Signals and Systems - Electrical Engineering

4.10 Other Properties of the Fourier Series 283

T 0 =MT 2 =NT 1 , and its Fourier coefficients are

Zk=αXk/N+βYk/M for integersksuch thatk/N,andk/Mare integers (4.45)

whereXkandYkare the Fourier coefficients ofx(t)andy(t).

Ifx(t)andy(t)are periodic signals of the same periodT 0 , the Fourier coefficients ofz(t)=αx(t)+
βy(t)(also periodic of periodT 0 ) are thenZk=αXk+βYkwhereXkandYkare the Fourier coefficients
ofx(t)andy(t), respectively.

In general, ifx(t)is periodic of periodT 1 , andy(t)is periodic of periodT 2 , their sumz(t)=αx(t)+
βy(t)is periodic if the ratioT 2 /T 1 is a rational number (i.e.,T 2 /T 1 =N/Mfor some nondivisible
integersNandM). If so, the period ofz(t)isT 0 =MT 2 =NT 1. The fundamental frequency ofz(t)
would be 0 = 1 /N= 2 /Mfor 1 the fundamental frequency ofx(t)and 2 the fundamental
frequency ofy(t). The Fourier series ofz(t)is then

z(t)=αx(t)+βy(t)=α

∑

k

Xkej^1 kt+β

∑

m

Ymej^2 mt

=α

∑

k

XkejN^0 kt+β

∑

m

YmejM^0 mt

=α

∑

n=0,±N,± 2 N,...

Xn/Nej^0 nt+β

∑

`=0,±M,± 2 M,...

Y`/Mej^0 `t

Thus, the coefficients are

Zk=αXk/N+βYk/M

for integersksuch thatk/Nandk/Mare integers.

nExample 4.15

Consider the sumz(t)of a periodic signalx(t)of periodT 1 =2, with a periodic signaly(t)with

periodT 2 =0.2. Find the Fourier coefficientsZkofz(t)in terms of the Fourier coefficientsXkand

Ykofx(t)andy(t).

Solution

The ratioT 2 /T 1 = 1 / 10 =N/Mis rational, soz(t)is periodic of periodT 0 =T 1 = 10 T 2 =2.

The fundamental frequency ofz(t)is 0 = 1 =π, and 2 = 10  0 = 10 πis the fundamental

frequency ofy(t). Thus, the Fourier coefficients ofz(t)are

Zk=

{

Xk+Yk/ 10 when k=0,±10,±20,...

Xk otherwise

n