Signals and Systems - Electrical Engineering

4.5 Trigonometric Fourier Series 251

4.5 Trigonometric Fourier Series

Thetrigonometric Fourier seriesof a real-valued, periodic signalx(t), of periodT 0 , is an equivalent

representation that uses sinusoids rather than complex exponentials as the basis functions. It is given by

x(t)=X 0 + 2

∑∞

k= 1

|Xk|cos(k 0 t+θk)

=c 0 + 2

∑∞

k= 1

[ckcos(k 0 t)+dksin(k 0 t)]  0 =

2 π

T 0

(4.19)

whereX 0 =c 0 is called theDC component, and{ 2 |Xk|cos(k 0 t+θk)}are thekthharmonicsfork=1, 2....

The frequencies{k 0 }are said to be harmonically related. The coefficients{ck,dk}are obtained fromx(t)as

follows:

ck=

1

T 0

t (^0) ∫+T 0
t 0
x(t)cos(k 0 t)dt k=0, 1,...
dk=
1
T 0
t (^0) ∫+T 0
t 0
x(t)sin(k 0 t)dt k=1, 2,... (4.20)
The coefficientsXk=|Xk|ejθkare connected with the coefficientsckanddkby
|Xk|=
√
c^2 k+d^2 k
θk=−tan−^1
[
dk
ck
]
The functions{cos(k 0 t), sin(k 0 t)}are orthonormal.
Using the relationXk=X−∗k, obtained in the previous section, we express the exponential Fourier
series of a real-valued periodic signalx(t)as
x(t)=X 0 +

∑∞

k= 1

[Xkejk^0 t+X−ke−jk^0 t]

=X 0 +

∑∞

k= 1

[

|Xk|ej(k^0 t+θk)+|Xk|e−j(k^0 t+θk)

]

=X 0 + 2

∑∞

k= 1

|Xk|cos(k 0 t+θk)

which is the top equation in Equation (4.19).