0195136047.pdf

632 SIGNAL PROCESSING

for all integersnandk, some positive integerN, and some integerm. From this it follows

that

2 πf 0 kN= 2 mπ or f 0 =m/(kN )

that is, the discrete sinusoidal signal is periodic only for rational values off 0.

Evenness and oddness are expressions of various types of symmetry present in signals. A

signalx(t)isevenif it has a mirror symmetry with respect to the vertical axis. A signal isodd

if it is symmetric with respect to the origin. The signalx(t) is even if and only if, for allt,it

satisfies

x(−t)=x(t) (14.1.3)

and is odd if and only if, for allt,

x(−t)=−x(t) (14.1.4)

Any signalx(t), in general, can be expressed as the sum of its even and odd parts,

x(t)=xe(t)+x 0 (t) (14.1.5)

xe(t)=

x(t)+x(−t)

2

(14.1.6)

xo(t)=

x(t)−x(−t)

2

(14.1.7)

Thehalf-wave symmetryis expressed by

x

(

t±

T

2

)

=−x(t) (14.1.8)

EXAMPLE 14.1.2

Discuss the nature of evenness and oddness of:

(a) The sinusoidal signalx(t)=Acos( 2 πf 0 t+θ).

(b) The complex exponential signalx(t)=ej^2 πf^0 t.

Solution

(a) The signal is, in general, neither even nor odd. However, for the special case ofθ=0,

it is even; for the special case ofθ=±π/2, it is odd. In general,

x(t)=Acosθ cos 2πf 0 t−Asinθ sin 2πf 0 t

Since cos 2πf 0 tis even and sin 2πf 0 tis odd, it follows that

xe(t)=Acosθ cos 2πf 0 t

and

xo(t)=−Asinθ sin 2πf 0 t