Angle Modulation

The voltage c(t) = Vc cos  = Vc cos(ct + )

may carry signal information  (t) by modulating the carrier
signal in different ways.

If d/dt vary with time, then, at any instant, we have an

instantaneous frequency i (t). Variation of i (t) by the
modulating signal gives a form of angle modulation
known as frequency modulation (FM).

Since  = (ct + ), another form of angle modulation

occurs when the phase () is varied by the modulating
signal. This results is phase modulation (PM).

ELEN 360, Dr. Shihab Jimaa

 Frequency Modulation (FM)

In FM the baseband (modulating) signal is

used to vary the FREQUENCY of the
carrier.

ELEN 360, Dr. Shihab Jimaa

 FM Signal Description

In this case the frequency modulated signal efm(t) can be

described by the expression
efm(t) = C Cos 
The frequency of the carrier is modulated and becomes at
any instance:

f = fc +k.em(t)
where fc is the carrier frequency without modulation.
now:
= 2f = d /dt
so
 =  2 f c + 2 k.e m (t) dt

ELEN 360, Dr. Shihab Jimaa

For simple cosinusoidal modulation:
em(t) = Em Cos ωmt
Hence
kEm
 = ct + sin  mt
fm
Where kEm is the peak deviation; f
Define the FM modulation index = f/fm , then:

e(t ) FM = C cos( c t +  sin  mt )

 = FM modulation index = k Em / fm
K = Modulator gain (a constant Hz/volt)
Em = Amplitude of baseband signal (Hz)
ωm = 2fm (frequency of baseband signal fm)
ωc = 2fc unmodulated carrier frequency (fc)
ELEN 360, Dr. Shihab Jimaa
Note that the carrier can deviate from its nominal frequency
(fc) by an amount +/- f Hz, ie: the instantaneous
frequency of the FM signal can be in the range
(fc- f) < fc < (fc + f) Hz. The form of this signal in the
time domain is shown below:

t
b(t)

t
e(t)

It can be seen that the rate of change of carrier frequency is governed by the frequency
of the baseband signal and the amount of deviation is a function of the amplitude of the
baseband signal.
ELEN 360, Dr. Shihab Jimaa
 Spectrum of an FM Signal

The FM signal may contain many sideband

pairs whose amplitudes are dependent on
the modulation index. The relationship is
not linear but is governed by a set of
Bessel Functions. Few examples of FM
frequency spectra for FM sinusoidal
modulation are shown below:
Copy from the white board the mathematical
spectral analysis of the FM signal
ELEN 360, Dr. Shihab Jimaa
1

2 > 1

3 > 2

ELEN 360, Dr. Shihab Jimaa

ELEN 360, Dr. Shihab Jimaa
 Design Procedure for Implementing FM
In practise we can implement FM by using a voltage-to-frequency
converter. A simple set of design steps can be followed:

* Identify the highest frequency component in the baseband signal

* Evaluate how many sideband pairs (n) will fit into available
channel width (W)
* Select the highest modulation index () from Bessel table that
ensures sideband pair outside channel width are insignificant
* Evaluate required peak deviation
* Evaluate required maximum amplitude (Em) for a given modulator
constant K (or vice versa)

ELEN 360, Dr. Shihab Jimaa

 Phase modulation

An unmodulated carrier signal is given by:

vc(t) = Vc cos  = Vc cos(ct + c)

If a binary pulse modulates c , then c will deviate from its

initial value each time the pulse changes amplitude.

ELEN 360, Dr. Shihab Jimaa

 binary input data BPSK symbols out
(R bits/sec) MOD (D=R baud)

data input
t

BPSK output
t

ELEN 360, Dr. Shihab Jimaa

If a voltage  (t) modulates c , then c = 0 + kp  (t). The
constant kp is thus the carrier phase deviation in radians per
volt of  (t).

