Page 496 - Advanced engineering mathematics

476 CHAPTER 14 The Fourier Integral and Transforms

The presence of the exponential factor e 2iω suggests use of the inverse version of the time
shifting theorem. Put t 0 =−2 and
1
ˆ
f (ω) =
5 + iω
into equation (14.12) to get
−1
F [e 2iω f (ω)](t) = f (t − (−2)) = f (t + 2)
ˆ
where
1

−1 −5t
f (t) = F = H(t)e
5 + iω
from Example 14.5. Then, by time shifting,
e
2iω
−1 −5(t+2)
F = f (t + 2) = H(t + 2)e .
5 + iω
Frequency Shifting If ω 0 is any real number then
ˆ
F[e iω 0 t f (t)]= f (ω − ω 0 ).
The Fourier transform of a function multiplied by e iω 0 t is the Fourier transform of f shifted right
by ω 0 .

Proof To prove this result, compute
∞

F[e −ω 0 t f (t)](ω) = e iω 0 t f (t)e −iωt dt
−∞
∞

ˆ
= e −i(ω−ω 0 )t dt = f (ω − ω 0 ).
−∞
The inverse version of frequency shifting is that
−1
F [ f (ω − ω 0 )](t) = e iω 0 t f (t).
ˆ
Time shifting and frequency shifting are reminiscent of the two shifting theorems for the Laplace
transform.

Scaling If c is any nonzero real number, then
1
ˆ
F[ f (ct)](ω) = f (ω/c).
|c|
Scaling can be verified by a change of variables u = ct in the integral for the transform
of f (ct).
The inverse version of the scaling theorem is

−1
F [ f (ω/c)]=|c| f (ct).
ˆ
Time Reversal
ˆ
F[ f (−t)](ω) = f (−ω).
Time reversal follows immediately from the scaling theorem upon putting c =−1.

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October 14, 2010 16:43 THM/NEIL Page-476 27410_14_ch14_p465-504