electronic modulation
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Narrow-band Frequency Modulation
KEEE343 Communication Theory
Lecture #16, May 3, 2011Prof. Young-Chai [email protected]
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Summary
·Narrowband Frequency Modulation
· Wideband Frequency Modulation
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Narrow-Band Frequency Modulation
• Narrow-Band FM means that the FM modulated wave has narrow
bandwidth.
• Consider the single-tone wave as a message signal:
• FM signal
• Instantaneous frequency
• Phase
m(t) = Am cos(2πf mt)
f i(t) = f c + kf Am cos(2πf mt)
= f c +∆f cos(2
π
f mt)
θi(t) = 2π
Z t
0
f i(τ ) dτ = 2π
f ct +
∆f
2πf msin(2πf mt)
= 2πf ct + ∆f
f msin(2πf mt)
∆f = kf Am
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• Definitions
• Phase deviation of the FM wave
• Modulation index of the FM wave:
• Then, FM wave is
• To be narrow-band FM, should be small.
• For small compared to 1 radian, we can rewrite
β = ∆f f m
s(t) = Ac cos[2πf ct + β sin(2πf mt)]
f m
s(t) ≈ Ac cos(2πf ct)− β Ac sin(2πf ct) sin(2πf mt)
s(t) = Ac cos(2πf ct)cos(β sin(2πf mt))−Ac sin(2πf ct)sin(β sin(2πf mt))
β cos[β sin(2πf mt)] ≈ 1, sin[β sin(2πf mt)] ≈ β sin(2πf mt)
β
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[Ref: Haykin & Moher, Textbook]
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• Consider the modulated signal
which can be rewritten as
where
and
Polar Representation from Cartesian
s(t) = a(t) cos(2πf ct + φ(t))
s(t) = sI (t) cos(2πf ct)−
sQ(t) sin(2πf ct)
sI (t) = a(t) cos(φ(t)), and, sQ(t) = a(t)sin(φ(t))
a(t) =s2
I (t) + s2
Q(t) 12 , and φ(t) = tan−1
sQ(t)
sI (t)
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• Approximated narrow-band FM signal can be written as
• Envelope
• Maximum value of the envelope
• Minimum value of the envelope
(t) ≈ Ac cos(2πf ct)− β Ac sin(2πf ct) sin(2πf mt)
for smallβ
a(t) = Ac
1 + β 2 sin2(2πf mt)
≈ Ac
1 +
1
2β 2 sin2(2πf mt)
Amax = Ac
1 +
1
2β 2
Amin = Ac
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• The ratio of the maximum to minimum value
• Rewriting the approximated narrow-band FM wave is
• Average power
• Average power of the unmodulated wave
Amax
Amin
=
1 +1
2β 2
s(t) ≈ Ac cos(2πf ct) +2β cos(2π(f c + f m)t)−
2β Ac cos(2π(f c − f m)t)
P av = 1
2A2
c +
1
2β Ac
2
+
1
2β Ac
2
= 1
2A2
c(1 + β 2)
P c =1
2A2
c
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• Ration of the average power to the power of the unmodulated wave
• Comparison with the AM signal
• Approximated Narrow-band FM signal
• AM signal
P av
P c= 1+ β 2
s(t) ≈ Ac cos(2πf ct) +2β cos(2π(f c + f m)t)−
2β Ac cos(2π(f c − f m)t)
sAM(t) = Ac cos(2πf ct) +2µAc {cos[2π(f c + f m)t)] + cos[2π(f c − f m)t]}
difference between the narrow-band FM and AM waves
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• Bandwidth of the narrow-band FM:
• Phasor interpretation
2f m
[Ref: Haykin & Moher, Textbook]
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• Angle
• Using the power series of the tangent function such as
θ(t) = 2
πf ct +
φ(t) = 2
πf ct + tan
−1
(β
sin(2πf mt
))
tan−1(x) ≈ x−
1
3x3 + · · ·
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• Angle can be approximated as
• Ideally, we should have
• The harmonic distortion value is
• The maximum absolute value of D(t) is
θ(t) ≈ 2πf ct + β sin(2πf mt)
D(t) = β 3
3 sin3(2πf mt)
θ(t)≈
2πf ct + β sin(2πf mt)−
1
3β 3
sin3
(2πf mt)
Dmax =β 3
3
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• For example for
which is small enough for it to be ignored in practice.
