Signal representation using Fourier series - Continuous and Discrete Time Signals and Systems - page 174

Using the results obtained in Example 4.14, the CTFS coefficients R n of r ( t )
are given by

= 2 1

n / 2
4

= 1

n
8

R n

4 sinc 2 sinc

, (4.63)

for −∞ < n < ∞ . The function r ( t ) can now be represented as an exponential

CTFS as follows:

n =−∞ R n e j n ω 0 t

∞

n =−∞ sinc

∞

= 1
2

n
2

e j2 nt , x ( t ) =

where the fundamental frequency ω 0 is set to 2.

Differentiation and integration The exponential CTFS coefficients of the
time-differentiated and time-integrated signal are expressed in terms of the
exponential CTFS coefficients of the original signal as follows:

D n
j n ω 0 .

d x
d t

CTFS
←−−→

CTFS
←−−→

CTFS
←−−→

j n ω 0 D n and x ( t )d t

if x ( t ) D n then

T 0

(4.64)

It may be noted that the signal obtained by differentiating or integrating a
periodic signal x ( t ) over one period T 0 has the same period T 0 as that of the
original signal.

Example 4.21
Calculate the exponential CTFS coefficients of the periodic signal g ( t ) shown
in Fig. 4.20.

Solution
The function g ( t ) can be obtained by differentiating x ( t ) shown in Fig. 4.14.
Therefore, the CTFS coefficients G n can be expressed in terms of the CTFS
coefficients D n as follows:

G n = j n ω 0 D n with ω 0 = 1 .

Substituting the value of

= 1

n
4

D n

4 sinc

g ( t )

3

t

−2 p − p 0 p 2 p

Fig. 4.20. Periodic signal g ( t )
for Example 4.21.

−3