NARROW-BAND FREQUENCY MODULATION
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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