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
carrier
I
Phase Modulation (PM):
- θi (t)
- =
- 2πfc t + kpm(t)
1 dθi (t)
kp dm(t)
message
- fi (t)
- =
=
= fc +
2π dt
Ac cos[2πfc t + kpm(t)]
2π dt
sPM (t)
amplitude modulation
I
Frequency Modulation (FM):
Z
t
θi (t) fi (t)
===
2πfc t + 2πkf
m(τ)dτ
0
phase modulation
1 dθi (t)
= fc + kf m(t)
2π dt
- ꢀ
- ꢁ
Z
t
sFM (t)
Ac cos 2πfc t + 2πkf
m(τ)dτ
frequency modulation
0
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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
- 4.2 Properties of Angle Modulation
- Constancy of Transmitted Power
- Constancy of Transmitted Power: AM
1
f0
I
Consider a sinusoid g(t) = Ac cos(2πf0t + φ) where T0 = or T0f0 = 1.
I
The power of g(t) (over a 1 ohm resister) is defined as:
ZZ
Z
T
/2
T
/2
- 0
- 0
- 1
- 1
P
===
g2(t)dt =
A2c cos2(2πf0t + φ)dt
- T0
- T0
−T /2 T
−T /2
- 0
- 0
/2
Ac2
T0
1
0
[1 + cos(4πf0t + 2φ)] dt
2
−T /2
0
ꢂꢂꢂꢂ
- ꢀ
- ꢁ
T
/2
Ac2
sin(4πf0t + 2φ)
0
t +
2T0
4f0
−T /2
0
- ꢀꢃ
- ꢄ
- ꢃ
- ꢄꢁ
Ac2
2T0
Ac2
- T0
- −T0
sin(4πf0T0/2 + 2φ) − sin(−4πf0T0/2 + 2φ)
=
=
−
+
- 2
- 2
4πf0
- ꢀ
- ꢁ
sin(2π + 2φ) − sin(−2π + 2φ)
Ac2
T0
- +
- =
2T0
4πf0
2
I
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
- Constancy of Transmitted Power: FM
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- 4.2 Properties of Angle Modulation
- 4.2 Properties of Angle Modulation
- Constancy of Transmitted Power
- Nonlinearity of Angle Modulation
Consider PM (proof also holds for FM).
I
Suppose
s1(t)
==
Ac cos [2πfc t + kpm1(t)]
s2(t)
Ac cos [2πfc t + kpm2(t)]
Therefore, angle modulated signals exhibit constancy of transmitted
power.
I
Let m3(t) = m1(t) + m2(t).
s3(t)
==
Ac cos [2πfc t + kp(m1(t) + m2(t))]
s1(t) + s2(t)
∵ cos(2πfc t + A + B) = 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
I
Zero-crossing: instants of time at which waveform changes amplitude from positive to negative or vice versa.
Therefore, angle modulation is nonlinear.
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- 4.2 Properties of Angle Modulation
- 4.2 Properties of Angle Modulation
- Zero-Crossings: AM
- 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
- 4.2 Properties of Angle Modulation
- Visualization Difficulty of Message
- Visualization: AM
I
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
- Visualization: FM
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- 4.2 Properties of Angle Modulation
- 4.2 Properties of Angle Modulation
- Visualization Difficulty of Message
- Bandwidth vs. Noise Trade-Off
I
Noise affects the message signal piggy-backed as amplitude modulation more than it does when piggy-backed as angle
modulation.
Therefore it is difficult to visualize the message in angle modulated
signals due to the nonlinear nature of the modulation process.
I
The more bandwidth that the angle modulated signal takes, typically the more robust it is to noise.
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4.2 Properties of Angle Modulation
carrier message amplitude modulation
phase modulation
frequency modulation
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ꢀ