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# Angle Modulation Frequency Modulation

## Consider again the general carrier ( ) ( )

c c c c
+ t V = t v cos
( )
c c
+ t represents the angle of the carrier.
There are two ways of varying the angle of the carrier.
a) By varying the frequency, e
c
Frequency Modulation.
b) By varying the phase, |
c
Phase Modulation
Frequency Modulation
In FM, the message signal m(t) controls the frequency f
c
of the carrier. Consider the
carrier
( ) ( ) t V = t v
c c c
cos
then for FM we may write:
FM signal
( ) ( ) ( ) t + f V = t v
c c s
deviation frequency 2 cos
,where the frequency deviation
will depend on m(t).
Given that the carrier frequency will change we may write for an instantaneous
carrier signal
( ) ( ) ( )
i c i c i c
V = t f V = t V cos 2 cos cos
where |
i
is the instantaneous angle =
t f = t
i i
2
and f
i
is the instantaneous
frequency.
Frequency Modulation
Since
t f =
i i
2 then
dt
d

= f f =
dt
d
i
i i
i
2
1
or 2
i.e. frequency is proportional to the rate of change of angle.
If f
c
is the unmodulated carrier and f
m
is the modulating frequency, then we may
deduce that
( )
dt
d

= t f + f = f
i
m c c i
2
1
cos
Af
c
is the peak deviation of the carrier.
Hence, we have
( ) t f + f =
dt
d

m c c
i
cos
2
1
,i.e.
( ) t f + f =
dt
d
m c c
i
cos 2 2
Frequency Modulation
After integration i.e.
( ) ( )
}
dt t f +
m c c
cos 2
( )
m
m c
c i

t f
+ t =
sin 2
( ) t
f
f
+ t =
m
m
c
c i
sin
Hence for the FM signal, ( ) ( )
i c s
V = t v cos
( ) ( )
|
|
.
|

\
|
t
f
f
+ t V = t v
m
m
c
c c s
sin cos
Frequency Modulation
The ratio
m
c
f
f
is called the Modulation Index denoted by | i.e.
frequency modulating
deviation frequency Peak
=
Note FM, as implicit in the above equation for v
s
(t), is a non-linear process i.e.
the principle of superposition does not apply. The FM signal for a message m(t) as a
band of signals is very complex. Hence, m(t) is usually considered as a 'single tone
modulating signal' of the form
( ) ( ) t V = t m
m m
cos
Frequency Modulation
The equation ( ) ( )
|
|
.
|

\
|
t
f
f
+ t V = t v
m
m
c
c c s
sin cos
may be expressed as Bessel
series (Bessel functions)
( ) ( ) ( )

= n
m c n c s
t n + J V = t v cos
where J
n
(|) are Bessel functions of the first kind. Expanding the equation for a few
terms we have:

+ + + +
+ + + =

+
t J V t J V
t J V t J V t J V t v
m c m c
m c m c
c
f f
m c c
f f
m c c
f f
m c c
f f
m c c
f
c c s
2 Amp
2
2 Amp
2
Amp
1
Amp
1
Amp
0
) 2 ( cos ) ( ) 2 ( cos ) (
) ( cos ) ( ) ( cos ) ( ) ( cos ) ( ) (
e e | e e |
e e | e e | e |
FM Signal Spectrum.
The amplitudes drawn are completely arbitrary, since we have not found any value for
J
n
(|) this sketch is only to illustrate the spectrum.
Generation of FM signals Frequency
Modulation.
An FM demodulator is:

