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14.3 The Fourier Transform 489

0.15

0.1

0.05

0
–15 –10 –5 0 5 10 15
t
–0.05

–0.1

–0.15

FIGURE 14.6 Graph of β j (t) for ω j = 2.2,
ω j−1 = 1.7.

SECTION 14.3 PROBLEMS

In each of Problems 1 through 15, find the Fourier trans- 12. e (20−4ω)i /(3 − (5 − ω)i)
form of the function and graph the amplitude spectrum. 13. e (2ω−6)i /(5 − (3 − ω)i)
Wherever k appears it is a positive constant. Use can be 14. 10sin(3ω)/(ω + π)
2
made of the following transforms: 15. (1+iω)/(6−ω +5iω) Hint: Factor the denominator
and use partial fractions.
π
F[e −kt 2 ](ω) = e −ω 2 /4k
k
In each of Problems 16, 17, and 18, use convolution to find
and
the inverse Fourier transform of the function.

1 π
F (ω) = e −k|ω|
2
k + t 2 k
16. 1/((1 + iω)(2 + iω))
17. 1/(1 + iω) 2
⎧ 18. sin(3ω)/ω(2 + iω)
⎪1 for 0 ≤ t ≤ 1
⎨ 19. Prove the following version of Parseval’s theorem:
1. f (t) = −1 for −1 ≤ t < 0

⎪ ∞ 1 ∞
0 for |t| > 1 | f (t)| dt = | ˆ f (ω)| dω.
2
2
⎩
2π
−∞ −∞

sin(t) for −k ≤ t ≤ k
2. f (t) = 20. Compute the total energy of the signal f (t) =
0 for |t| > k
H(t)e −2t .
3. f (t) = 5[H(t − 3) − H(t − 11)] 21. Compute the total energy of the signal f (t) = (1/t)
4. f (t) = 5e −3(t−5) 2 sin(3t). Hint: Use Parseval’s theorem, Problem 19.
5. f (t) = H(t − k)e −t/4 22. Use the Fourier transform to solve
6. f (t) = H(t − k)t 2

2
7. f (t) = 1/(1 + t ) y + 6y + 5y = δ(t − 3).
8. f (t) = 3H(t − 2)e −3t
9. f (t) = 3e −4|t+2| In each of Problems 23 through 28, compute the windowed
10. f (t) = H(t − 3)e −2t Fourier transform of f for the given window function
w. Also compute the center and RMS bandwidth of the
In each of Problems 11 through 15, find the inverse Fourier window function.
transform of the function.

1for −5 ≤ t ≤ 5,
2
23. f (t) = t , w(t) =
11. 9e −(ω+4) 2 /32 0for |t| > 5.
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October 14, 2010 16:43 THM/NEIL Page-489 27410_14_ch14_p465-504