Generation of FM wave in MATLAB

1. 1
Statement:
Generate FM wave (Frequency Modulation) at a different modulation
index. Verify whether the modulation index depends on the band width of FM
modulated wave or the power of entire FM wave including side bands?
Literature Background:
Frequency Modulation:
The process in which the frequency of the carrier is changed according to
the instantaneous amplitude of the base band signal.
Mathematically:
( ) (1)
Figure 1 : Baseband signal and modulated wave signal.
Deviation:
The amount by which the signal frequency varies is termed as daviation.
Normally measured in kiloHertz (kHz).

2. 2
Narrowband FM:
Narrow band FM is defined as an FM transmission where the value of
modulation index is small enough that the terms in the Bessel expansion, i.e.
sidebands are negligible.
Narrowband FM, NBFM often uses deviation figures of around ±3kHz or
possibly slightly more. If quality is not as important for radio communications
applications, the much narrower bandwidth has advantages in terms of radio
spectrum efficiency.
Wideband FM:
Wideband FM is defined as the situation where the modulation index is
above 0.5. Under these circumstances the sidebands beyond the first two terms
are not insignificant.
Broadcast stations in the VHF portion of the frequency spectrum between
88.5 and 108 MHz uses large values of deviation, typically ±75kHz. This is
known as wideband FM (WBFM).
These signals are capable of supporting high quality transmissions, but
occupy a large amount of bandwidth. Usually 200 kHz is allowed for each
wideband FM transmission.
Band width:
The difference of highest and lowest frequencies which contain 90%
energy of the signal is called bandwidth.
Where bandwidth of an FM is given by:
B.W=2 x x No. of side bands (2)
Power of a Signal:
Power is a time average of energy.
∫ | | (3)
In MATLAB we can use this formula to calculate power:
∑ | |
(4)

3. 3
Procedure:
Matlab Coding:
clc
clear all
close all
t=0:1:300;
fm=100; %modulated freq
fc=20*fm; %carrier freq
fs=5*fc; %sampling freq
ts=t/fs; %sampling time
Vm=5;
Vc=10;
Mf=1;
Fm =Vc*cos(2*pi*fc*ts+Mf*sin(2*pi*fm*ts));
plot(ts*1000,Fm)
title('FM wave in time domain')
xlabel('time (ms)')
ylabel('Vc')
grid on;
figure
z=fft(Fm);
z=(z(1:length(z)/2+1));
frq=(0:length(z)-1)*fs/length(z)/2;
plot(frq/1000,z)
title('FM wave in Frequency domain')
xlabel(['F= ' num2str(fc/1000) 'kHz'])
grid on
power=sum(Fm.^2)/length(t) %for calculating power in entire FM wave

4. 4
Analysis:
Output:
= 0
Figure 2: FM Modulated Wave in time domain
Figure 3: FM Modulated Wave in frequency domain
power =
50.1661

5. 5
= 0.5
Figure 4: FM Modulated Wave in time domain
Figure 5: FM Modulated Wave in frequency domain
power =
50.1661

6. 6
= 1
Figure 6: FM Modulated Wave in time domain
Figure 7: FM Modulated Wave in frequency domain
power =
50.1661

7. 7
= 1.5
Figure 8: FM Modulated Wave in time domain
Figure 9: FM Modulated Wave in frequency domain
power =
50.1661

8. 8
= 2
Figure 10: FM Modulated Wave in time domain
Figure 11: FM Modulated Wave in frequency domain
power =
50.1661

9. 9
Table no 1:
S.No Modulation Index Power in entire FM
wave
Band width in Hertz
1 0 50.1661W 1Hz
2 0.5 50.1661W 400Hz
3 1 50.1661W 600 Hz
4 2 50.1661W 800 Hz
Questions
Q1: What will happen to the power of FM wave including side-bands
(increasing or decreasing) with the increase in modulation index? Give
mathematical reasoning.
Ans: Power is a time average of energy. By increasing the Modulation index,
power of FM wave will remains constant because power is independent of
Modulation index.
Mathematically:
(5)
And from eq(5) it is clear that power only depends upon .
Q2: What will happen to the band-width of FM wave (increasing or
decreasing) with the increase in modulation index? Give mathematical
reasoning.
Ans: The difference of highest and lowest frequencies which contain 90%
energy of the signal is called bandwidth. If modulation index is increased,
number of sidebands also increases. If number of sidebands increases, the
bandwidth of signal also increases.
Mathematically:
(6)
In Basel chart we can see that no. of side bands increases with increase in
modulation index.

10. 10
Q3: At what modulation index of FM, the band-width of AM and FM are
same?
Ans: The bandwidth of AM and FM is same when M 1, because number of
sidebands are equal for M 1.
Mathematically:
If n=1 (M<1)
=
Conclusion:
 Power of entire FM band remains constant if is constant. As:
 Band width of FM depends upon modulation index. If M is increased,
number of sidebands increases and bandwidth also increases because
 Modulation index has no effect on change in power.
 Modulation index is low for higher frequencies of baseband signal and it
increases for lower frequencies by the relation