0195136047.pdf

14.1 SIGNALS AND SPECTRAL ANALYSIS 633

(b) From the sketches of Figure E14.1.1(b), forθ= 0 , x(t)=Aej^2 πf^0 t, the real part and

the magnitude are even; the imaginary part and the phase are odd. Noting that a complex

signalx(t) is calledhermitianif its real part is even and its imaginary part is odd, the

signal and symmetry are then said to be hermitian.

A signalx(t) is said to becausalif, for allt<0,x(t) = 0; otherwise, the signal is noncausal.
An anticausal signal is identically equal to zero fort>0. A discrete-time signal is a causal signal
if it is identically equal to zero forn<0. Note that the unit step multiplied by any signal produces
a causal version of the signal.
Signals can also be classified asenergy-typeandpower-typesignals based on the finiteness
of their energy content and power content, respectively. A signalx(t) is an energy-type signal if
and only if the energyExof the signal,

Ex=

∫∞

−∞

|x(t)|^2 dt=lim

T→∞

∫T/ 2

−T/ 2

|x(t)|^2 dt (14.1.9)

is well defined and finite. A signal is a power-type signal if and only if the powerPxof the signal,

Px=lim

T→∞

1

T

∫T/ 2

−T/ 2

|x(t)|^2 dt (14.1.10)

is well defined and 0≤Px<∞. For real signals, note that|x(t)|^2 can be replaced byx^2 (t).

EXAMPLE 14.1.3

(a) Evaluate whether the sinusoidal signalx(t)=Acos( 2 πf 0 t+θ)is an energy-type or a

power-type signal.

(b) Show that any periodic signal is not typically energy type, and the power content of any

periodic signal is equal to the average power in one period.

Solution

(a)Ex= lim

T→∞

∫T/ 2

−T/ 2

A^2 cos^2 ( 2 πf 0 t+θ) dt=∞

Therefore, the sinusoidal signal is not an energy-type signal. However, the power of this

signal is

Px= lim

T→∞

1

T

∫T/ 2

−T/ 2

A^2 cos^2 ( 2 πf 0 t+θ) dt

= lim

T→∞

1

T

∫T/ 2

−T/ 2

A^2

2

[ 1 +cos( 4 πf 0 t+ 2 θ)]dt

= lim

T→∞

{

A^2 T

2 T

+

[

A^2

8 πf 0 T

sin( 4 πf 0 t+ 2 θ)

]+T/ 2

−T/ 2

}

=

A^2

2

<∞