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Chapter 5AM, FM, and Digital Modulated Systems

Phase Modulation (PM) Frequency Modulation (FM) Generation of PM and FM Spectrum of PM and FM Carson’s Rule Narrowband FM

Huseyin BilgekulEeng360 Communication Systems I

Department of Electrical and Electronic Engineering Eastern Mediterranean University

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AAM and FM M and FM Modulation Modulation

(a) Carrier wave. (a) Carrier wave.

((bb) Sinusoidal modulating signal. ) Sinusoidal modulating signal.

(c) (c) Amplitude-modulated signal.Amplitude-modulated signal.

(d) Frequency modula(d) Frequency modulated signal.ted signal.

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Angle Modulation Angle Modulation

We have seen that an AM signal can be represented as

ttmAts c ccos )](1[)(

Now we will see that information can also be carried in the angle of the signal as

Note that in this type of modulation the amplitude of signal carries information.

ttAts cc cosHere the amplitude Ac remains constant and the angle is modulated.

This Modulation Technique is called the Angle Modulation

Angle modulation: Vary either the Phase or the Frequency of the carrier signal

Phase Modulation and Frequency Modulation are special cases of Angle Modulation

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Angle Modulation Angle Modulation Representation of PM and FM signals:

The Complex Envelope for an Angle Modulation is given by tjceAtg

cAtgtR Is a constant Real envelope,

θ(t) - linear function of the modulating signal m(t)

The Angle-modulated Signal in time domain is given by ttAts cc cos

g(t) - Nonlinear function of the modulation.

Special Case 1: For PM the phase is directly proportional to the modulating signal. i.e.;

Where Dp is the Phase sensitivity of the phase modulator, having units of radians/volt.

Special Case 2:For FM, the phase is proportional to the integral of m(t) so that

where the frequency deviation constant Df has units of radians/volt-sec.

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Angle Modulation Angle Modulation

)](cos[)( tmDtAts pcc Resulting PM wave:

Phase Modulation occurs when the instantaneous phase varied in proportion to that of the message signal.

tmDt p Dp is the phase sensitivity of the modulator

Frequency Modulation occurs when the instantaneous frequency is varied linearly with the message signal.

)()( tmDt fci

t

f dmDt

] )([cos)(

t

fcc dmDtAts Resulting FM wave:

Df is the frequency deviation constant

Instantaneous Frequency (fi) of a signal is defined by

dtdt

tdtt

ii )(

)( where ttt c

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Phase and Frequency ModulationsPhase and Frequency Modulations

• Phase Modulation • Frequency Modulation

Comparing above two equations , we see that if we have a PM signal modulated by mp(t), there is also FM on the signal, corresponding to a different modulation wave shape that is given by:

Similarly if we have a FM signal modulated by mf(t),the corresponding phase modulation on this signal is:

Where f and p denote frequency and phase respectively.

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t

fp

fp dm

DD

tm

dttdm

DD

tm p

f

pf

Integrator Phase Modulator (Carrier Frequency fc)

Differentiator Frequency Modulator (Carrier Frequency fc)

Generation of FM from PM and vice versa Generation of FM from PM and vice versa

tm p

tm f tm p

tm f

ts

ts

FM Signal

PM signal

Generation of FM using a Phase Modulator:

Generation of PM using a Frequency Modulator:

Gain f

p

DD

Gain f

p

DD

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FM with sinusoidal modulating signal FM with sinusoidal modulating signal

The Instantaneous Frequency of the FM signal is given by:

dttdftftf cid

21

The Peak Frequency Deviation is given by:

dttdF

21max

The Frequency Deviation from the carrier frequency:

The Peak-to-peak Deviation is given by

dttd

dttdFpp

2

1min21max

∆F is related to the peak modulating voltage by pf VDF21

tmV p maxWhere

12i c

d tf t f

dt

If a bandpass signal is represented by: )( ttt c

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FM with sinusoidal modulating signal FM with sinusoidal modulating signal

dttdftf ci

21

But,

Vp BW

Average Power does not change with modulation

2PowerAverage

2cA

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Angle Modulation Angle Modulation

Advantages: Constant amplitude means Efficient Non-linear Power Amplifiers can be used.

Superior signal-to-noise ratio can be achieved (compared to AM) if bandwidth is sufficiently high.

Disadvantages: Usually require more bandwidth than AM

More complicated hardware

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Modulation Index Modulation Index

The Peak Phase Deviation is given by: t max

∆θ is related to the peak modulating voltage by: ppVD tmVp maxWhere

The Phase Modulation Index is given by: pWhere ∆θ is the peak

phase deviation

The Frequency Modulation Index is given by:

BF

f ∆F Peak Frequency Deviation

B Bandwidth of the modulating signal

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Spectra of Angle modulated signals Spectra of Angle modulated signals

Spectra for AM, DSB-SC, and SSB can be obtained with simple formulas relating S(f) to M(f).

