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

Professor Deepa Kundur (University of Toronto)

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Angle Modulation

carrier

Phase Modulation (PM):

1 dθ_i(t)

k_pdm(t)

message

=
= f_c+

2π dt

A_ccos[2πf_ct + k_pm(t)]

2π dt

s_PM(t)

amplitude modulation

Frequency Modulation (FM):

θ_i(t) f_i(t)
===
2πf_ct + 2πk_f

m(τ)dτ

phase modulation

1 dθ_i(t)
= f_c+ k_fm(t)

2π dt

s_FM(t)

A_ccos 2πf_ct + 2πk_f

m(τ)dτ

frequency modulation

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carrier

message

amplitude modulation

phase modulation

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Consider a sinusoid g(t) = A_ccos(2πf₀t + φ) where T₀= or T₀f₀= 1.

The power of g(t) (over a 1 ohm resister) is deﬁned as:

g²(t)dt =

A²_ccos²(2πf₀t + φ)dt

−T /2 T
−T /2

[1 + cos(4πf₀t + 2φ)] dt
2

−T /2

ꢂꢂꢂꢂ

sin(4πf₀t + 2φ)

−T /2

sin(4πf₀T₀/2 + 2φ) − sin(−4πf₀T₀/2 + 2φ)
=

4πf₀

sin(2π + 2φ) − sin(−2π + 2φ)

4πf₀

Therefore, the power of a sinusoid is NOT dependent on f₀, just its envelop

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Consider PM (proof also holds for FM).

Suppose

s₁(t)

==
A_ccos [2πf_ct + k_pm₁(t)]

s₂(t)

A_ccos [2πf_ct + k_pm₂(t)]

Therefore, angle modulated signals exhibit constancy of transmitted

power.

Let m₃(t) = m₁(t) + m₂(t).

s₃(t)

==
A_ccos [2πf_ct + k_p(m₁(t) + m₂(t))]

s₁(t) + s₂(t)

∵ cos(2πf_ct + A + B) = cos(2πf_ct + A) + cos(2πf_ct + B)

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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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Therefore angle modulated signals exhibit irregular zero-crossings

because they contain information about the message (which is irregular in general).

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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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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.

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

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Properties of Angle Modulation

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