Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 198

X 1 ( w ) < X 1 ( w )

p /2

1/ a

x 1 ( t )

t w w

0

0

0

− p /2

(a) (b) (c)

equation reduces to

Fig. 5.3. CTFT of the causal
decaying exponential function
x ( t ) = e − at u ( t ). (a) x ( t );
(b) magnitude spectrum;
(c) phase spectrum.

∞ ∞

− a t − at cos( ω t )d t

X 2 ( ω ) = e cos( ω t )d t = 2 e

−∞ 0

2
a 2 + ω 2 [ − a e

2 a
a 2 + ω 2 .

− at cos( ω t ) + ω e − at sin( ω t )] ∞

0

= =

Since X 2 ( ω ) is positive real-valued, the magnitude and phase of X 2 ( ω ) are given
by

2 a
a 2 + ω 2

2 a
a 2 + ω 2 .

magnitude X 2 ( ω ) = =

phase < X 2 ( ω ) = 0 .

The non-causal exponentially decaying function x 2 ( t ) and its magnitude and
phase spectra are plotted in Fig. 5.4.

We note from Example 5.1 that the magnitude spectrum is symmetric along
the vertical axis while the phase spectrum is symmetric about the origin. The
magnitude spectrum is, therefore, an even function of ω , while the phase spec-
trum is an odd function of ω . This is a consequence of the symmetry properties
observed by real-valued functions. The symmetry properties are discussed in
detail in Section 5.3.

Fig. 5.4. CTFT of the causal
decaying exponential function
x 2 ( t ) = exp(− a t ). (a) x 2 ( t );
(b) Magnitude spectrum;
(c) phase spectrum for a > 0.

Example 5.2
Calculate the CTFT of a constant function x ( t ) = 1.

< X 2 ( w ) = 0
p /2

X 2 ( w )

x 2 ( t ) 2 a

ω w

t

0

0

0

− p /2

(a) (b) (c)