a voltage-to-frequency converter V/F
a voltage controlled oscillator VCO
In these devices (V/F or VCO), the output frequency is dependent on the input voltage
amplitude.
V/F Characteristics.
Apply V
IN
, e.g. 0 Volts, +1 Volts, +2 Volts, -1 Volts, -2 Volts, ... and measure the
frequency output for each V
IN
. The ideal V/F characteristic is a straight line as
shown below.
f
c
, the frequency output when the input is zero is called the undeviated or nominal
carrier frequency.
V
f
is called the Frequency Conversion Factor,
denoted by o per Volt.
V/F Characteristics.
Consider now, an analogue message input,
( ) ( ) t V = t m
m m
cos
As the input m(t) varies from
m m
V V + 0
the output frequency will vary from a
maximum, through f
c
, to a minimum
frequency.
V/F Characteristics.
For a straight line, y = c + mx, where c = value of y when x = 0, m = gradient, hence
we may say
IN OUT
V + f = f
c
and when V
IN
= m(t)
( ) t m + f = f
c OUT
,i.e. the deviation depends on m(t).
Considering that maximum and minimum input amplitudes are +V
m
and -V
m

respectively, then
m c
m c
V f = f
V + f = f

min
max
on the diagram on the previous slide.
The peak-to-peak deviation is f
max
f
min
, but more importantly for FM the peak
deviation Af
c
is
Peak Deviation,
m c
V = f Hence, Modulation Index,
m
m
m
c
f
V
=
f
f
=
Summary of the important points of FM
In FM, the message signal m(t) is assumed to be a single tone frequency,
( ) ( ) t V = t m
m m
cos
The FM signal v
s
(t) from which the spectrum may be obtained as
( ) ( ) ( )

= n
m c n c s
t n + J V = t v cos
where J
n
(|) are Bessel coefficients and Modulation Index,
m
m
m
c
f
V
=
f
f
=
o Hz per Volt is the V/F modulator, gradient or Frequency Conversion Factor,
o per Volt
o is a measure of the change in output frequency for a change in input amplitude.
Peak Deviation (of the carrier frequency from f
c
)
m c
V = f
FM Signal Waveforms.
The diagrams below illustrate FM signal waveforms for various inputs
At this stage, an input digital data
sequence, d(t), is introduced
the output in this case will be FSK,
(Frequency Shift Keying).
FM Signal Waveforms.
Assuming
s 0' for
s 1' for ) (
V
V t d
=
+ =
)
`

= =
+ = =
s 0' for
s 1' for
0
1
V f f f
V f f f
c OUT
c OUT
o
o the output switches
between f
1
and f
0
.
FM Signal Waveforms.
The output frequency varies gradually from f
c
to (f
c
+ oV
m
), through f
c
to
(f
c
- oV
m
) etc.
FM Spectrum Bessel Coefficients.
In the series for v
s
(t), n = 0 is the carrier component, i.e. ) cos( ) (
0
t J V
c c
e | , hence the
n = 0 curve shows how the component at the carrier frequency, f
c
, varies in amplitude,
with modulation index |.
|
J
n
(|)
| = 2.4 | = 5
FM Spectrum Bessel Coefficients.
Hence for a given value of modulation index |, the values of J
n
(|) may be read off the
graph and hence the component amplitudes (V
c
J
n
(|)) may be determined.
A further way to interpret these curves is to imagine them in 3 dimensions
Examples from the graph
| = 0: When | = 0 the carrier is unmodulated and J
0
(0) = 1, all other J
n
(0) = 0, i.e.
| = 2.4: From the graph (approximately)
J
0
(2.4) = 0, J
1
(2.4) = 0.5, J
2
(2.4) = 0.45 and J
3
(2.4) = 0.2
Effect of increase
in modulation
index , , due to
increasing
amplitude
Effect of increase
in modulation
index , , due to
decrease in f
m
Significant Sidebands Spectrum.
As may be seen from the table of Bessel functions, for values of n above a certain
value, the values of J
n
(|) become progressively smaller. In FM the sidebands are
considered to be significant if J
n
(|) > 0.01 (1%).
Although the bandwidth of an FM signal is infinite, components with amplitudes
V
c
J
n
(|), for which J
n
(|) < 0.01 are deemed to be insignificant and may be ignored.
Example: A message signal with a frequency f
m
Hz modulates a carrier f
c
to produce
FM with a modulation index | = 1. Sketch the spectrum.
n J
n
(1) Amplitude Frequency
0 0.7652

