Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 195

follows:

∞

− j ω t d t .

X ( ω ) = x ( t )e (5.5)

−∞

In terms of X ( ω ), Eq. (5.4) can, therefore, be expressed as follows:

1
T 0

D n = lim

T 0 →∞

X ( n ω 0 ) . (5.6)

Using the exponential CTFS definition, x ( t ) can be evaluated from the CTFS
coefficients D n as follows:

n =−∞ D n e j n ω 0 t =

∞ ∞

1
T 0

X ( n ω 0 )e j n ω 0 t . x ( t ) = lim

T 0 →∞

(5.7)

n =−∞

→∞ , the fundamental frequency ω 0 approaches a small value denoted by
ω . The fundamental period T 0 is therefore given by T 0

As T 0

= 2 π / ω . Substituting

T 0 = 2 π / ω as ω 0 → ω in Eq. (5.7) yields

n =−∞ X ( n ω )e j n ω t ω

∞

1
2 π

x ( t ) = lim

ω→ 0

. (5.8)

A

In Eq. (5.8), consider the term A as illustrated in Fig. 5.2. In the limit ω → 0,
term A represents the area under the function X ( ω )exp(j ω t ). Therefore Eq.
(5.8) can be rewritten as follows:

∞

1
2 π

− j ω t d t ,

CTFT synthesis equation x ( t ) = = X ( ω )e (5.9)

−∞

which is referred to as the synthesis equation for the CTFT used to express
any aperiodic signal in terms of complex exponentials, exp(j ω t ). The analysis
equation of the CTFT is given by Eq. (5.5), which, for convenience of reference,
is repeated below.

∞

− j ω t d t .

CTFT analysis equation X ( ω ) = x ( t )e (5.10)

−∞

X ( w )e j w t

A = X ( n w )e j n w t w

Fig. 5.2. Approximation of the
term

n =−∞ X ( n ω)
e j n ω t ω as the area under the
function X (ω)exp( jω t ).

∞

w

0

n w w ( n + 1) w