Signals and Systems - Electrical Engineering

454 C H A P T E R 8: Discrete-Time Signals and Systems

8.2.1 Periodic and Aperiodic Signals

A discrete-time signalx[n]isperiodicif

n It is defined for all possible values ofn,−∞<n<∞.

n There is a positive integerN, the period ofx[n], such that

x[n+kN]=x[n] (8.2)

for any integerk.

Periodic discrete-time sinusoids, of periodN, are of the form

x[n]=Acos

(

2 πm

N

n+θ

)

−∞<n<∞ (8.3)

where the discrete frequency isω 0 = 2 πm/Nrad, for positive integersmandN, which are not divisible by

each other, andθis the phase angle.

The definition of a discrete-time periodic signal is similar to that of continuous-time periodic signals,

except for the period being an integer. That discrete-time sinusoids are of the given form can be easily

shown: Shifting the sinusoid in Equation (8.3) by a multiplekof the periodN, we have

x[n+kN]=Acos

(

2 πm

N

(n+kN)+θ

)

=Acos

(

2 πm

N

n+ 2 πmk+θ

)

=x[n]

since we add to the original angle a multiplemk(an integer) of 2π, which does not change the angle.

Remarks

n The units of the discrete frequencyωis radians. Moreover, discrete frequencies repeat every 2 π(i.e.,

ω=ω+ 2 πk for any integer k), and as such we only need to consider the range−π≤ω < π. This is

in contrast with the analog frequency, which has rad/sec as units, and its range is from−∞to∞.

n If the frequency of a periodic sinusoid is

ω=

2 π

N

m

for nondivisible integers m and N> 0 , the period is N. If the frequency of the sinusoid cannot be written

like this, the discrete sinusoid is not periodic.

nExample 8.3

Consider the sinusoids

x 1 [n]=2 cos(πn−π/ 3 )

x 2 [n]=3 sin( 3 πn+π/ 2 ) −∞<n<∞