comm 601: modulation i lecture 7 - wideband fm - generation...

Dr. Ahmed El-Mahdy COMM 601: Modulation I

COMM 601: Modulation I

Lecture 7

- Wideband FM - Generation of FM signals


Wideband FM (WBFM) Now we determine the spectrum of the single tone FM signal for an

arbitrary value of the modulation index

signal. FM of envelope

complex theis)(where

)(Re

)(Re

Re

2sin2cos

2sin

2

22sin

2sin2

tfj

c

tfj

tfjtfj

c

tftfj

c

mcc

m

c

cm

mc

eAtc

etc

eeA

eA

tftfAts


Wideband FM (WBFM)

Where:


Wideband FM (WBFM) Bessel function


Wideband FM (WBFM)


Transmitted Power of WBFM

Since

T

T

22

)(2

2

1

2

1

)(where)(2

1

:as signals) bandpass oftion representa of form three the(fromeasily deduced becan thisNote

cc

tj

cT

AA

eAtctcP


Spectrum of WBFM

Spectrum of WBFM


8

.

.

.

.

442

332

222

2

442

332

222

2

2

2

22cos

44

33

22

11

44

33

22

11

mcmcc

mcmcc

mcmcc

mcmcc

mcmcc

mcmcc

mcmcc

mcmcc

cococ

mcmc

n

nc

FM

mc

n

ncFM

fffJfffJA

fffJfffJA

fffJfffJA

fffJfffJA

fffJfffJA

fffJfffJA

fffJfffJA

fffJfffJA

ffJffJA

nfffnfffJA

fS

tnftfJAts

n = 2

n = 1

n = 0

n = 3

n = 4

n = -2

n = -1

n = -3

n = -4


9

0 100 200 300 400 500 600 700 800 900 1000

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

XFM

(f)

FREQUENCY [Hz]

VAc 10

[Hz]f

[Hz]f

m

c

50

500

14.1239.02

10

22 c

o

AJ

835.25767.02

10

221 cA

J 764.13528.02

10

222 cA

J

644.01289.02

10

222 cA

J

2

0 100 200 300 400 500 600 700 800 900 1000

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

XFM

(f)

FREQUENCY [Hz]

5.0

6955.49385.02

10

25.0 c

o

AJ

215.12423.02

10

25.01 cA

J

1530.003060.02

10

25.02 cA

J


Spectrum of WBFM

(1)The spectrum of an FM signal contains a carrier component

and an infinite set of sideband frequencies located

symmetrically on either side of the carrier at frequency

separation of ,...3,2, mmm fff

(2) For the special case of , Only the Bessel

coefficients have significant values so that the

spectrum of the FM is composed of a carrier and a single

pair of side frequencies at

1

)(and)( 10 JJ

mc ff


Spectrum of WBFM

(3) Unlike the AM signal, the amplitude of the

carrier component varies with the modulation

index


FM with Single Tone Modulating SignalRemember:

signal. FM theofindex modulation theis

where

m

m

fβΔff

Δfβ

)1(]sincos[ ttAs(t) mcc

signalmodulatingtheofamplitudetheismA

mf Akf


Example: Spectrum of WBFM

In this example, we investigate

the ways in which variations in

the amplitude and frequency of a

sinusoidal modulating signal

affect the spectrum of WBFM

signal.

Case I: The frequency of the

modulating Signal is fixed but its

amplitude is varied producing a

frequency deviation f

See the amplitude

variation of the spectrum

for different

Normalized amplitude

increasesthenincreasesfincreasesAAs m ,,,


Example: (Cont.)

Case II: The amplitude of

the modulating Signal is

fixed (producing constant

frequency deviation )but

its frequency is varied.

We see that when the

frequency deviation is

constant and the

modulation index is

increased, we have an

increasing number of

spectral lines crowding

into a fixed frequency

interval:

constantisSincedecreases;increases,As mm Afβ


Example (Cont.):

That is when approaches infinity, the bandwidth of the

FM signal approaching the limiting value of which is an

important point to keep in mind for later discussion.

f2


16

Transmission Bandwidth of FM Signals

It can be shown that 98 percent of the normalized total signal power is

contained in the bandwidth

HzfBW

fBW

m

m

2

121

HzfBW 12 m

NARROW-BAND ANGLE-MODULATED

SIGNALS

WIDE-BAND ANGLE-MODULATED

SIGNALS

Usually a value of β<0.2 is

sufficient to satisfy this

condition.

Hzfff

ffBW

fBW

m

m

m

m

222

121

In theory, an FM signal contains an infinite number of side frequencies so that

the bandwidth required to transmit such a signal is similarly infinite in extent.

Carson’s rule

This expression can

represents the general

case where is the max.

frequency in the signal mf

The larger the modulation

Index, the larger the

bandwidth


Transmission Bandwidth of FM Signals

Another form of FM Bandwidth:


Example:


Example (Cont.)

)(2m

fnB

kHzkHzB 40)102(2

kHzkHzB 4010)11(2

HzfBW 12 m


Example (Cont.)

Calculate the magnitudes ????


Generation of FM Signals: Indirect Method

(1) Direct Method:

In this method, the carrier frequency is directly varied in

accordance with the input baseband signal, which is

performed using voltage controlled oscillator.

VCO ))(2cos( tcxtfA

cc)(tx


Generation of FM Signals: Indirect Method

(2) Indirect Method: NBFM WBFM

This method is preferred when the carrier frequency stability is

of major concern as in commercial radio broadcasting


Frequency Multiplier

]sincos[

]sincos[:Remember

ttA

tf

ftAs(t)

mcc

m

m

cc


Example:

Dr. Ahmed El-Mahdy COMM 601: Modulation I 26

Received signal S ( t )

Limited signal SL ( t )

+VL

+VL

+VL

+VL

The received FM signal is passed to a BPF to remove

the out of band noise and then passed to a limiter to

remove any amplitude fluctuations due to noise. The next

step, the FM signal is demodulated to extract the signal.

Demodulation of FM Signals


Demodulation of FM Signals

)(2 tmkt fc

Envelope

detector

Dr. Ahmed El-Mahdy COMM 601: Modulation I 28

• The DC offset can be removed with a capacitor placed in series to the demodulator. The varying portion of the signal is proportional to the original signal.

• By passing the differentiated signal through an ideal envelope detector and low-pass filter, we can recover the original signal. The carrier frequency determines the DC offset of this signal, which will be much larger than the varying portion of the signal.


Difference between AM and Angle Modulation

(1)Zero crossings (instants of time at which a waveform

changes from a negative to a a positive value) no longer

have a perfect regularity in their spacing.

(2)The envelope of FM or PM signal is constant (equal to the

carrier amplitude), where as the envelope of an AM signal

is dependent on the message signal.

(3) The FM signal s(t) is a nonlinear function of the modulating

signal m(t) which makes FM to be a nonlinear modulation

process.

(4)Unlike AM, the spectrum of an FM signal is not related in a

simple manner to that of the modulating signal, rather, its

analysis is much more difficult than that of an AM signal.


Advantages of FM over AM


Disadvantages of FM over AM

comm 601: modulation i lecture 7 - wideband fm - generation...

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