EE 261 The Fourier Transform and its
Applications
Fall 2011
Problem Set Six
Due Friday, November 11
1. (35 points)
Frequency Modulation and Music
A frequency modulated (FM) signal is one whose frequency is a function of time:
x
(
t
) =
A
cos(2
πf
(
t
))
.
FM signals are central to many scientific fields. Most notably, they are used in com
munications where
f
(
t
) =
f
c
t
+
k
integraldisplay
t
−∞
m
(
t
)
dt.
Here
f
c
is the
carrier frequency
(typically a large value is necessary for the physics of
wave propagation),
k
is a constant known as the
frequency modulation index
, and
m
(
t
)
is the function with the information and doing the modulating. You set your receiver
to
f
c
to listen to the signal.
Another famous (and profitable) application of FM signals is in the digital synthesis of
music, pioneered by John Chowning at Stanford. We’ll take a closer look at how this
done later in this problem.
Let’s start by taking the case where we modulate a pure tone,
x
(
t
) =
A
cos(2
πf
c
t
+
k
sin 2
πf
m
t
)
.
Here’s a plot for 0
≤
t
≤
2 with
A
= 1,
f
c
= 5,
k
= 2, and
f
m
= 20.
1
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What is the spectrum? Remember Bessel functions, introduced in an earlier problem?
The answer depends on these. Let’s recall: The Bessel equation of order
n
is
x
2
y
′′
+
xy
′
+ (
x
2

n
2
)
y
= 0
.
A solution of the equation is the
Bessel function of the first kind of order
n
, given by
the integral
J
n
(
x
) =
1
2
π
integraldisplay
2
π
0
cos(
x
sin
θ

nθ
)
dθ .
You showed in Problem Set 1 that
e
ix
sin
θ
=
∞
summationdisplay
n
=
−∞
J
n
(
x
)
e
inθ
and
e
ix
cos
θ
=
∞
summationdisplay
n
=
−∞
i
n
J
n
(
x
)
e
inθ
.
(a) Show the Fourier series relationship
exp(2
πif
c
t
+
ik
sin(2
πf
m
t
)) =
∞
summationdisplay
n
=
−∞
J
n
(
k
) exp(2
πi
(
f
c
+
nf
m
)
t
)
.
Use this result to show that the Fourier transform of
x
(
t
) is
F
x
(
s
) =
A
2
∞
summationdisplay
n
=
−∞
J
n
(
k
)[
δ
(
s

(
f
c
+
nf
m
)) +
δ
(
s
+ (
f
c
+
nf
m
)]
.
Hint: What is the real part of the Fourier series relationship?
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 '08
 BOYD,S
 Frequency, Nyquist–Shannon sampling theorem, Bessel function, Poisson summation formula

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