Signals and Systems - Electrical Engineering

5.5 Linearity, Inverse Proportionality, and Duality 309

FIGURE 5.2
(a) Pulsex(t)and its compressed
versionx 1 (t)=x( 2 t), and (b) the
magnitude of their Fourier
transforms. Notice that when the
signal contracts in time it expands
in frequency.

0 0.2 0.4 0.6 0.8 1

0

0.5

1

x(

t),

x^1

(t

) x(t)

x 1 (t)

(a)

t

− 50 0 50

0

0.5

1

|X

(Ω

)|, |

X^1

(Ω

)| |X(Ω)|

|X 1 (Ω)|

(b)

Ω

The Fourier transforms can be found from the integral definitions. Thus, forx(t),

X()=

∫^1

0

1 ejtdt=

e−jt

−j

|^10 =

sin(/ 2 )

/ 2

e−j/^2

Likewise, forx 1 (t),

X 1 ()=

∫0.5

0

1 ejtdt=0.5

sin(/ 4 )

/ 4

e−j/^4

n

nExample 5.4

Apply the reflection property to find the Fourier transform ofx(t)=e−a|t|,a>0. Fora=1, plot

using MATLAB the signal and its magnitude and phase spectra.

Solution

The signalx(t)can be expressed asx(t)=e−atu(t)+eatu(−t)=x 1 (t)+x 1 (−t). The Fourier trans-

form ofx 1 (t)is

X 1 ()=

1

s+a

∣

∣s=j=^1

j+a