Signals and Systems - Electrical Engineering

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

FIGURE 8.5

Discrete frequenciesω.

ω=π/2, 5π/2, 9π/2 · · · =π/2

ω= 3 π/2, 7π/2, 11π/2 , ··· = −π/2

ω=π, 3π, 5π ··· = π = −π ω=0, 2π, 4π, ··· = 0

nExample 8.14

Consider the following four sinusoids:

(a)x 1 [n]=sin(0.1πn)

(b)x 2 [n]=sin(0.2πn)

(c)x 3 [n]=sin(0.6πn)

(d)x 4 [n]=sin(0.7πn)

Find if they are periodic, and if so, determine their periods. Are these signals harmonically

related? Use MATLAB to plot these signals fromn=0,..., 40. Comment on which of these signals

resemble sampled analog sinusoids.

Solution

To find if they are periodic, rewrite the given signals as follows indicating that the signals are

periodic of periods 20, 10, 10, and 20:

(a)x 1 [n]=sin(0.1πn)=sin

(

2 π

20

n

)

(b)x 2 [n]=sin(0.2πn)=sin

(

2 π

10

n

)

(c)x 3 [n]=sin(0.6πn)=sin

(

2 π

10

3 n

)

(d)x 4 [n]=sin(0.7πn)=sin

(

2 π

20

7 n

)

If we letω 1 = 2 π/20, the frequencies ofx 2 [n],x 3 [n], andx 4 [n] are 2ω 1 , 6ω 1 , and 7ω 1 , respectively;

thus they are harmonically related. Also, one could consider the frequencies ofx 1 [n] andx 4 [n]

harmonically related (i.e., the frequency ofx 4 [n] is seven times that ofx 1 [n]), and likewise the

frequencies ofx 2 [n] andx 3 [n] are also harmonically related, with the frequency ofx 3 [n] being