Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 17

x ( t ) s ( t )

1

1

t t

−1 0 1 2 3 4 0 1.5 3 4.5

Fig. 1.11 Two square waves

and

;

s ð t Þ¼ 1

2 þ 2

cos ð x o t Þ 1

3 cos ð 3x o t Þþ 1 5 cos ð 5x o t Þþ

ð 1 : 13 Þ

p

where x o ¼ 2 T o ¼ p.

Complex Fourier Series

Using Euler's formula, one can write the trigonometric Fourier series in complex
exponential form. This form is:

x ð t Þ¼ X

1

X n e þ jnx o t ¼ X

1

X n e þ j2pnf o t

ð 1 : 14 Þ

n ¼1 n ¼1

where,

Z Z

T o T o

X n ¼ 1
T o

e jnx o t dt ¼ 1
T o

e j2pnf o t dt ; n ¼ 0 ; 1 ; 2 ; ... ð 1 : 15 Þ

0 0

Relationship Between CFS and TFS Coefficients

The coefficients of the CFS and the TFS are related as follows:

X o ¼ a o ; ð 1 : 16 Þ

X n ¼ 1

2 ð a n jb n Þ;

ð 1 : 17 Þ

and,

X n ¼ 1

2 ð a n þ jb n Þ:

ð 1 : 18 Þ

Exercise: Verify the above relations using Euler's formula.