0.7652V
c
f
c
1 0.4400 0.44V
c
f
c
+f
m
f
c
- f
m
2 0.1149 0.1149V
c
f
c
+2f
m
f
c
- 2f
m

3 0.0196 0.0196V
c
f
c
+3f
m
f
c
-3 f
m

4 0.0025 Insignificant
5 0.0002 Insignificant

Significant Sidebands Spectrum.
As shown, the bandwidth of the spectrum containing significant
components is 6f
m
, for | = 1.
Significant Sidebands Spectrum.
The table below shows the number of significant sidebands for various modulation
indices (|) and the associated spectral bandwidth.
| No of sidebands > 1% of
unmodulated carrier
Bandwidth
0.1 2 2f
m

0.3 4 4f
m

0.5 4 4f
m

1.0 6 6f
m

2.0 8 8f
m

5.0 16 16f
m

10.0 28 28f
m

e.g. for | = 5,
16 sidebands
(8 pairs).
Carsons Rule for FM Bandwidth.
An approximation for the bandwidth of an FM signal is given by
BW = 2(Maximum frequency deviation + highest modulated
frequency)
) ( 2 Bandwidth
m c
f f + A =
Carsons Rule
Narrowband and Wideband FM
From the graph/table of Bessel functions it may be seen that for small |, (| s 0.3)
there is only the carrier and 2 significant sidebands, i.e. BW = 2fm.

FM with | s 0.3 is referred to as narrowband FM (NBFM) (Note, the bandwidth is
the same as DSBAM).
For | > 0.3 there are more than 2 significant sidebands. As | increases the number of
sidebands increases. This is referred to as wideband FM (WBFM).
Narrowband FM NBFM
Wideband FM WBFM
VHF/FM
m c
V f o = A
VHF/FM (Very High Frequency band = 30MHz 300MHz) radio transmissions, in the
band 88MHz to 108MHz have the following parameters:
Max frequency input (e.g. music) 15kHz
f
m

Deviation 75kHz
Modulation Index | 5
m
c
f
f A
= |
For | = 5 there are 16 sidebands and the FM signal bandwidth is 16fm = 16 x 15kHz
= 240kHz. Applying Carsons Rule BW = 2(75+15) = 180kHz.
The FM spectrum contains a carrier component and an infinite number of sidebands
at frequencies f
c
nf
m
(n = 0, 1, 2, )
FM signal,

=
+ =
n
m c n c s
t n J V t v ) cos( ) ( ) ( e e |
In FM we refer to sideband pairs not upper and lower sidebands. Carrier or other
components may not be suppressed in FM.
The relative amplitudes of components in FM depend on the values J
n
(|), where
m
m
f
V o
| =
thus the component at the carrier frequency depends on m(t), as do all the
other components and none may be suppressed.
Components are significant if J
n
(|) > 0.01. For |<<1 (| ~ 0.3 or less) only J
0
(|) and
J
1
(|) are significant, i.e. only a carrier and 2 sidebands. Bandwidth is 2f
m
, similar to
DSBAM in terms of bandwidth - called NBFM.
Large modulation index
m
c
f
f A
= | means that a large bandwidth is required called
WBFM.
The FM process is non-linear. The principle of superposition does not apply. When
m(t) is a band of signals, e.g. speech or music the analysis is very difficult
(impossible?). Calculations usually assume a single tone frequency equal to the
maximum input frequency. E.g. m(t) band 20Hz 15kHz, fm = 15kHz is used.
Power in FM Signals.
From the equation for FM

=
+ =
n
m c n c s
t n J V t v ) cos( ) ( ) ( e e |
we see that the peak value of the components is V
c
J
n
(|) for the n
th
component.
Single normalised average power =
2
2
) (
2
RMS
pk
V
V
=
|
|
.
|

