Signals and Systems - Electrical Engineering

10.3 Fourier Series of Discrete-Time Periodic Signals 611

FIGURE 10.12
Periodic convolution of the Fourier
series coefficientsX[k]andY[k].

m= 1

m= 0

Y[1] X[1] X[0] Y[0]

Y[0] X[1] X[0] Y[1]

V[0]=X[0]Y[0]+X[1]Y[1]

V[1]=X[0]Y[1]+X[1]Y[0]

while keepingX[k] stationary. The circular representation ofX[k] is given by the internal circle

with the values ofX[0] andX[1] in the clockwise direction, whileY[m−k] is represented in the

outer circle with the two values of a period in the counterclockwise direction (corresponding to

the reflection of the signal orY[−k] form=0). Multiplying the values opposite to each other and

adding them we getV[0]=X[0]Y[0]+X[1]Y[1]. If we shift the outer circle 180 degrees clock-

wise form=1 and multiply the values opposite to each other and add their product, we get

V[1]=X[0]Y[1]+X[1]Y[0]. There is no need for further shifting, as the results would coincide

with the ones obtained before. The process is similar to the linear convolution but implemented

circularly. n

For periodic signalsx[n]andy[n]of periodN, we have

(a) Duality in time and frequency circular shifts:The Fourier series coefficients of the signals on the left are

the terms on the right:

x[n−M]⇔X[k]e−j^2 πMk/N

x[n]ej^2 πMn/N⇔X[k−M] (10.38)

(b) Duality in multiplication and periodic convolution sum:The Fourier series coefficients of the signals on

the left are the terms on the right:

z[n]=x[n]y[n]⇔Z[k]=

N∑− 1

m= 0

X[m]Y[k−m]

v[n]=

N∑− 1

m= 0

x[m]y[n−m]⇔V[k]=NX[k]Y[k] (10.39)