ece 4710: lecture #21 1 frequency modulation overview: mathematical analysis of fm signal &...

ECE 4710: Lecture #21 1

Frequency Modulation

Overview: Mathematical analysis of FM signal & spectrum is very complicated FM is normally a wideband modulation method where

BFM >> baseband signal B» Spectrally inefficient

FM is a non-linear modulation method» AM & DSB-SC are linear methods

Non-linear method allows one to trade RF signal BW for non-linear increase in Rx S/N» Power for bandwidth tradeoff

FM is the most widely used analog modulation method for mobile radio used extensively from 1940-present day

Spectrally efficient digital methods have replaced FM in many mobile radio applications (e.g. cellular)» Congested spectrum requires spectral efficiency for large user populations



Complex envelope for phase modulation (PM) and frequency modulation (FM) is: The real envelope R(t) = | g(t) | = Ac constant value PM and FM are constant envelope modulation methods

» AM & DSB-SC are linear since R(t) is linearly to m(t) Bandpass signal is: Relationship between (t) & m(t) determines type of

modulation PM or FM For PM we have

» Linear relationship between phase and modulating signal» Still nonlinear modulation since:

)()( tjceAtg

)](2cos[)( ttfAts cc

}Re{)( )(2 tmjDtfjc

pc eeAts

)()( tmDt p



In the constant Dp is the phase sensitivity of the phase modulator with units of rad/V Larger Dp means greater phase change in s(t) for every

volt of m(t) assuming m(t) is voltage signal FM is a special case of PM

Frequency is the time derivative of phase: So

Let Df = frequency deviation constant with units rad/V•s then

)()( tmDt p

dtdf

dtftdtfd )( then and

dmDtt

f )()(

rad/s are units )(

abledummy vari

mD f


PM & FM Circuits

PM circuit pass unmodulated RF signal through circuit that introduces time variation in phase

FM circuit vary frequency of tuned RF oscillator circuit by varying resonant frequency of the circuit

Varactor diodes have variable capacitance controlled by amount of voltage applied across diode Time varying capacitance used to generate time-varying

phase for PM and time-varying frequency for FM


PM & FM Circuits

PM Circuit FM Circuit


PM & FM

PM: and FM: So and

Integrator and differentiator circuits (op-amp) can be used on baseband m(t) to generate PM from FM and vice versa

Integrator + PM circuit = FM output Differentiator + FM circuit = PM output

dmDtt

ff )()( )()( tmDt pp

dmDtmDt

ffpp )()(

t

fp

fp

p

f

pf dm

D

Dtm

dt

tdm

D

Dtm )()( and

)()(



For this class we are mainly interested in FM since it is the most widely used analog modulation method in the world analog mobile radio

Bandpass signal & Instantaneous frequency of s(t) is

and for FM we have

)](2cos[)( ttfAts cc dtdf

dttd

ftf ci

)(21

)(

)(21

)( tmDftf fci

Instantaneous frequency varies in time about assigned carrier frequency by an amount determined by Df and modulating

information signal m(t)



For illustration purposes assume m(t) is sinusoidal:

F = peak frequency deviation



fc + F

fc F

Note Constant FM Signal Envelope!!

Non-linear Class C or D power amplifiers with high DC to RF efficiency can

be used for FM since amplitude of RF signal does not contain any

information

fc

)(ts



Instantaneous frequency Frequency of s (t) at specific point in time

Fourier Transform frequency

FT spectrum for s (t) is evaluated for interval < t < + FT spectrum represents frequency content of s (t) over all

time **average** frequency content Peak frequency deviation is given by

dttstsfS tfj 2exp)]([)]([)(

)](max[ where21

tmVVDF ppf



Peak frequency deviation related to RF signal BW

Increasing amplitude of modulating signal (Vp) increases F and RF

signal BW

» Spectral components appear farther an farther away from fc

Note that average power of FM signal is constant value =» Independent of FM signal BW

As BW the spectral components near fc must decrease in strength

since the average power remains constant

AM or DSB-SC increase in m(t) affects Tx output power but not RF signal BW

FM increase in m(t) affects RF signal BW but not Tx output power

)](max[ where21

tmVVDF ppf

2/2cA



Frequency modulation index: B is absolute bandwidth of m(t) For sinusoidal m(t) B = fm sinusoid frequency Sinusoidal modulation is often used for demonstration

purposes and simplified calculations Actual m(t) is usually non-deterministic (e.g. voice)

FM signal spectrum

and g(t) is nonlinear function of m(t) no general relationship

relating G(f ) to M(f ) !! evaluated on case by case basis for deterministic waveforms only

B

VD

BF pf

f

2

)()()( *21

cc ffGffGfS

t

f dmDj

c

tj

ceAeAtgfG

)()( ][)]([)(


FM Signal Spectrum

Assume sinusoidal modulation signal:where

Complex envelope is:

Baseband signal spectrum can be shown to be:

Infinite series line spectrum (e.g. ) spaced by fm with amplitudes determined by first order Bessel functions Jn ()» Jn () can only be evaluated numerically

)2cos()( tfAtm mmf

2 and

2mf

fmfm

mff

ADfBF

f

AD

BF

)2sin()2cos()()()( tfjc

dfADj

c

dmDj

ctj

cmf

t

mmf

t

feAeAeAeAtg

n

mnc nffJAfG )()()(


Bessel Functions

n

mnc nffJAfG )()()(

1. Note that for f = fc n = 0

2. Carrier spectral amplitude is J0 ()

3. Modulation index determines carrier

strength

4. can be selected such that J0 ()

e.g. = 2.4, 5.5, etc.


FM Signal Spectrum

2.0BF

f

0.1BF

f

0.2BF

f


FM Signal Spectrum

0.5f

0.8f


FM Signal Spectrum

RF signal BW depends on f and B Computation of RF signal BW using standard

definitions (3-dB BW) is very difficult for anything but simplified m(t) sinusoidal Computer computations for non-deterministic waveforms

(data, voice, etc.) Carson’s rule approximate BW for 98% total

power Simple and therefore very useful Widely used rather than computational approach for

estimating signal BW

BB fT )1(2

ece 4710: lecture #21 1 frequency modulation overview: mathematical analysis of fm signal &...

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