Signal representation using Fourier series - Continuous and Discrete Time Signals and Systems - page 153

an arbitrary periodic signal can also be expressed as a linear combination of
sinusoidal signals:

∞

x ( t ) = a 0 + ( a n cos( n ω 0 t ) + b n sin( n ω 0 t )) . (4.28)

n = 1

Corollary 4.1 can then be applied to calculate the output y ( t ). Expressing a
periodic signal as a linear combination of sinusoidal signals leads to the trigono-
metric CTFS. The trigonometric and exponential CTFS representations of CT
periodic signals are covered in Sections 4.4 and 4.5.

4.4 Trigonometric CTFS

Definition 4.6 An arbitrary periodic function x ( t ) with fundamental period T 0
can be expressed as follows:

∞

x ( t ) = a 0 + ( a n cos( n ω 0 t ) + b n sin( n ω 0 t )) , (4.29)

n = 1

where ω 0 = 2 π/ T 0 is the fundamental frequency of x ( t ) and coefficients a 0 ,a n ,

and b n are referred to as the trigonometric CTFS coefficients. The coefficients
are calculated as follows:

1
T 0

a 0 = x ( t )d t , (4.30)

T 0

2
T 0

=

x ( t ) cos( n ω 0 t )d t , a n (4.31)

T 0

and

= 2
T 0

b n x ( t ) sin( n ω 0 t )d t . (4.32)

T 0

From Eqs. (4.29)-(4.32), it is straightforward to verify that coefficient a 0 rep-
resents the average or mean value (also referred to as the dc component) of
x ( t ). Collectively, the cosine terms represent the even component of the zero
mean signal ( x ( t )- a 0 ). Likewise, the sine terms collectively represent the odd
component of the zero mean signal ( x ( t )- a 0 ).

Example 4.6
Calculate the trigonometric CTFS coefficients of the periodic signal x ( t ) defined
over one period T 0 = 3 as follows:

t + 1 − 1 ≤ t ≤ 1

x ( t ) = (4.33)

< t < 2 .