The expressions for the PM signal are:

vc = Vc cos (ct + 0 + kp Vp), for t1< t <t2, and

vc = Vc cos (ct + 0 ), for t1> t or t >t2

ELEN 360, Dr. Shihab Jimaa

If a tone t = Vt cos(t t) modulates c,
then c = 0 + kp Vt cos(t t). The maximum carrier phase
deviation (d m) will be kp Vt , so

 c = 0 + dm cos(t t)

The initial phase (0) is arbitrary and, therefore, carries no

information, so we can ignore the initial phase without
affecting the results of our analysis. Thus, the effective
expression for a continuous wave phase modulated by a tone
is

c = Vc cos {ct + dm cos(t t)}

ELEN 360, Dr. Shihab Jimaa

Comparing the PM and FM signals, it •
could be said that  represents the
maximum phase deviation of an FM
signal.

e(t ) FM = C cos( ct +  sin  mt )

e(t ) PM = C cos[ct + dm cos(t t )]

ELEN 360, Dr. Shihab Jimaa

The expressions derived for PM and FM •
suggest that an FM modulated wave can be
achieved using phase modulator.
vt = [sin(tt)] / t

Tone Phase
Integrator modulator
Vt cos(tt)
Vc cos [ct + kf Vt [sin (tt)] / t

Carrier

Vc cos (ct + )

ELEN 360, Dr. Shihab Jimaa

Another frequency modulator, based on a •
voltage-controlled oscillator (VCO), is shown
below. A VCO has a frequency-determining
component that can be varied by a voltage,
such as a capacitor.

Modulating Output freq. RF power

voltage VCO amplifier
v(t) Proportional
to v(t)

ELEN 360, Dr. Shihab Jimaa

 FM Generator

This is a simple FM transmitting system

Capacitor FM
microphone

Sound Oscillator
waves Oscillator

The intelligence amplitude determines the amount of carrier frequency deviation.

The intelligence frequency determines the rate of carrier frequency deviation.

ELEN 360, Dr. Shihab Jimaa

A key difference between AM and FM:
All frequency components, including the
carrier, vary in magnitude making FM a
more efficient information carrying
system.
There are approximately (+1) sidebands
separated by fm in the spectrum, giving
a bandwidth of 2(+1)fm = 2(f+fm),
which is known as Carson’s Rule.

ELEN 360, Dr. Shihab Jimaa

Example (1)
Use the Bessel functions graph (table) to
sketch the frequency spectra of an FM
signal modulated by a tone, when  =0.5
and then 1.5.

ELEN 360, Dr. Shihab Jimaa

Example (2)
A commercial FM broadcasts are restricted
to a frequency deviation of 75 kHz.
Calculate the modulation index values for
tones of frequency 20 Hz and 20 kHz.
Estimate the bandwidth required for the
transmission of high-quality sound
covering this range and comment on the
effect of a reduction in bandwidth on the
sound quality.

ELEN 360, Dr. Shihab Jimaa

 Example
An FM signal is given by

FM (t ) = 10cos[10  t + 8sin(10  t )]

6 3

Determine the following: (a) c , (b) ; (c) ; (d) fm

Solution

a) c = 106  rad/s= 500 kHz

b)  = 8
c)   = 8 X l03 rad/s= 4 kHz

d) fm = 103  rad/s = 103  / 2 = 500 Hz

ELEN 360, Dr. Shihab Jimaa

 Demodulation of FM Signals

Any FM demodulator must be able to convert

frequency variations at its input into amplitude
variations at its output.

There are several possible alternatives but we will

discuss here the frequency discriminator circuit
to demodulate an FM signal.

FM signals can be detected by a differentiation

circuit followed by an envelope detector.

ELEN 360, Dr. Shihab Jimaa

FM Receiver (Discriminator)

FM(t)
d
FM*(t) A[c + k f v(t )]
Envelope
dt Detector

t
For FM(t) = A cos[ c t + k f  v( )d +  0 ]
0

t
d / dt [φFM (t)] = − A{  c + k f v(t)} sin [ωct + k f  v( )
0
d +  0 ]

ELEN 360, Dr. Shihab Jimaa

 t
d / dt [φFM (t)] = A{  c + k f v(t)} cos [ωc t + k f 
0
v(τ )dτ + θ0 +   ]

The Envelope detector output is A{c + k f v(t)} and the

term Ac can be removed by a blocking capacitor.

