EELE 3370

Communications I

Angle Modulation and Demodulation

Dr. Talal Skaik 2016

Islamic University of Gaza

Electrical Engineering Department

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Introduction

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Introduction

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In AM signals, the amplitude of a carrier is modulated by a signal m(t), and, hence, the information content of m(t) is in the amplitude variations of the carrier.

Angle Modulation: The generalized angle θ(t) of a sinusoidal signal is varied in proportion the message signal m(t).

Two types of Angle Modulation

• Frequency Modulation: The frequency of the carrier signal is varied in proportion to the message signal.

• Phase Modulation: The phase of the carrier signal is varied in proportion to the message signal

The Concept of Instantaneous Frequency A general sine wave signal can be expressed as θ(t) is the generalized angle. For a sine wave with fixed

frequency and phase, it can be represented as a linear function of time with a slope ω=2πf:

In general ω is the derivative of the angle. That is

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Phase Modulation (PM) The message signal is modulating the phase of the carrier signal: without loosing generalization, we can omit the initial phase θ0 and we get the following PM signal:

ωi is called the instantaneous frequency of the modulated signal and Kp is a constant. 5

Frequency Modulation (FM) The message signal is modulating the frequency of the carrier signal: Kf is constant

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Relationship between FM and PM In FM:

In PM: 7

Example Sketch FM and PM waves for the modulating signal m(t) shown. The

constants kf and kp are 2π x105 and 10π, respectively, and the carrier

frequency fc is 100 MHz.

Solution

→ FM Signal

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Example Solution

→ PM Signal

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Example

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Example Sketch FM and PM waves for digital modulating signal m(t) shown. The

constants kf and kp are 2π x105 and π/2, respectively, and fc is 100 MHz.

Solution

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Example

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Bandwidth of Angle Modulated Waves To determine the bandwidth of an FM wave :

Define

and define

Then

Expanding the factor in power series

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2c

nj tn nf f

f

k kFM ( t ) A jk a( t ) a ( t ) j a ( t ) e

! n !

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Bandwidth of Angle Modulated Waves The FM signal is expressed as an un-modulated carrier plus spectra of

a(t), a2(t), … an(t), … centered at ωc.

Let M(ω) be the spectrum of m(t) with bandwidth B.

The bandwidth of a(t) is also B.

a2(t) has a bandwidth of 2B [M(ω)*M(ω)]

a3(t) has a bandwidth of 3B

an(t) has a bandwidth of nB

Conclusion: FM signal has infinite bandwidth!!. (theoretically)

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Bandwidth of Angle Modulated Waves

Special cases:

Narrow-Band Angle Modulation

The angle modulation is not linear in general. However, if |kf a(t)| << 1

→ only the 1st two terms are important in the above equation.

This is a linear modulation. It is like an AM wave with bandwidth =

2B. This is called Narrow Band FM (NBFM).

* However the waveform is entirely different from AM.

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Bandwidth of Angle Modulated Waves

Narrow-Band Angle Modulation

Similarly the Narrow Band PM (NBPM) is given by:

•NBPM also has approximate bandwidth of 2B.

•The narrow band angle modulation is similar to AM (same bandwidth,

carrier plus spectrum centered on ωc).

The difference: in angle modulation the sideband spectrum is π/2 phase

shifted with respect to the carrier.

•However, the waveforms of AM and FM are completely different. In

AM signal, the frequency is constant and the amplitude varies with time,

whereas in FM amplitude is constant and the frequency varies with time. 16

Bandwidth of Angle Modulated Waves

Wide-Band FM (WBFM)

This is the situation where we cannot ignore the higher order terms

because (|kf a(t)| << 1) is not satisfied. (can be due to high kf ).

Consider a low-pass m(t) with bandwidth B Hz.

The signal m(t) is now approximated by a staircase signal as

shown in Fig. (next slide).

The signal m(t) is now approximated by pulses of constant amplitude.

Each pulse will be called a cell.

The FM spectrum for consists of the sum of the Fourier

transforms of these sinusoidal pulses corresponding to all the cells.

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m( t )

m( t )

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Bandwidth of Angle Modulated Waves

Wide-Band FM (WBFM)

Define peak frequency deviation in hertz as ∆f:

Then, #

which is overestimation due to staircase approximation.

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Bandwidth of Angle Modulated Waves

Wide-Band FM (WBFM)

Considering NBFM Δf ≈ 0. Then above reduces to, BFM ≈ 4B Hz.

However we previously found for NBFM, BNBFM ≈ 2B Hz.

Therefore a better approximation is:

This result is known as Carson’s rule. Carson’s rule can be also be expressed in terms of the deviation ratio

β (also called modulation index for FM)

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Bandwidth of Angle Modulated Waves

Wide-Band PM (WBPM)

All the analysis developed for the FM can be applied to the PM by

replacing mp by and kf by kp. That is:

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pm

Example 5.3 (a) Estimate the BFM and BPM for the modulating signal m(t) shown for

kf =2π x105 and kp =5π. Assume the essential bandwidth of the

periodic m(t) as the frequency of the third harmonic.

(b) Repeat the problem if the amplitude of m(t) is doubled [ if m(t) is

multiplied by 2].

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Example 5.3 - solution For FM

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Example 5.3 - solution For PM

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Example 5.4 Repeat example 5.3 if m(t) is time expanded by a factor of 2, that is, if

the period of m(t) is 4 x 10-4

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Example 5.4 - solution

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Example 5.5 An angle-modulated signal with carrier frequency ωc =2π x105 is

described by the equation:

(a) Find the power of the modulated signal.

(b) Find the frequency deviation ∆f.

(c) Find the deviation ratio β.

(d) Find the phase deviation ∆φ.

(e) Estimate the bandwidth of φEM(t).

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10 5 3000 10 2000EM c( t ) cos t sin t sin t

Example 5.5 - solution

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