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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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ꢀ

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  • Constant Envelope OFDM Transmission Over Impulsive Noise Power-Line Communication Channels

    Constant Envelope OFDM Transmission Over Impulsive Noise Power-Line Communication Channels

    Constant Envelope OFDM Transmission Over Impulsive Noise Power-Line Communication Channels K. M. Rabie∗, E. Alsusa∗, A. D. Familuaz, and L. Chengz ∗School of Electrical & Electronic Engineering, The University of Manchester, Manchester, UK, M13 9PL Email: {khaled.rabie, e.alsusa}@manchester.ac.uk zSchool of Electrical & Information Engineering, University of Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Email: [email protected]; [email protected] Abstract—Signal blanking is a simple and efficient method to into distortionless techniques such as coding [3], tone reservation reduce the effect of impulsive noise over power-line channels. The [4] and selective mapping (SLM) [5]; and non-distortionless efficiency of this method, however, is found to be not only impacted techniques such as signal clipping and peak cancellation [6], by the threshold selection but also by the average peak-to-average ratio (PAPR) value of the orthogonal frequency division multiplexing [7]. In general, although distortionless techniques are relatively (OFDM) signals. As such, the blanking capability can be further more complex than the non-distortionless ones, they are still more enhanced by reducing the PAPR value. With this in mind, in this attractive because the distortion caused by non-distortionless paper we evaluate the performance of constant envelope OFDM schemes can outweigh the benefit of the reduced PAPR. The other (CE-OFDM) which has inherently the lowest achievable PAPR of alternative technique is based on transforming the OFDM signal 0 dB; therefore, the proposed system is expected to provide the lower bound performance of the blanking-based method. In order prior to transmission and applying the inverse transformation at to characterize system performance, we consider the probability the receiver prior to demodulation.