Signals and Systems - Electrical Engineering

10.4 Discrete Fourier Transform 625

FIGURE 10.16
Circular convolution of lengthL= 8 ofx[n]
andy[n]. The signalx[k]is stationary with a
circular representation given by the inside
circle, whiley[n−k]is represented by the
outside circle and rotated in the clockwise
direction. The shown circular convolution
sum corresponds ton= 0.

y[3]

y[2]

y[1]

y[0]

y[7]

y[6]

y[5]

y[4]

x[3]x[2]

x[1]

x[0]

x[6]x[7]

x[5]

x[4]

n= 0

FIGURE 10.17
Circular versus linear
convolutions: (a) Plot
corresponds to linear
convolution. (b) and (c)
Plots are circular
convolutions wih
L< 2 N− 1. (d) Plot is
circular convolution with
L> 2 N− 1 coinciding
with the linear
convolution.

0 10 20 30 40

0

5

10

15

20

z(n)

Linear convolution

0 10 20 30 40

0

5

10

15

20

y(

n)

Circular convolution (L=20)

0 10 20 30 40

0

5

10

15

20

n

Circular convolution (L=49)

y^2

(n

)

0 10 20 30 40

0

5

10

15

20

n

Circular convolution (L=30)

y^1

(n

)

(a) (b)

(c) (d)

Solution

We know that the length of the linear convolutionz[n]=(x∗x)[n] isN+N− 1 =39. If we use

the functioncirconv2shown below to compute the circular convolution ofx[n] with itself with

lengthN< 2 N−1, for instanceL=20 as shown in Figure 10.17(b), the result will not equal the

linear convolution. Likewise, if the circular convolution is of lengthN+ 10 = 30 < 2 N−1, only

part of the result resembles the linear convolution (see Figure 10.17(c)). If we let the length of the

circular convolution be 2N+ 9 = 49 > 2 N−1, the result is identical to the linear convolution

(see Figure 10.17(d)). The script is given as follows.