Signals and Systems - Electrical Engineering

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

FIGURE 4.17

Even and odd components of the period ofy(t),

− 1 ≤t≤ 1.

t

y 1 e(t) y 1 o(t)

− 1 1

− 1

− 1

1

0

2

t

1

1

Thus, the mean value ofye(t)is the area undery 1 e(t)divided by 2 or 1.5, and fork6=0,

Yek=

1

T 0

Y 1 e(s)

∣

∣s=jk 0 =^1

2

[

1

s

(es−e−s)+

1

s^2

(es− 2 +e−s)

]

s=jkπ

=

sin(kπ)

πk

+

1 −cos(kπ)

(kπ)^2

= 0 +

1 −cos(kπ)

(kπ)^2

=

1 −(− 1 )k

(kπ)^2

The mean value ofyo(t)is zero, and fork6=0,

Yok=

1

T 0

Y 1 o(s)

∣

∣s=jk 0 =^1

2

[

es−e−s

s^2

−

es+e−s

s

]

s=jkπ

=−j

sin(kπ)

(kπ)^2

+j

cos(kπ)

kπ

= 0 +j

cos(kπ)

kπ

=j

(− 1 )k

kπ

Finally, the Fourier series coefficients ofy(t)are

Yk=

{

Ye 0 +Yo 0 =1.5+ 0 =1.5 k= 0

Yek+Yok=( 1 −(− 1 )k)/(kπ)^2 +j(− 1 )k/(kπ) k6= 0

n

4.10.2 Linearity of Fourier Series—Addition of Periodic Signals

n Same fundamental frequency: Ifx(t)andy(t)are periodic signals with the same fundamental frequency

 0 , then the Fourier series coefficients ofz(t)=αx(t)+βy(t)for constantsαandβare

Zk=αXk+βYk (4.44)

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

n Different fundamental frequencies: Ifx(t)is periodic of periodT 1 , andy(t)is periodic of periodT 2 such

thatT 2 /T 1 =N/M, for nondivisible integersNandM, thenz(t)=αx(t)+βy(t)is periodic of period