ELEN 360, Dr. Shihab Jimaa

 Demodulation of PM Signals

PM signals can be detected by a differentiation circuit

followed by an envelope detector and an integrator.

A[c + k p v* (t )]

PM(t) PM*(t) v(t)


Envelope
d Detector
dt

ELEN 360, Dr. Shihab Jimaa

 For PM (t) = A cos[ct + kp v(t) + 0]


t PM (t ) = − A{c + k p t v(t )}sin[ct + k p v(t ) + 0 ]


t PM (t ) = A{c + k p t v(t )}cos[ct + k pv(t ) + 0 + 2 ]

The envelope detector output is A [c + kp (/t) v(t) ] and the term A c can be
removed by a blocking capacitor to obtain Ak p (/t) v(t) which upon integration
yields the desired output signal v(t).

ELEN 360, Dr. Shihab Jimaa

 Power in FM and PM signals

Since the amplitude of FM signals is constant, the

total power must be constant for all values of .
Bessel functions graph indicates that, as 
increases, the carrier amplitude decreases. This
means that power is transferred from the carrier
to the sidebands as  increases, so extra
information is sent as this power transfer takes
place. This suggests that the received signal-
to-noise ratio (SNR) can be improved by
increasing  rather than the transmitter power.

ELEN 360, Dr. Shihab Jimaa

Here's how increasing β improves SNR

1. Noise Reduction: In FM, noise primarily affects the amplitude of the

signal. By increasing β, the signal's frequency variations become more
pronounced, making it easier for the receiver to distinguish the signal from
noise. This leads to a higher SNR.

2. Capture Effect: FM receivers exhibit a phenomenon called the "capture

effect," where the receiver locks onto the stronger of two signals on the
same frequency. A higher β increases the signal's strength relative to noise,
improving the receiver's ability to capture the desired signal and reject
noise.

3. Bandwidth Expansion: Increasing β results in a wider signal bandwidth.

While this might seem counterintuitive, a wider bandwidth allows the signal
to be less affected by narrowband noise sources, effectively improving the
SNR.

ELEN 360, Dr. Shihab Jimaa

In summary, increasing the modulation index (β) in FM
enhances the received SNR by making the signal more
distinguishable from noise, leveraging the capture
effect, and expanding the signal bandwidth to mitigate
narrowband noise sources. This approach is often more
efficient than simply increasing the transmitter power,
which can lead to higher energy consumption and
potential interference with other signals.

ELEN 360, Dr. Shihab Jimaa

ELEN 360, Dr. Shihab Jimaa
 Comparison of modulation techniques
The actual modulation method used in a particular application
will depend upon the criteria governing the system.

For instance if bandwidth is at a premium then clearly SSB

would appear as the most attractive. Full AM on the other
hand offers what is probably the cheapest receivers if an
envelope detector is used.

For best quality operation FM is best, but only at the expense

of considerably increased bandwidth. It is nevertheless
interesting to note that in poor S/N environment FM
improvement diminishes and there may be no advantage
in using it compared with AM.

ELEN 360, Dr. Shihab Jimaa

 Type of Bandwidth Reception Comment
modulation
Full AM 2W Envelope Simple and cheap.
detector Carrier transmission
wasted power. Use –
broadcast.
DSBSC 2W Coherent Requires synchronous
detection. SSB preferred
due to reduced B.W.
requirements.
SSB W Coherent Synchronous detection.
Spectrally efficient.
Widely used for
telephony.
FM >>W Frequency B.W. expansion system.
discriminator Improves S/N. Use-
terrestrial, satellite
microwave
communication.
PM >>W Non-linear Similar performance to
FM. More complex
reception.

ELEN 360, Dr. Shihab Jimaa