β = 0.3,
Dmax = 0.33
3 = 0.009 ≈ 1%
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Amplitude Distortion of Narrow-band FM
• Ideally, FM wave has a constant envelope
• But, the modulated wave produced by the narrow-band FM differ from
this ideal condition in two fundamental respects:
•The envelope contains a residual amplitude modulation that varies
with time
• The angle contains harmonic distortion in the form of third-
and higher order harmonics of the modulation frequencyθi(t)
f m
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Wide-Band Frequency Modulation
• Spectral analysis of the wide-band FM wave
or
where is called “complex envelope”.
Note that the complex envelope is a periodic function of time with a
fundamental frequency which means
where
s(t) = Ac cos[2πf ct + β sin(2πf mt)]
s(t) = < [Ac exp[ j2πf ct + jβ sin(2πf mt)]] = <[s̃(t)exp( j2πf ct)]
s̃(t) = Ac exp [ jβ sin(2πf mt)]
m
s̃(t) = s̃(t + kT m) = s̃(t + kf m
)
T m = 1/f m
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• Then we can rewrite
• Fourier series form
where
s̃(t) =∞X
n=−∞
cn exp( j2πnf mt)
cn = f mZ 1/(2f m)
−1/(2f m) s̃(t) exp(− j2πnf mt) dt
= f mAc
Z 1/(2f m)
−1/(2f m)
exp[ jβ sin(2πf mt)− j2πnf mt] dt
s̃(t) = s̃(t + k/f m)
= Ac exp[ jβ sin(2πf m(t + k/f m))]
= Ac exp[ jβ sin(2πf mt + 2kπ)]
= Ac exp[ jβ sin(2πf mt)]
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• Define the new variable:
Then we can rewrite
• nth order Bessel function of the first kind and argument
• Accordingly
which gives
x = 2πf mt
cn = Ac
2π
Z π
−π
exp[ j(β sinx− nx)] dx
β
J n(β ) = 1
2π
Z π
−π
exp[ j(β sinx− nx)] dx
cn = AcJ n(β )
s̃(t) = Ac
∞X
n=−∞
J n(β ) exp( j2πnf mt)
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• Then the FM wave can be written as
• Fourier transform
which shows that the spectrum consists of an infinite number of deltafunctions spaced at for
s t = < s̃ t exp j2πf ct
= <
"Ac
∞Xn=−∞
J n(β ) exp[ j2πn(f c + f m)t]
#
= Ac
∞Xn=−∞
J n(β ) cos[2π(f c + nf m)t]
S (f ) = Ac
2
∞X
n=−∞
J n(β ) [δ (f − f c − nf m) + δ (f + f c + nf m)]
f = f c ± nf m n = 0, +1, +2, ...
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1. For different values of n
2. For small value of
6. The equality holds exactly for arbitrary
Properties of Single-Tone FM for Arbitrary ModulationIndex β
J n(β ) = J −n(β ), for n even
J n(β ) = −J
−n(β ), for n odd
J 0(β ) ≈ 1,
J 1(β ) ≈
β
2
J n(β ) ≈
0, n > 2
β
β
∞X
n=−∞
J 2
n(β ) = 1
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[Ref: Haykin & Moher, Textbook]
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1. The spectrum of an FM wave contains a carrier component and and an infiniteset of side frequencies located symmetrically on either side of the carrier at
frequency separations of ...
2. The FM wave is effectively composed of a carrier and a single pair of side-
frequencies at .
3. The amplitude of the carrier component of an FM wave is dependent on the
modulation index . The averate power of such as signal developed across a 1-ohm resistor is also constant:
The average power of an FM wave may also be determined from
f m, 2
f m, 3
f m
f c ± f
m
β
P av =1
2A2
c
P av = 1
2A2
c
∞X
n=−∞
J 2
n(β )