\
|
then the n
th
component is
( )
2
) (
2
) (
2
2
| |
n c n c
J V J V
= |
.
|

\
|
Hence, the total power in the infinite spectrum is
Total power

=
=
n
n c
T
J V
P
2
)) ( (
2
|
2
)) ( (
2
2
2
2
c
n
n
c
V
J
V
= =

=
|
Power in FM Signals.
Now consider if we generate an FM signal, it will contain an infinite number of
sidebands. However, if we wish to transfer this signal, e.g. over a radio or cable, this
implies that we require an infinite bandwidth channel. Even if there was an infinite
channel bandwidth it would not all be allocated to one user. Only a limited bandwidth
is available for any particular signal. Thus we have to make the signal spectrum fit into
the available channel bandwidth. We can think of the signal spectrum as a train and
the channel bandwidth as a tunnel obviously we make the train slightly less wider
than the tunnel if we can.
Power in FM Signals.
However, many signals (e.g. FM, square waves, digital signals) contain an infinite
number of components. If we transfer such a signal via a limited channel bandwidth,
we will lose some of the components and the output signal will be distorted. If we put
an infinitely wide train through a tunnel, the train would come out distorted, the
question is how much distortion can be tolerated?
Generally speaking, spectral components decrease in amplitude as we move away
from the spectrum centre.
Power in FM Signals.
In general distortion may be defined as
spectrum in total Power
spectrum d Bandlimite in Power - spectrum in total Power
= D
T
BL T
P
P P
D

=
With reference to FM the minimum channel bandwidth required would be just wide
enough to pass the spectrum of significant components. For a bandlimited FM
spectrum, let a = the number of sideband pairs, e.g. for | = 5, a = 8 pairs
(16 components). Hence, power in the bandlimited spectrum P
BL
is

=
=
a
a n
n c
BL
J V
P
2
)) ( (
2
|
= carrier power + sideband powers.
Power in FM Signals.
Since
2
2
c
T
V
P =

=
=
=

=
a
a n
n
c
a
a n
n
c c
J
V
J
V V
D
2
2
2
2 2
)) ( ( 1
2
)) ( (
2 2
|
|
Distortion
Also, it is easily seen that the ratio

=
= = =
a
a n
n
T
BL
J
P
P
D
2
)) ( (
spectrum in total Power
spectrum d Bandlimite in Power
|
= 1 Distortion
i.e. proportion p
f
power in bandlimited spectrum to total power =

=
a
a n
n
J
2
)) ( ( |
Example
Consider NBFM, with | = 0.2. Let V
c
= 10 volts. The total power in the infinite

2
2
c
V
spectrum
= 50 Watts, i.e.

=
a
a n
n
J
2
)) ( ( |
= 50 Watts.
From the table the significant components are
n J
n
(0.2) Amp = V
c
J
n
(0.2)
Power =
2
) (
2
Amp

0 0.9900 9.90 49.005
1 0.0995 0.995 0.4950125
P
BL
= 49.5 Watts

i.e. the carrier + 2 sidebands contain 99 . 0
50
5 . 49
= or 99% of the total power
Example
Distortion =
01 . 0
50
5 . 49 50
=

T
BL T
P
P P
or 1%.
Actually, we dont need to know V
c
, i.e. alternatively
Distortion =

1
1
2
)) 2 . 0 ( ( 1
n
n
J
(a = 1)
D = 01 . 0 ) 0995 . 0 ( ) 99 . 0 ( 1
2 2
=
Ratio
99 . 0 1 )) ( (
1
1
2
= = =

