angle modulation fm1

8/2/2019 Angle Modulation FM1

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Bessel Functions and their

Relationship to FM

The equation of the FM wave cannot be solved with

algebra or trigonometric identities.

However, certain Bessel-function identities are

available that will yield solutions to the equation andallow us to determine the frequency components of

an FM wave.

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t

f

f+tV=tv m

m

cccs sincos


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Frequency ModulationFrequency Modulation

The equation

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t

f

f+tV=tv m

m

cccs sincos may be expressed as Bessel

series (Bessel functions)Theimage cannotbe displayed.Your computer may nothaveenough memory toopen theimage,or theimagemay havebeen corrupted.Restartyour computer,and then open thefileagain. Ifthe red x stillappears,you may havetodeletethe imageandthen insertitagain.

g

g=nmcncs tn+JV=tv cos

where Jn(F) are Bessel functions of the first kind. Expanding the equation for a few

terms we have:

]

.

!

tJVtJV

tJVtJVtJVtv

mcmc

mcmcc

ff

mcc

ff

mcc

ff

mcc

ff

mcc

f

ccs

2Amp

2

2Amp

2

Amp

1

Amp

1

Amp

0

)2(cos)()2(cos)(

)(cos)()(cos)()(cos)()(

[[F[[F

[[F[[F[F

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FM Signal Spectrum.FM Signal Spectrum.

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The amplitudes drawn are completely arbitrary, since we have not found any value for

Jn(F) this sketch is only to illustrate the spectrum.

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FM SpectrumFM Spectrum Bessel Coefficients.Bessel Coefficients.

The FM signal spectrum may be determined from

g

g!

!n

mcncs tnJVtv )cos()()( [[F

The values for the Bessel coefficients, Jn(F) may be found from

graphs or, preferably, tables of Bessel functions of the first kind.

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In the series forvs(t), n = 0 is the carrier component, i.e. )cos()(0 tJV cc [F , hence the

n = 0 curve shows how the component at the carrier frequency, fc, varies in amplitude,

with modulation index F.19

F

Jn(F)

F = 2.4 F = 5


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Table showing Bessel Functions of theTable showing Bessel Functions of the

First kindFirst kind


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Hence for a given value of modulation index F, the values ofJn(F) may be read off the

graph and hence the component amplitudes (VcJn(F)) may be determined.

A further way to interpret these curves is to imagine them in 3 dimensions

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Examples from the graphExamples from the graph

F = 0: WhenF = 0 the carrier is unmodulated and J0(0) = 1, all otherJn(0) = 0,i.e.

F = 2.4:From the graph (approximately)

J0(2.4) = 0, J1(2.4) = 0.5, J2(2.4) = 0.45 and J3(2.4) = 0.2

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Significant SidebandsSignificant Sidebands Spectrum.Spectrum.

As may be seen from the table of Bessel functions, for values ofn above a certain

value, the values ofJn(F) become progressively smaller. In FM the sidebands are

considered to be significant ifJn(F) u 0.01 (1%).

Although the bandwidth of an FM signal is infinite, components with amplitudesVcJn(F), for which Jn(F) < 0.01 are deemed to be insignificant and may be ignored.

Example:A message signal with a frequency fm Hz modulates a carrierfcto produce

FM with a modulation index F = 1. Sketch the spectrum.

n Jn(1) Amplitude Frequency

0 0.7652 0.7652Vc fc1 0.4400 0.44Vc fc+fm fc -fm2 0.1149 0.1149Vc fc+2fm fc - 2fm

3 0.0196 0.0196Vc fc+3fm fc -3fm

4 0.0025 Insignificant

5 0.0002 Insignificant

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As shown, the bandwidth of the spectrum containing significant

components is 6fm, forF = 1.

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The table below shows the number of significant sidebands for various modulation

indices (F) and the associated spectral bandwidth.

F Noofsidebandsu 1% of

unmodulatedcarrier

Bandwidth

0.1 2 2fm0.3 4 4fm

0.5 4 4fm

1.0 6 6fm

2.0 8 8fm

5.0 16 16fm

10.0 28 28fm

e.g. forF = 5,16 sidebands

(8 pairs).

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Carsons Rule for FM Bandwidth.Carsons Rule for FM Bandwidth.

An approximation for the bandwidth of an FM signal is given by

BW = 2(Maximum frequency deviation + highest modulated

frequency)

)(2Bandwidth mc ff (! Carsons Rule

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Narrowband and Wideband FMNarrowband and Wideband FM

From the graph/table of Bessel functions it may be seen that for small F, (F e 0.3)

there is only the carrier and 2 significant sidebands, i.e. BW = 2fm.

FM with F e 0.3 is referred to as narrowband FM (NBFM) (Note, the bandwidth is

the same as DSBAM).

ForF > 0.3 there are more than 2 significant sidebands. As F increases the number of

sidebands increases. This is referred to as wideband FM (WBFM).

Narrowband FM NBFM

Wideband FM WBFM

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VHF/FMVHF/FM

mc Vf E!(

VHF/FM (Very High Frequency band = 30MHz 300MHz) radio transmissions, in the

band 88MHz to 108MHz have the following parameters:

Max frequency input (e.g. music) 15kHz fm

Deviation 75kHz

Modulation Index F 5m

c

f

f(!F

ForF = 5 there are 16 sidebands and the FM signal bandwidth is 16fm = 16 x 15kHz

= 240kHz.Applying Carsons Rule BW

= 2(75+15) = 180kHz.

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Comments FMComments FM

The FM spectrum contains a carrier component and an infinite number of sidebands

at frequencies fcs nfm (n = 0, 1, 2, )

FM signal, g

g!

!

n

mcncs tnJVtv )cos()()( [[F

In FM we refer to sideband pairs not upper and lower sidebands. Carrier or other

components may not be suppressed in FM.

The sidebands at equal distances from fc have equal amplitudes so that the

sideband distribution is symmetrical about the carrier frequency.

The J coefficients occasionally have negative values, signifying a 180 degree

phase change for that particular pair of sidebands.

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Components are significant ifJn(F) u 0.01. ForF


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In FM,unlike in AM the amplitude of the carrier component does not remain

constant. The relative amplitudes of components in FM depend on the values Jn(F).

Keeping the overall amplitude of the FM wave constant would be very

difficult, if the amplitude of the carrier were not reduced when the amplitude

of the various sidebands increased.

It is possible for the carrier component of FM wave to disappear completely.

This happens for certain values of the modulation index, called eigenvalues.

These are approximately, 2.4, 5.5, 8.6, 11.8 and so on.

angle modulation fm1

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