Signals and Systems - Electrical Engineering

5.5 Linearity, Inverse Proportionality, and Duality 307

− 3 − 2 − 1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

t

− 3 − 2 − 1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

t

−− (^150050)
0
1
2
3
4
5
Ω
X^2
(Ω
)
x^1
(t)
x^2
(t
)
− 50 − 40 − 30 − 20 − 10 0 10 20 30 40 50
− 1
0
1
2
3
4
5
Ω
X^1
(Ω
)
(a)
(b)
FIGURE 5.1
Fourier transform of pulses (a)x 1 (t)withA= 1 andτ= 1 , and (b)x 2 (t)withA= 1 andτ= 4. Notice the wider
the pulse the more concentrated in frequency its Fourier transform, and thatXi( 0 )=Aτ,i=1, 2, is the area
under the pulses.