Signals and Systems - Electrical Engineering

284 C H A P T E R 4: Frequency Analysis: The Fourier Series

4.10.3 Multiplication of Periodic Signals

Ifx(t)andy(t)are periodic signals of same periodT 0 , then their product

z(t)=x(t)y(t) (4.46)

is also periodic of periodT 0 , and with Fourier coefficients that are theconvolution sumof the Fourier

coefficients ofx(t)andy(t):

Zk=

∑

`

X`Yk−` (4.47)

Ifx(t)andy(t)are periodic with the same periodT 0 , thenz(t)=x(t)y(t)is also periodic of periodT 0 ,

sincez(t+kT 0 )=x(t+kT 0 )y(t+kT 0 )=x(t)y(t)=z(t). Furthermore,

x(t)y(t)=

∑

k

Xkejk^0 t

∑

`

Y`ej`^0 t=

∑

k

∑

`

XkY`ej(k+`)^0 t

=

∑

m

[

∑

k

XkYm−k

]

ejm^0 t=z(t)

where we letm=k+`. The coefficients of the Fourier series ofz(t)are then

Zm=

∑

k

XkYm−k

or the convolution sum of the sequencesXkandYk, to be formally defined in Chapter 8.

nExample 4.16

Consider the train of rectangular pulsesx(t)shown in Figure 4.4. Letz(t)=0.25x^2 (t). Use the

Fourier series ofz(t)to show that

Xk=α

∑

m

XmXk−m

for some constantα. Determineα.

Solution

The signal 0.5x(t)is a train of pulses of unit amplitude, so thatz(t)=(0.5x(t))^2 =0.5x(t). Thus,

Zk=0.5Xk, but also as a product of 0.5x(t)with itself we have that

Zk=

∑

m

[0.5Xm][0.5Xk−m]