To define and explain frequency modulation (FM) and phase modulation (PM)

To analyze the FM in terms of Mathematical analysis

To analyze the Bessel function for FM and PM

To analyze the FM bandwidth and FM power distribution

Frequency modulation (FM) and phase modulation (PM)

Analysis of FM Bessel function for FM and PM FM bandwidth Power distribution of FM

Angle modulation is the process by which the angle (frequency or Phase) of the carrier signal is changed in accordance with the instantaneous amplitude of modulating or message signal.

also known as “Exponential modulation"

classified into two types such as› Frequency modulation (FM)› Phase modulation (PM)

Used for :› Commercial radio broadcasting› Television sound transmission› Two way mobile radio› Cellular radio› Microwave and satellite communication

system

FM is the process of varying the frequency of a carrier wave in proportion to a modulating signal.

The amplitude of the carrier wave is kept constant while its frequency and a rate of change are varied by the modulating signal.

Fig 3.1 : Frequency Modulated signal

The important features about FM waveforms are :

i. The frequency varies.ii. The rate of change of carrier frequency changes

is the same as the frequency of the information signal.

iii. The amount of carrier frequency changes is proportional to the amplitude of the information signal.

iv. The amplitude is constant.

The FM modulator receives two signals,the information signal from an external source and the carrier signal from a built in oscillator.

The modulator circuit combines the two signals producing a FM signal which passed on to the transmission medium.

The demodulator receives the FM signal and separates it, passing the information signal on and eliminating the carrier signal.

Federal Communication Coporation (FCC) allocation for a standard broadcast FM station is as shown in Fig.3.2

Mathematical analysis: Let message signal:

(3.1) And carrier signal: (3.2) Where carrier frequency is very much

higher than message frequency.

tVt mmm cos

]cos[ tVt ccc

In FM, frequency changes with the change of the amplitude of the information signal. So the instantenous frequency of the FM wave is;

(3.3)

K is constant of proportionality

tKVtKv mmCmci cos

Thus, we get the FM wave as:

(3.4)

Where modulation index for FM is given by

)sincos(cos)( 1 tKVtVVctv mm

mCCFM

)sincos()( tmtVtv mfCCFM

m

mfKVm

Frequency deviation: ∆f is the relative placement of carrier frequency (Hz) w.r.t its unmodulated value. Given as:

mC KVmax

mC KVmin

mCCd KV minmax

22md KVf

Therefore:

mf

m

m

ffm

HzKVfor

KVf

)(

2;mVf

FM broadcast station is allowed to have a frequency deviation of 75 kHz. If a 4 kHz (highest voice frequency) audio signal causes full deviation (i.e. at maximun amplitude of information signal) , calculate the modulation index.

Determine the peak frequency deviation, , and the modulation index, mf, for an FM modulator with a deviation Kf = 10 kHz/V. The modulating signal to be transmitted is Vm(t) = 5 cos ( cos 10kπt).

f

Thus, for general equation:

)coscos()( tmtVtv mfCCFM

)sinsin()( tmtVtv mfCCFM

)sinsin(cos)sincos(sin tmttmtV mfcmfCC

ttmJtmJVtv mCmCfCfCFM )sin()sin()(sin)([ 10

ttmJV mCmCfC )2sin()2sin()([ 2

..])3sin()3sin()([ 3 ttmJV mCmCfC

It is seen that each pair of side band is preceded by J coefficients. The order of the coefficient is denoted by subscript m. The Bessel function can be written as

N=number of the side frequency M=modulation index

....

!2!22/

!1!12/

!1

2

42

nm

nm

nm

mJ ffn

ffm

Theoretically, the generation and transmission of FM requires infinite bandwidth. Practically, FM system have finite bandwidth and they perform well.

The value of modulation index determine the number of sidebands that have the significant relative amplitudes

If n is the number of sideband pairs, and line of frequency spectrum are spaced by fm, thus, the bandwidth is:

For n=>1

mfm nfB 2

Estimation of transmission b/w; Assume mf is large and n is approximate mf +2; thus Bfm=2(mf +2)fm

=

(1) is called Carson’s rule

mm

fff )2(2

)1)........(2(2 mfm ffB

The worse case modulation index which produces the widest output frequency spectrum.

Where › ∆f(max) = max. peak frequency deviation› fm(max) = max. modulating signal frequency

(max)

(max)

mffDR

An FM modulator is operating with a peak frequency deviation =20 kHz.The modulating signal frequency, fm is 10 kHz, and the 100 kHz carrier signal has an amplitude of 10 V. Determine :

a) The minimum bandwidth using Bessel Function table.

b)The minimum bandwidth using Carson’s Rule.

c)Sketch the frequency spectrum for (a), with actual amplitudes.

For an FM modulator with a modulation index m=1, a modulating signal Vm(t)=Vm sin(2π1000t), and an unmodulated carrier Vc(t) = 10sin(2π500kt), determine :

a) Number of sets of significant side frequencies

b) Their amplitudesc) Draw the frequency spectrum showing

their relative amplitudes.

a) From table of Bessel function, a modulation index of 1 yields a reduced carrier component and three sets of significant side frequencies.

b) The relative amplitude of the carrier and side frequencies are

Jo = 0.77 (10) = 7.7 V J1 = 0.44 (10) = 4.4 V J2 = 0.11 (10) = 1.1 V J3 = 0.02 (10) = 0.2 V

Frequency spectrum

497 498 499 500 501 502 503 J3 J2 J1 JO J1 J2 J3

7.7 V

4.4V4.4V

1.1V1.1V

0..2 0..2

As seen in Bessel function table, it shows that as the sideband relative amplitude increases, the carrier amplitude,J0 decreases.

This is because, in FM, the total transmitted power is always constant and the total average power is equal to the unmodulated carrier power, that is the amplitude of the FM remains constant whether or not it is modulated.

In effect, in FM, the total power that is originally in the carrier is redistributed between all components of the spectrum, in an amount determined by the modulation index, mf, and the corresponding Bessel functions.

At certain value of modulation index, the carrier component goes to zero, where in this condition, the power is carried by the sidebands only.

The average power in unmodulated carrier

The total instantaneous power in the angle modulated carrier.

The total modulated power

RVP c

c

2

RVtt

RVP

ttRV

RtmP

cc

ct

cc

t

2)](22cos[

21

21

)]([cos)(

22

222

RV

RV

RV

RVPPPPP nc

nt 2)(2..

2)(2

2)(2

2..

222

21

2

210 RV

RV

RV

RVPPPPP nc

nt 2)(2..

2)(2

2)(2

2..

222

21

2

210

a) Determine the unmodulated carrier power for the FM modulator and condition given in example 3.4, (assume a load resistance RL = 50 Ώ)

b) Determine the total power in the angle modulated wave.

a) Pc = (10)(10)/(2)(50) =1 W

b) )50(2)2.0(2

)50(2)1.1(2

)50(2)4.4(2

)50(27.7 2222

tP

WPt 0051.10008.00242.03872.05929.0

To define and explain frequency modulation (FM) and phase modulation (PM)

To analyze the FM in terms of Mathematical analysis

To analyze the Bessel function for FM and PM

To analyze the FM bandwidth and FM power distribution

END OF CHAPTER 3 PART 1: ANGLE MODULATION

(INTRODUCTION)