Signals and Systems - Electrical Engineering

10.3 Fourier Series of Discrete-Time Periodic Signals 613

(a)

01

1

2345

n

x[n]

......

(b)

0 1

1

2

2

3

n

v[n]

1 ......

1

0, 0, 1, 1 0

1, 0, 0, 1

0, 1, 1, 0

1, 1, 0, 0

0

1

1

0

0

x[n]

(c)

n

1

2

0123

z[n]

0, 0, 1, 1 ......

1, 0, 0, 1

0, 1, 1, 0

1, 1, 0, 0

1

1

0

0

FIGURE 10.13
Periodic convolution sum ofx[n]with itself to getv[n]: (a) linear and circular representations ofx[n]; (b) periodic
convolution sum givingv[n]. (c) Circular representation of periodic convolution sum ofx[n]andy[n]=x[n−2],
the result isz[n]=v[n−2].

As before, the Fourier series coefficients ofz[n] are given by

Z[k]= 4

X 1 (z)Y 1 (z)

4 × 4

|z=ej 2 πk/ 4 =

z−^2 + 2 z−^3 +z−^4

4

|z=ej 2 πk/ 4

=

1

4

(e−j^2 π^2 k/^4 + 2 e−j^2 π^3 k/^4 +e−j^2 π^4 k/^4 )=

1

4

( 1 +e−j^2 π^2 k/^4 + 2 e−j^2 π^3 k/^4 )