Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 233

In part (iii), we proved that the CTFT does not exist for a periodic cosine
function. This appears to be in violation of Table 5.2, which lists the following
CTFT pair for the periodic cosine function:

CTFT
←−−→ cos( ω 0 t ) π [ δ ( ω − ω 0 ) + δ ( ω + ω 0 )] .

Actually, the two statements do not contradict each other. The condition for the
existence of the CTFT assumes that the CTFT must be finite for all values of
ω . The above CTFT pairs indicate that the CTFT of the periodic cosine func-
tion consists of two impulses at ω =ω 0 . From the definition of the impulses,
we know that the magnitudes of the two impulse functions in the aforemen-
tioned CTFT pair are infinite at ω =ω 0 , and therefore that the periodic cosine
function violates the condition for the existence of the CTFT.
In Section 5.7, we show that the CTFTs of most periodic signals are derived
from the CTFS representation of such signals, not directly from the CTFT
definition. Therefore, we make an exception for periodic signals and ignore the
condition of CTFT existence for periodic signals.

5.7 CTFT of period ic functions

Consider a periodic function x ( t ) with a fundamental period of T 0 . Using the
exponential CTFS, the frequency representation of x ( t ) is obtained from the
following expression:

n =−∞ D n e j n ω 0 t ,

∞

x ( t ) = (5.60)

where ω 0 = 2 π/ T 0 is the fundamental frequency of the periodic signal and D n

denotes the exponential CTFS coefficients D n , given by

= 1
T 0

− j n ω 0 t d t .

D n x ( t )e (5.61)

T 0

Calculating the CTFT of both sides of Eq. (5.60), we obtain

n =−∞ D n e j n ω 0 t

∞

X ( ω ) =ℑ x ( t ) =ℑ

.

Using the linearity property, the above expression is simplified to

n =−∞ D n δ ( ω − n ω 0 ) .
In other words, the CTFT of a periodic function x ( t )isgivenby

n =−∞ D n

∞ ∞

ℑ e j n ω 0 t = 2 π X ( ω ) =

n =−∞ D n δ ( ω − n ω 0 ) .

∞

CTFT
←−−→

x ( t ) 2 π (5.62)