
5.20. Consider Example 5.7, where p (t) and q (t) are square pulses.
(a) Find the Fourier transforms of p (t) and q (t) .
(b) Find the Fourier transform of the convolution p ∗ q.
(c) Verify that the convolution theorem holds in this case, that is, F (p (t)) F (q (t)) = F (p (t) ∗ q (t)) .
Solution: (a) The Fourier transform of a rectangular pulse, p (t) , is
F (p (t)) =
1
2π
∞
−∞
p (t) e
−iωt
dt
=
1
2π
1
−1
1 × e
−iωt
dt
=
1
2π
1
−1
cos ωtdt
=
sin ω
ωπ
.
Then, the Fourier transform of a rectangular pulse q (t) is
F (q (t)) = 2
sin ω
ωπ
.
(b) The convolution p ∗q is given by
p ∗ q =
∞
−∞
p (t − τ) q (τ) dτ
=
0 for t < −2
2 (t + 2) for −2 < t < 0
2 (−t + 2) for 0 < t < 2
0 for ...