Signals and Systems - Electrical Engineering

4.3 Complex Exponential Fourier Series 247

where the Fourier coefficientsXkare found according to

Xk=

1

T 0

t (^0) ∫+T 0
t 0
x(t)e−jk^0 tdt (4.12)
fork=0,±1,±2,...,and anyt 0. The form of Equation (4.12) indicates that the information needed for the
Fourier series can be obtained from any period ofx(t).
Remarks
n The Fourier series uses the Fourier basis{ejk^0 t,k integer}to represent the periodic signal x(t)of period
T 0. The Fourier basis functions are also periodic of period T 0 (i.e., for an integer m,
ejk^0 (t+mT^0 )=ejk^0 tejkm^2 π=ejk^0 t
as ejkm^2 π= 1 ).
n The Fourier basis functions areorthonormalover a period—that is,

1

T 0

t (^0) ∫+T 0
t 0
ejk^0 t[ej`^0 t]∗dt=

{

1 k=`

0 k6=`

(4.13)

That is, ejk^0 tand ej`^0 tare said to beorthogonal when for k6=`the above integral is zero, and they

arenormal(or normalized) when for k=`the above integral is unity. The functions ejk^0 tand ej`^0 t

are orthogonal since

1

T 0

t (^0) ∫+T 0
t 0
ejk^0 t[ej`^0 t]∗dt=

1

T 0

t (^0) ∫+T 0
t 0
ej(k−`)^0 tdt

=

1

T 0

t (^0) ∫+T 0
t 0
[cos((k−) 0 t)+jsin((k−) 0 t)]dt
= 0 k6= The above integrals are zero given that the integrands are sinusoids and the limits of the integrals cover one or more periods of the integrands. The normality of the Fourier functions is easily shown when for k=the above integral is

1

T 0

t (^0) ∫+T 0
t 0
ej^0 tdt= 1