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Spectral Analysis for Narrowband FM
For Tone Modulation Only
Using Trig Identities we can write u p ( t ) as:
u p (t )
A cos ( 2p f c t ) cos ( β sin ( 2p f mt ) ) − A sin ( 2p f c t ) sin ( β sin ( 2p f mt ) )
In Complex Envelope Notation, this can be represented as:
From the Complex Envelope, it is easy to see we a large unmodulated carrier in the Inphase
component, whereas the message information is now in a time-varying Quadrature component.
From here, we can proceed in two ways. Madhow likes to look at this from the Frequency Domain,
whereas I like to proceed from the time Domain.
Frequency Domain Approach
Note: We have a Carrier, USB, and LSB, just like we did with AM.
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Time Domain Approach
=
u p ( t ) Ac cos ( 2p f c t ) − Acθ ( t ) sin ( 2p f c t )
Because β 1 :
Thus, our signal reduces to:
Using the sin A sin B trig identity:
Note the Following:
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Spectral Analysis for Wideband FM
For Tone Modulation Only
Consider the complex envelope of our FM Signal, if we have tone modulation:
u p ( t ) Ac cos ( 2p f c t ) − Acθ ( t ) sin ( 2p f c t )
=
u ( t=
) Ac + jθ ( t )
Because the argument of the exponential ( β sin ( 2π f mt ) ) is periodic with T =
1
fm
, that means that
the modulated signal u ( t ) is periodic. That means we can use a Complex Exponential Fourier
Series Expansion to write u ( t ) .
Thus:
Where the u [ n ] terms are given by:
We can then do a change of variables to get:
x = f mt
dt =
u [ n]
=
1
fm
dx
1
2
j β sin ( 2π x ) − j 2π nx
e
e
dx
∫=
− 12
Jn ( β )
J n ( β ) is a_____________________________________________________________________.
It is tabulated and we can easily compute the integral. What then falls out is:
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J n ( β ) is a function of two variables, n and β . Fortunately for us, the values of J n ( β ) are
tabulated, and it is clear that the magnitude of J n ( β ) decreases rapidly as n increases.
Madhow Figure 3.26
Note the Following:
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Question: What is the Bandwidth?
We have two answers: The Absolute Bandwidth (based on the number of significant Bessel
functions) and the Carson’s Rule Bandwidth (an approximation based on the range of frequencies
that capture approximately 95% of the signal energy).
Carson’s Rule
Absolute (Bessel)