But for angle modulation signaling, because g(t) is a nonlinear function of m(t). Thus, a general formula relating G(f) to M(f) cannot be obtained.

To evaluate the spectrum for angle-modulated signal, G(f) must be evaluated on a case-by-case basis for particular modulating waveshape of interest.

cc ffGffGfS

21

tjceAtgfG Where

Spectrum of Angle modulated signal

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Spectrum of PM or FM Signal with Sinusoidal Modulating Signal Spectrum of PM or FM Signal with Sinusoidal Modulating Signal Assume that the modulation on the PM signal is

tAtm mmp sin tt m sinThen

mpp ADWhere is the phase Modulation Index.

Same θ(t) could also be obtained if FM were used

tAtm mmf coswhere

mmff AD /

mf ADF21

The Complex Envelope is:

and

The peak frequency deviation would be

tjc

tjc

meAeAtg sin which is periodic with periodm

m fT 1

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Using discrete Fourier series that is valid over all time, g(t) can be written as

n

n

tjnn

mectg

2

2

sinm

m

mmT

T

tjntj

m

cn dtee

TAc

Where

nc

njcn JAeAc

sin

21

Which reduces to

Jn(β) – Bessel function of the first kind of the nth order

Taking the fourier transform of the complex envelope g(t), we get

nn

n JJ 1 Is a special property of Bessel Functions

Spectrum of PM or FM Signal with Sinusoidal Modulating Signal Spectrum of PM or FM Signal with Sinusoidal Modulating Signal

n

nmn nffcfG or

n

c n mn

G f A J f nf

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Bessel Functions of the First Kind Bessel Functions of the First Kind

J0(β)=0 at β=2.4, 5.52 & so on

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Bessel Functions of the First Kind Bessel Functions of the First Kind

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])cos[()()( tnJAtS mn

cnc

The FM modulated signal in time domain

From this equation it can be seen that the frequency spectrum of an FM waveform with a sinusoidal modulating signal is a discrete frequency spectrum made up of components spaced at frequencies of c± nm.

By analogy with AM modulation, these frequency components are called sidebands.

We can see that the expression for s(t) is an infinite series. Therefore the frequency spectrum of an FM signal has an infinite number of sidebands.

The amplitudes of the carrier and sidebands of an FM signal are given by the corresponding Bessel functions, which are themselves functions of the modulation index

Observations:

Frequency spectrum of FM Frequency spectrum of FM

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Spectra of an FM Signal with Sinusoidal ModulationSpectra of an FM Signal with Sinusoidal Modulation

BT

cA

fS

21

)(

f

1.0

The following spectra show the effect of modulation index, , on the bandwidth of an FM signal, and the relative amplitudes of the carrier and sidebands

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BT

J0(1.0)

J1(1.0)

J2(1.0)

cA

fS

21

)(

f

1.0

Spectra of an FM Signal with Sinusoidal ModulationSpectra of an FM Signal with Sinusoidal Modulation

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BT

cA

fS

21

)(

f

1.0

Spectra of an FM Signal with Sinusoidal ModulationSpectra of an FM Signal with Sinusoidal Modulation

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Although the sidebands of an FM signal extend to infinity, it has been found experimentally that signal distortion is negligible for a bandlimited FM signal if 98% of the signal power is transmitted.

Based on the Bessel Functions, 98% of the power will be transmitted when the number of sidebands transmitted is 1+ on each side.

Carson’s ruleCarson’s rule

(1+fm

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Carson’s ruleCarson’s rule Therefore the Bandwidth required is given by

β – phase modulation index/ frequency modulation index

B – bandwidth of the modulating signal

mT fB 12

For sinusoidal modulation mfB

Carson’s rule : Bandwidth of an FM signal is given by

Note: When β =0 i.e. baseband signals mT fB 2

2 1TB B

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Narrowband Angle ModulationNarrowband Angle Modulation Narrowband Angle Modulation is a special case of angle modulation where θ(t) is restricted to a small value.

rad 2.0)( t

The complex envelope can be approximated by a Taylor's series in which only first two terms are used.

tjAtg c 1

jceAtg

ttAtAts cccc sincos

]1xfor 1 because [ xe xbecomes

The Narrowband Angle Modulated Signal is

The Spectrum of Narrowband Angle Modulated Signal is

ccccc ffffjffff

AfS

2

.2

,

fMfj

D

fMDtf f

p

where

PM

FM

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Indirect method of generating WBFMIndirect method of generating WBFM

Balanced Modulator ttAtAts cccc sincos

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Wideband Frequency Wideband Frequency modulationmodulation

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FM Stero SystemFM Stero System

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FM Stero SystemFM Stero System