Signals and Systems - Electrical Engineering

608 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

nExample 10.17

To visualize the difference between a linear shift and a circular shift consider the periodic signal

x[n] of periodN=4 with a first period

x 1 [n]=n n=0,..., 3

Plotx[−n] andx[n−1] as functions ofnusing the linear and the circular representations.

Solution

In the circular representation ofx[n], the samplesx[0],x[1],x[2], andx[3] of the first period are

located in a clockwise direction in the E(ast), S(outh), W(est), and N(orth) directions in the circle.

Considering the E direction the origin of the representation, circular shifting is done analogously

to linear shifting with respect to the origin. Delaying byMin the circular representation corre-

sponds to shifting circularly, or rotating,Mpositions in the clockwise direction. Advancing byM

corresponds to shifting circularlyMpositions in the counterclockwise direction. Reflection corre-

sponds to reversing the order of the samples ofx 1 [n] to E, N, W, and S (i.e., placing the entries of

x 1 [n] counterclockwise starting withx[0] in E).

The circular representation ofx[−n] starts withx[0] in the first quadrant and then follows withx[1],

x[2], andx[3] in a counterclockwise direction (Figure 10.11). Looking at the circular representation

in a clockwise direction we obtain the linear representation that as expected coincides with the

reflection ofx[n].

FIGURE 10.11

Circular representation ofx[−n]

andx[n−1].^0

0

1

1

2

2

3

3

x[−n]

n

......

0

01

1

2

2

3

3

x[n−1]

n

......

x[1]

x[2] x[0]

x[3]

x[1]

x[2]

x[3]

x[0]