MATH 348 - Advanced Engineering Mathematics October 29, 2008
Homework 8, Fall 2008 Due: November 3, 2008
Fourier Transforms
1. Calculate the following Fourier sine/cosine transformations. Please include the domain which the transformation is valid.
(a) Fc(e−ax), a ∈R+
(b) F−1
c1
1 + ω2
(c) Fs(e−ax), a ∈R+
(d) F−1
s r2
π
ω
a2+ω2!, a ∈R+
2. Calculate the following transforms:
(a) F{f}where f(x) = δ(x−x0), x0∈R.1
(b) F{f}where f(x) = e−k0|x|, k0∈R+.
(c) F−1nˆ
fowhere ˆ
f(ω) = 1
2(δ(ω+ω0) + δ(ω−ω0)) , ω0∈R.
(d) F−1nˆ
fowhere ˆ
f(ω) = 1
2(δ(ω+ω0)−δ(ω−ω0)) , ω0∈R.
(e) Find ˆ
f(ω) where f(x+c), c ∈R.
3. The convolution hof two functions fand gis defined as2,
h(x)=(f∗g)(x) = Z∞
−∞
f(p)g(x−p)dp =Z∞
−∞
f(x−p)g(p)dp. (1)
(a) Show that F{f∗g}=√2πF{f}F{g}.
(b) Find the convolution h(x)=(f∗g)(x) where f(x) = δ(x−x0) and g(x) = e−x.
4. Given the ODE,
y0+y=f(x),−∞ <x<∞.(2)
Let f(x) = δ(x) and then:
(a) Calculate the frequency response associated with (2). 3
(b) Calculate the Green’s function associated with (2).
(c) Using convolution find the steady-state solution to the (2).
5. List three questions you have associated with Fourier series and three questions you have associated with Fourier transforms
and submit them as the leading page to this homework assignment.4
1Here the δis the so-called Dirac, or continuous, delta function. This isn’t a function in the true sense of the term but instead what is called a
generalized function. The details are unimportant and in this case we care only that this Dirac-delta function has the property Z∞
−∞
δ(x−x0)f(x)dx =
f(x0). For more information on this matter consider http://en.wikipedia.org/wiki/Dirac_delta_function. To drive home that this function is
strange, let me spoil the punch-line. The sampling function f(x) = sinc(ax) can be used as a definition for the Delta function as a→0. So can a
bell-curve probability distribution. Yikes!
2Here wee keep the same notation as Kreysig pg. 523
3this is often called the steady-state transfer function
4I will write up a Q+A sheet addressing both large and shared misunderstandings associated with our sections questions and post them on the ticc
website.
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