Signals and Systems - Electrical Engineering

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

Solution

The analog frequency is 0 = 2 π/T 0 soT 0 = 2 π/ 0 is the period. WheneverT 0 >0 (or 0 >0)

these sinusoidals are periodic. For instance, consider

x(t)=2 cos( 2 t−π/ 2 ) −∞<t<∞

Its period is found by noticing that this signal has an analog frequency 0 = 2 = 2 πf 0 (rad/sec),

or a hertz frequency off 0 = 1 /π= 1 /T 0 , so thatT 0 =πis the period in seconds. That this is the

period can be seen for an integerN,

x(t+NT 0 )=2 cos( 2 (t+NT 0 )−π/ 2 )=2 cos( 2 t+ 2 πN−π/ 2 )

=2 cos( 2 t−π/ 2 )=x(t)

since adding 2πN(a multiple of 2π) to the angle of the cosine gives the original angle. If 0 =0—

that is, dc frequency—the period cannot be defined because of the division by zero when finding

T 0 = 2 π/ 0. n

nExample 1.9

Consider a periodic signalx(t)of periodT 0. Determine whether the following signals are periodic,

and if so, find their corresponding periods:

(a) y(t)=A+x(t).

(b) z(t)=x(t)+v(t)wherev(t)is periodic of periodT 1 =NT 0 , whereNis a positive integer.

(c) w(t)=x(t)+u(t)whereu(t)is periodic of periodT 1 , not necessarily a multiple ofT 0.

Determine under what conditionsw(t)could be periodic.

Solution

(a) Adding a constant to a periodic signal does not change the periodicity, soy(t)is periodic of

periodT 0 —that is, for an integerk,y(t+kT 0 )=A+x(t+kT 0 )=A+x(t)sincex(t)is periodic

of periodT 0.

(b) The periodT 1 =NT 0 ofv(t)is also a period ofx(t), and soz(t)is periodic of periodT 1 since

for any integerk,

z(t+kT 1 )=x(t+kT 1 )+v(t+kT 1 )=x(t+kNT 0 )+v(t)=x(t)+v(t)

given thatv(t+kT 1 )=v(t), and thatkNis an integer so thatx(t+kNT 0 )=x(t). The peri-

odicity can be visualized by considering that in one period ofv(t)we can placeNperiods

ofx(t).

(c) The condition forw(t)to be periodic is that the ratio of the periods ofx(t)and ofu(t)be

T 1

T 0

=

N

M