Properties of Angle Modulation
Properties of Angle Modulation
Reference:
Sections 4.2 and 4.3 of
Professor Deepa Kundur
S. Haykin and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., John Wiley & Sons, Inc., 2007. ISBN-13
978-0-471-43222-7.
University of Toronto
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Angle Modulation
I
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Properties of Angle Modulation
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carrier
Phase Modulation (PM):
θi (t)
= 2πfc t + kp m(t)
1 dθi (t)
kp dm(t)
fi (t) =
= fc +
2π dt
2π dt
sPM (t) = Ac cos[2πfc t + kp m(t)]
I
message
amplitude
modulation
Frequency Modulation (FM):
Z
θi (t) = 2πfc t + 2πkf
t
m(τ )dτ
phase
modulation
0
fi (t)
=
sFM (t)
=
1 dθi (t)
= fc + kf m(t)
2π dt
Z t
Ac cos 2πfc t + 2πkf
m(τ )dτ
frequency
modulation
0
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Professor Deepa Kundur (University of Toronto)
carrier
carrier
message
message
amplitude
modulation
amplitude
modulation
phase
modulation
phase
modulation
frequency
modulation
frequency
modulation
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4.2 Properties of Angle Modulation
I
Consider a sinusoid g (t) = Ac cos(2πf0 t + φ) where T0 =
I
The power of g (t) (over a 1 ohm resister) is defined as:
=
=
=
=
=
I
T0 /2
1
T0
Z
A2c
T0
Z
g 2 (t)dt =
−T0 /2
1
T0
Z
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4.2 Properties of Angle Modulation
Constancy of Transmitted Power
P
Professor Deepa Kundur (University of Toronto)
T0 /2
−T0 /2
Constancy of Transmitted Power: AM
1
f0
or T0 f0 = 1.
A2c cos2 (2πf0 t + φ)dt
T0 /2
1
[1 + cos(4πf0 t + 2φ)] dt
2
A2c
sin(4πf0 t + 2φ) T0 /2
t+
2T0
4f0
−T0 /2
2
T0
sin(4πf0 T0 /2 + 2φ) − sin(−4πf0 T0 /2 + 2φ)
Ac
−T0
−
+
2T0
2
2
4πf0
A2c
sin(2π + 2φ) − sin(−2π + 2φ)
A2
T0 +
= c
2T0
4πf0
2
−T0 /2
Therefore, the power of a sinusoid is NOT dependent on f0 , just its envelop
Ac .
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4.2 Properties of Angle Modulation
4.2 Properties of Angle Modulation
Constancy of Transmitted Power: PM
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Properties of Angle Modulation
Constancy of Transmitted Power: FM
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4.2 Properties of Angle Modulation
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4.2 Properties of Angle Modulation
Constancy of Transmitted Power
Nonlinearity of Angle Modulation
Consider PM (proof also holds for FM).
I
Suppose
Therefore, angle modulated signals exhibit constancy of transmitted
power.
I
s1 (t)
=
Ac cos [2πfc t + kp m1 (t)]
s2 (t)
=
Ac cos [2πfc t + kp m2 (t)]
Let m3 (t) = m1 (t) + m2 (t).
s3 (t)
= Ac cos [2πfc t + kp (m1 (t) + m2 (t))]
6= s1 (t) + s2 (t)
∵ cos(2πfc t + A + B) 6= cos(2πfc t + A) + cos(2πfc t + B)
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4.2 Properties of Angle Modulation
4.2 Properties of Angle Modulation
Nonlinearity of Angle Modulation
Irregularity of Zero-Crossings
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Therefore, angle modulation is nonlinear.
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4.2 Properties of Angle Modulation
Properties of Angle Modulation
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4.2 Properties of Angle Modulation
Zero-Crossings: AM
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Zero-crossing: instants of time at which waveform changes
amplitude from positive to negative or vice versa.
Zero-Crossings: PM
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4.2 Properties of Angle Modulation
4.2 Properties of Angle Modulation
Zero-Crossings: FM
Irregularity of Zero-Crossings
Therefore angle modulated signals exhibit irregular zero-crossings
because they contain information about the message (which is
irregular in general).
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4.2 Properties of Angle Modulation
Properties of Angle Modulation
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4.2 Properties of Angle Modulation
Visualization Difficulty of Message
I
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Visualization: AM
Visualization of a message refers to the ability to glean insights
about the shape of m(t) from the modulated signal s(t).
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4.2 Properties of Angle Modulation
4.2 Properties of Angle Modulation
Visualization: PM
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Visualization: FM
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4.2 Properties of Angle Modulation
Bandwidth vs. Noise Trade-Off
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Noise affects the message signal piggy-backed as amplitude
modulation more than it does when piggy-backed as angle
modulation.
I
The more bandwidth that the angle modulated signal takes,
typically the more robust it is to noise.
Therefore it is difficult to visualize the message in angle modulated
signals due to the nonlinear nature of the modulation process.
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4.2 Properties of Angle Modulation
Visualization Difficulty of Message
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4.2 Properties of Angle Modulation
carrier
message
amplitude
modulation
phase
modulation
frequency
modulation
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