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Demodulation of FM SignalsKEEE343 Communication Theory
Lecture #18, May 17, 2011Prof. Young-Chai [email protected]
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Summary
BandwidthofWidebandFrequencyModulationCarlsonsrule
DemodulationofFMSignals
BasebandrepresentationofPassbandsignal
DemodulationofFMsignals
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Recall the single-tone frequency modulated wave given as
and its FT is given as
where for the message signal
To see the bandwidth let us consider two different cases
1. Case 1: Fix and vary (phase deviation is varied but the BW
of message signal is fixed.)
2. Case 2: Fix and vary (phase deviation is fixed but the BWof message signal is varied.)
Transmission Bandwidth of FM Waves
s(t) = Ac cos[2fct + sin(2fmt)]
S(f) = Ac
2
1X
n=1
Jn()[(f fc nfm) + (f+ fc + nfm)]
m(t) = Am cos(2fmt)
fm Am
Am fm
=kfAm
fm=
f
fm
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Case 1
[Ref: Haykin & Moher, Textbook]
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Case 2
[Ref: Haykin & Moher, Textbook]
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In theory, an FM wave contains an infinite number of side-frequencies.
However, we find that the FM wave is effectively limited to a finite numberof significant side-frequencies compatible with a specified amount of
distortion.
Observations of two limiting cases
1. For large values of the modulation index , the bandwidth
approaches, and is only slightly greater than the total frequencyexcursion .
2. For small values of the modulation index, the spectrum of the FM
wave is effectively limited to one pair of side-frequencies atso that the bandwidth approaches .
Transmission Bandwidth of the FM Wave
2f
fc fm2fm
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Carsons rule is the approximate rule for the transmission bandwidth of anFM wave
Single-tone case
Arbitrary modulating wave
where is deviation ratio.
Carsons Rule
BT 2f+ 2fm = 2f
1 +
1
BT 2(f+ W) = 2f
1 +
1
D
D =f
W
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Carsons rule is simple but unfortunately it does not always provide a goodestimate of the transmission bandwidth, in particular, for the widebandfrequency modulation.
Universal Curve for FM TransmissionBandwidth
[Ref: Haykin & Moher, Textbook]
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[Ref: Haykin & Moher, Textbook]
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Baseband Representation of Modulated Waves andBandpass Filters
Band-pass representation (Modulated waves)
Let
and
Then
s(t) = sI(t)cos(2fct) sQ(t) sin(2fct)
s t = sI t + jsQ t
c(t = c(t + jc(t
= cos(2fct) + j sin(2fct)
= exp(j2fct)
s(t) = < [s(t)c(t)]
= < [s(t) exp(j2fct)]
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We can derive the baseband representation of the modulated wave infrequency domain as follows
Define a signal, called analytical signal that contains only the positivefrequencies in s(t)
Then its inverse Fourier transform is
|S(f |
ffcfc 0
S+(f = 2u(f S(f
1
2S+(f)
s+(t) =
Z1
1
S+(f)ej2ft df
= 2F1 [2u(f)] F1[S(f)]
=
(t) +
j
t
s(t)
= s(t) +j
t s(t)
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Using the Hilbert transform given as
the analytical signal can be written as
Now define the equivalent lowpass representation by performing a frequency
translation of such as
The equivalent time-domain relation is
or equivalently
s
(t) =
1
t s
(t)
s+(t = s(t + js(t
S+(f
S(f) = S+(f+ fc)
s
(t) =
s+(
t)ej2fct
= [s
(t) +
js
(t)]ej2fct
s(t) + js(t) = s(t)ej2fct
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In general,
Then
Complex envelope representation
s(t) = sI(t) + jsQ(t)
s(t) = sI(t) cos(2fct) sQ(t) sin(2fct)
s(t) = sI(t) sin(2fct) + sQ(t) cos(2fct)
s(t) = 00, f < 0
H(f fc)
Then H(f fc) = 0, f > 0H
(f
), f