NARROW-BAND FREQUENCY MODULATION

For small values of ,

)2sin())2sin(sin(

1))2sin(cos(

tftf

tf

mm

m

Thus the expression for FM signal

))2cos(2cos()( tmfmApktcfcAtx

can be expanded as

)2sin()2sin()2cos()( tmftcfcAtcfcAtx

since BABABA sinsincoscoscos

])(2cos[])(2cos[21)2cos()( tmfcftmfcfcAtcfcAtx

Then

since BABABA coscos2

1sinsin

fc

Ac

cA21

ffc +fmfc -fm

cA21

Bandwidth=2fm

Amplitude spectrum (single-sided plot)

WIDE-BAND FREQUENCY MODULATION

Wide-band FM signal.

In order to compute the spectrum of an angle-

modulated signal with a sinusoidal message signal, let

)2sin()( tmfmfft

Where (t) = Phase deviation

The corresponding FM signal is given by

))2sin(2cos()( tmftcfcAtx

It may alternatively be written as

tmfj

etcj

ecAtx 2sin

Re)(

where Re(x) denotes the real part of x.

The parameter is known as the modulation index and

is the maximum value of phase deviation of the FM

signal.

Consider the function z(t) given by

tmfjetz 2sin)(

It is periodic with frequency fm and can therefore be

expanded in a Fourier series as follows.

)2exp()( tmnfjnctz

mf

mf

dttmnfjtzmfnc2/1

2/1

)2exp()(

mf

mf

dttmnfjtmfjmfnc2/1

2/1

)2)2sin(exp[

The Fourier coefficient are given by

Hence we can rewrite cn as

dynyyjnc )])sin((exp[

21

The integral on the right hand side is a function of “n”

and and is known as the Bessel function of the first

kind of order n and argument .

It is conventionally denoted by Jn( ). That is,

dynyyjnJ )])sin((exp[

21)(

)(nJnc Thus

)2exp()()( tmnfjnJtz

we get

x(t) is accordingly given by

ntmnfcfjnJcAtx ])(2exp[)(Re)(

The discrete spectrum of x(t) is, therefore, given by

)()()(

2)(

mnf

cff

mnf

cffnJcA

fX

PROPERTIES OF BESSEL FUNCTIONS

Property - 1:

For n even,

we have Jn () = J-n ()

For n odd,

we have Jn () = (-1) J-n ()

Thus,

Jn () = (-1)n J-n ()

Property - 2:

For small values of the modulation index

we have

J0 () 1

J1 () /2

J3 () 0 for n > 2

n nJ 1)(2 Property - 3:

TABLE OF BESSEL FUNCTIONS

AMPLITUDE SPECTRUM

With the increase in the modulation index, the carrier

amplitude decreases while the amplitude of the

various sidebands increases. With some values of

modulation index, the carrier can disappear

completely.

POWER IN ANGLE-MODULATED SIGNAL

The power in an angle-modulated signal is easily

computed

2

2

)(2221

cA

n

nnJcAP

Thus the power contained in the FM signal is

independent of the message signal. This is an

important difference between FM and AM.

The time-average power of an FM signal may also be

obtained from

))(2cos()( ttfAtx cc

22

222

222

2

1)(

))(2(2cos2

1

2

1)(

))(2(cos)(

))(2cos()(

c

ccc

cc

cc

Atx

ttfAAtx

ttfAtx

ttfAtx

dttxT

txT

1where

TRANSMISSION BANDWIDTH OF FM SIGNALS

Theoretically, a FM signal contains an infinite number

of side frequencies so that the bandwidth required to

transmit such signal is infinite.

However, since the values of Jn() become negligible

for sufficiently large n, the bandwidth of an angle-

modulated signal can be defined by considering only

those terms that contain significant power.

In practice, the bandwidth of a FM signal can be

determined by knowing the modulation index and

using the Bessel function table.

Example:

Determine bandwidth with table of bessel functions

Calculate the bandwidth occupied by a FM signal with

a modulation index of 2 and a highest modulating

frequency of 2.5 kHz.

Referring to the table, we can see that this produces

six significant pairs of sidebands. The bandwidth can

then be determined with the simple formula

max2. NfWB

where N is the number of significant sidebands.

Using the example above and assuming a highest

modulating frequency of 2.5 kHz, the bandwidth of

the FM signal is

kHz

WB

30

5.262..

DETERMINE BW WITH CARSON'S RULE

An alternative way to calculate the bandwidth of a FM

signal is to use Carson's rule. This rule takes into

consideration only the power in the most significant

sidebands whose amplitudes are greater than 2 percent

of the carrier. These are the sidebands whose values

are 0.02 or more.

Carson's rule is given by the expression

BT BW f fm 2 2

In this expression, f is the maximum frequency

deviation, and fm is the maximum modulating

frequency.

We may thus define an approximate rule for the

transmission bandwidth of an FM signal generated by

a single of frequency fm as follows:

BT BW f fm f 2 2 2 1 1 ( )

Example:

Assuming a maximum frequency deviation of 5 kHz

and a maximum modulating frequency of 2.5 kHz, the

bandwidth would be

kHz

kHz

kHzkHzWB

15

5.72

)55.2(2..

Comparing the bandwidth with that computed in the

preceding example, you can see that Carson's rule

gives a smaller bandwidth.

FM SIGNAL GENERATION

They are two basic methods of generating frequency-

Modulated signals

•Direct Method

•Indirect Method

DIRECT FM

fi fc kf

m t ( )

In a direct FM system the instantaneous frequency is

directly varied with the information signal. To vary

the frequency of the carrier is to use an Oscillator

whose resonant frequency is determined by

components that can be varied. The oscillator

frequency is thus changed by the modulating signal

amplitude.

For example, an electronic Oscillator has an output

frequency that depends on energy-storage devices.

There are a wide variety of oscillators whose

frequencies depend on a particular capacitor value.

By varying the capacitor value, the frequency of

oscillation varies. If the capacitor variations are

controlled by m(t), the result is an FM waveform

INDIRECT FM

))(2cos()( ttfAtx cc

)(2)( tmpkt

t

dmktf0

)(2)(

Angle modulation includes frequency modulation FM

and phase modulation PM.

FM and PM are interrelated; one cannot change

without the other changing. The information signal

frequency also deviates the carrier frequency in PM.

Phase modulation produces frequency modulation.

Since the amount of phase shift is varying, the effect is

that, as if the frequency is changed.

Since FM is produced by PM , the later is referred to

as indirect FM.

The information signal is first integrated and then used

to phase modulate a crystal-controlled oscillator,

which provides frequency stability.

In order to minimize the distortion in the phase

modulator, the modulation index is kept small, thereby

is resulting in a narrow-band FM-signal

The narrow-band FM signal is multiplied in frequency

by means of frequency multiplier so as to produce the

desired wide-band FM signal.

The frequency multiplier is used to perform narrow

band to wideband conversion.

The frequency deviation of this new waveform is “M”

times that of the old, while the rate at which the

instantaneous frequency varies has not changed