Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 16

Note that although the Fourier series reveals the frequency content of the signal,
it is not strictly speaking a frequency transform as the representation is still in
the time domain.

Trigonometric Fourier Series

If x(t) is a periodic signal with fundamental period T o , then it can be expanded as
follows:

x ð t Þ¼ a o þ a 1 cos ð x o t Þþ a 2 cos ð 2x o t Þþþ b 1 sin ð x o t Þþ b 2 sin ð 2x o t Þþ
¼ a o þ X

1

½ a n cos ð x n t Þþ b n sin ð x n t Þ;

n ¼ 1

ð 1 : 8 Þ

where x o ¼ 2pf o ¼ 2 T o , and:

Z

T o

a o ¼ 1
T o

x ð t Þ dt ; ð the constant term Þ ð 1 : 9 Þ

0

Z

T o

a n ¼ 2
T o

x ð t Þ cos ð nx o t Þ dt ; ð 1 : 10 Þ

0

Z

T o

b n ¼ 2
T o

x ð t Þ sin ð nx o t Þ dt : ð 1 : 11 Þ

0

Special Cases

1. If x(t) is odd, then x(t)cos(nx o t) is odd, hence a o = a n = 0, and the Fourier
series is a series of sines without a constant (zero frequency) term.
2. If x(t) is even, then x(t)sin(nx o t) is odd, hence, b n = 0 and the Fourier series is
a series of cosines.

Example Consider the signals x(t) and s(t) depicted in Fig. 1.11 . The signal
x(t) - 1/2 is odd, so one can use results related to odd functions in deducing the
Fourier series. The signal s(t) is even. The fundamental period of both signals is
T o = 2. Using the formulae in ( 1.9 )-( 1.11 ), the Fourier series of these two signals
are found to be:

;

x ð t Þ¼ 1

2 þ 2

sin ð x o t Þþ 1

3 sin ð 3x o t Þþ 1

5 sin ð 5x o t Þþ

ð 1 : 12 Þ

p