=
D J
P
P
n
n
T
BL
|
FM Demodulation General Principles.
An FM demodulator or frequency discriminator is essentially a frequency-to-voltage
converter (F/V). An F/V converter may be realised in several ways, including for
example, tuned circuits and envelope detectors, phase locked loops etc.
Demodulators are also called FM discriminators.
Before considering some specific types, the general concepts for FM demodulation
will be presented. An F/V converter produces an output voltage, V
OUT
which is
proportional to the frequency input, f
IN
.
FM Demodulation General Principles.
If the input is FM, the output is m(t), the analogue message signal. If the input is FSK,
the output is d(t), the digital data sequence.
In this case f
IN
is the independent variable and V
OUT
is the dependent variable (x and
y axes respectively). The ideal characteristic is shown below.
We define V
o
as the output when f
IN
= f
c
, the nominal input frequency.
FM Demodulation General Principles.
f
V
A
A
is called the voltage conversion factor
i.e. Gradient = Voltage Conversion Factor, K volts per Hz.
Considering y = mx + c etc. then we may say V
OUT
= V
0
+ Kf
IN
from the frequency
modulator, and since V
0
= V
OUT
when f
IN
= f
c
then we may write
IN OUT
V K V V o + =
0
where V
0
represents a DC offset in V
OUT
. This DC offset may be removed by level
shifting or AC coupling, or the F/V may be designed with the characteristic shown next
FM Demodulation General Principles.
The important point is that V
OUT
= KoV
IN
. If V
IN
= m(t) then the output contains the
message signal m(t), and the FM signal has been demodulated.
FM Demodulation General Principles.
Often, but not always, a system designed so that
o
1
= K
, so that Ko = 1 and
V
OUT
= m(t). A complete system is illustrated.
FM Demodulation General Principles.
Methods
Tuned Circuit One method (used in the early days of FM) is to use the slope of a
tuned circuit in conjunction with an envelope detector.
Methods
The tuned circuit is tuned so the f
c
, the nominal input frequency, is on the slope, not at
the centre of the tuned circuits. As the FM signal deviates about f
c
on the tuned circuit
slope, the amplitude of the output varies in proportion to the deviation from f
c
. Thus
the FM signal is effectively converted to AM. This is then envelope detected by the
diode etc to recover the message signal.
Note: In the early days, most radio links were AM (DSBAM). When FM came along,
with its advantages, the links could not be changed to FM quickly. Hence, NBFM was
used (with a spectral bandwidth = 2fm, i.e. the same as DSBAM). The carrier
frequency fc was chosen and the IF filters were tuned so that fc fell on the slope of the
filter response. Most FM links now are wideband with much better demodulators.
A better method is to use 2 similar circuits, known as a Foster-Seeley Discriminator
Foster-Seeley Discriminator
This gives the composite characteristics shown. Diode D
2
effectively inverts the f
2

tuned circuit response. This gives the characteristic S type detector.
Phase Locked Loops PLL
A PLL is a closed loop system which may be used for FM demodulation. A full
analytical description is outside the scope of these notes. A brief description is
presented. A block diagram for a PLL is shown below.
Note the similarity with a synchronous demodulator. The loop comprises a multiplier,
a low pass filter and VCO (V/F converter as used in a frequency modulator).
Phase Locked Loops PLL
The input f
IN
is applied to the multiplier and multiplied with the VCO frequency output
f
O
, to produce E = (f
IN
+ f
O
) and A = (f
IN
f
O
).

The low pass filter passes only (f
IN
f
O
) to give VOUT which is proportional to (f
IN
f
O
).

If f
IN
~ f
O
but not equal, V
OUT
= V
IN
, of
IN
f
O
is a low frequency (beat frequency) signal to
the VCO.

This signal, V
IN
, causes the VCO output frequency f
O
to vary and move towards f
IN
.

When f
IN
= f
O
, V
IN
(f
IN
f
O
) is approximately constant (DC) and f
O
is held constant, i.e
locked to f
IN
.

As f
IN
changes, due to deviation in FM, f
O
tracks or follows f
IN
. V
OUT
= V
IN
changes to drive
f
O
to track f
IN
.

V
OUT
is therefore proportional to the deviation and contains the message signal m(t).
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