 |^2 =| 1 +j|^2 =2. Thus,x(t)is an infinite-energy sig-

nal whiley(t)andz(t)are finite-energy signals. The power ofy(t)andz(t)are zero because they

have finite energy. The power ofx(t)can be calculated by using the symmetry of the signal squared

and lettingT=NT 0 :

Px=lim

T→∞

2

2 T

∫T

0

cos^2 (πt/ 2 +π/ 4 )dt= lim

N→∞

1

NT 0

∫NT^0

0

cos^2 (πt/ 2 +π/ 4 )dt

= lim

N→∞

1

NT 0



N

∫T^0

0

cos^2 (πt/ 2 +π/ 4 )dt



=^1

T 0

∫T^0

0

cos^2 (πt/ 2 +π/ 4 )dt

Using the trigonometric identity

cos^2 (πt/ 2 +π/ 4 )=

1

2

[cos(πt+π/ 2 )+ 1 ]