MATH 348 - Advanced Engineering Mathematics February 4, 2008
Homework 4, Spring 2008 Due: February 13, 2008
Continuous Fourier Transform - Signal Processing - Green’s Functions
If a function is not periodic nor can it be periodically extended then the function has no Fourier series represen-
tation. However, in this case the function can have a Fourier integral representation. Analysis of the Fourier integral
representation1reveals the complex Fourier transform pairs:
f(x) = 1
√2πZ∞
−∞
ˆ
f(ω)eiωxdω (1)
ˆ
f(ω) = 1
√2πZ∞
−∞
f(x)e−iωxdx (2)
We can connect (1)-(2) to the complex Fourier series. If a function is periodic then there exists a representation
of the function in the countably-infinite basis of imaginary-exponential functions. In this case the coefficients2of
this expansion are given by an integral whose limits of integration are bounded.3These coefficients quantify the
amplitude of oscillation for each discrete frequency of oscillation.4In the case of (1) we have that the function f
has a representation in the uncountably-infinite basis of imaginary-exponential functions. In this case the coefficients
are given by an integral (2) whose limits of integration are unbounded.5These coefficients quantify the amplitude of
oscillation for each continuous frequency of oscillation.
1. Now, we want to connect all of this to sections 11.7 and 11.8 in our text. We have that the Fourier integral
represents functions in the sine/cosine basis without requiring the function to be periodic. Now, what role does
symmetry play in this representation? With minimal work we see that if a function is even/odd then the Fourier
integral reduces to a Fourier cosine/sine integral.6As with our original derivation of transform, if we look at
the interplay between the coefficient functions A(ω)/B(ω) and f(x) we find the transform pairs:
fc(x) = r2
πZ∞
0
ˆ
f(ω) cos(ωx)dω ˆ
fc(ω) = r2
πZ∞
0
f(x) cos(ωx)dx (3)
fs(x) = r2
πZ∞
0
ˆ
f(ω) sin(ωx)dω ˆ
fs(ω) = r2
πZ∞
0
f(x) sin(ωx)dx (4)
We call (3) the Fourier cosine transform pair and, surprisingly, we call (4) the Fourier sine transform.
(a) Show that fc(x) and ˆ
fc(ω) are even functions and that fs(x) and ˆ
fs(ω) are odd functions.7
(b) Show that if we assume that f(x) is an even function then (1)-(2) defines the transform pair given by (3).
Also, show that if f(x) is an odd function then (1)-(2) defines the transform pair given by (4). 8
Given,
f(x) = A, 0< x < a
0,otherwise , A, a ∈R+.(5)
(c) On the same graph plot the even and odd extensions of f.
(d) Find the Fourier cosine and sine transforms of f.
(e) Using the Fourier cosine transform show that Z∞
−∞
sin(πω)
πω dω = 1.
1See classnotes or Kreyszig pgs. 518-519
2also called weights
3Recall that our derivation lead to cn=1
2πZL
−L
f(x)eiωnxdx where ωn=nπ
L.
4That is, for each ωnthere is a corresponding cnwhere |cn|2is a measure of the power of the sinusoids associated with ωn.
5In this case the behavior of fmust be known everywhere instead of on the interval (−L, L).
6Kreyszig pg. 511
7Thus, if an input function has an even or odd symmetry then the transformed function shares the same symmetry.
8Thus, if an input function has symmetry then the Fourier transform is real-valued.
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