EEM 308 Lecture telecommunications introduction

telecommunications

1. EEM 308 Introduction to Communications
Lecture 8
Dr. Can Uysal
Department of Electrical and Electronics Engineering,
Eskisehir Technical University
April 22, 2025

2. Angle Modulation
So far we have modulated the
amplitude of the carrier to transmit
the message signal.
Another class of modulation methods
includes frequency modulation (FM)
and phase modulation (PM),
In FM systems, the frequency of the
carrier is changed by the message
signal; in PM systems, the phase of
the carrier is changed according to
the variations in the message signal.
Both FM and PM are non-linear and
are often calledangle modulation
methods.
Both techniques are forms of
angle modulation because they
involve varying the phase
angle of the carrier signal to
convey information.
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3. Angle Modulation
Angle modulation, due to its inherent nonlinearity, is more
complex to implement and much more di!cult to analyze
Poor spectral e!ciency
Requires much wider bandwidth compared to amplitude
modulation schemes (DSB, SSB, etc.)
The major benefit of these systems is their high degree of
noise immunity.
these systems sacrifice bandwidth for high-noise immunity
FM systems are widely used in high-fidelity music
broadcasting and point-to-point communication systems,
where the transmitter power is quite limited.
Can use non-linear amplifiers
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4. Concept of Instantaneous Frequency
Instantaneous frequencyis a concept commonly used in the
analysis of non-stationary signals. In simpler terms, it refers to
the frequency of a signal at a particular point in time.
For a stationary signal, like a sine wave, the frequency remains
constant over time. However, many real-world signals, are
non-stationary, meaning their frequency content changes over
time. Instantaneous frequency helps describe how the
frequency of such signals varies.
While AM signals carry a message with their varying
amplitude, FM signals can vary theinstantaneous frequencyin
proportion to the message signalm(t).
This means that the carrier frequency is changing
continuously every instant.
Let us consider a generalized sinusoidal signals(t) given by
s(t)=Accos[ω(t)]
whereω(t) is the generalized angle and is a function oft.
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5. Concept of Instantaneous Frequency
Figure shows a hypothetical case ofω(t). The generalized angle for
a conventional sinusoidAccos (2εfct+ω0) is a straight line
2εfct+ω0, as shown in Figure. The hypothetical case general
angle ofω(t) is tangential to the angle (2εfct+ω0) at some
instantt. The crucial point is that, aroundt, over a small interval
”t→0, the signals(t)=Accosω(t) and the sinusoid
Accos (2εfct+ω0)areidentical;thatis,
s(t)=Accos (2εfct+ω0)t1<t<t2
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6. Concept of Instantaneous Frequency
Over this small interval” t, the angular frequency ofs(t)is
ϑc=2εfc.
Because (2εfct+ω0) is tangential toω(t), the angular
frequency ofs(t) is the slope of its angleω(t) over this small
interval.
We can generalize this concept at every instant and define
that the instantaneous frequencyfiat any instanttis the
slope ofω(t)att. Thus, the instantaneous frequency and the
generalized angle are related,
fi(t)=
1
2ε
dω(t)
dt
ω(t)=2ε
!
t
→↑
fi(ϖ)dϖ
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7. Angle Modulation
As a generalized angle-modulation, the angleω(t)is
ω(t)=2εfct+ϱ(t)
We will use two methods to modulate message signal with the
angle,
1.Phase Modulation (PM): Ifm(t) is the message signal, the
angle is proportional to the message as,
ω(t)=2εfct+kpm(t)
wherekpis a phase deviation constant (phase sensitivity). So,
the phase-modulated signal is described as,
uPM(t)=Accos(2εfct+kpm(t))
The instantaneous frequencyfi(t) in this case is given by
fi(t)=
1
2ε
dω(t)
dt
=fc+
kp
2ε
dm(t)
dt
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8. Angle Modulation
2.Frequency Modulation (FM):instantaneous frequencyis
proportional to the message signal,
fi(t)=fc+kfm(t)
wherekfis the frequency deviation constant (frequency
sensitivity). The angleω(t)isnow
ω(t)=
!
t
→↑
[2εfc+2εkfm(ϖ)]dϖ
=2εfct+2εkf
!
t
→↑
m(ϖ)dϖ
Here we have assumed the constant term inω(t)tobezero
without loss of generality. The FM wave is
uFM(t)=Accos
"
2εfct+2εkf
!
t
→↑
m(ϖ)dϖ
#
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9. FM - PM Comparison
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10. Angle Modulation
Representation of FM and PM Signals
From the preceding relationships, we have
ϱ(t)=
$
kpm(t), PM
2εkf
%
t
→↑
m(ϖ)dϖ,FM
d
dt
ϱ(t)=
&
kp
d
dt
m(t),PM
2εkfm(t),FM
,
The maximum phase deviation in a PM system is given by
”ϱmax=kpmax[|m(t)|],
and the maximum frequency deviation in an FM system is given by
”fmax=kfmax[|m(t)|].
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11. Frequency Deviation
In FM, the carrier frequency (fc) changes
based on the instantaneous level of the
modulating signal.
The current distance (in Hz) of the carrier
from the nominal center frequency (f0)isthe
frequency deviation (” f)
Can be negative or positive
A deviation of +15 kHz means that the
carrier is currently 15 kHz abovef0
Di#erence between the carrier frequency (for
the frequency at maximum amplitude of the
message signal) and nominal carrier
frequency is termed as maximum frequency
deviation,” fmax
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12. Example
Message signal is given in the figure. This signal will be modulated
using FM. Carrier frequency is 3000 Hz. Frequency deviation
constant is 40 Hz/V. Plot instantaneous frequency graphic with
respect to time, find maximum frequency deviation.
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14. Angle Modulation
Relation Between FM and PM
From PM and FM equations, it is apparent that PM and FM
not only are very similar but are inseparable.
Replacingm(t)inPMeq.with
%
m(ς)dςchanges PM into
FM.
Thus, a signal that is an FM wave corresponding to m(t) is
also the PM wave corresponding to
%
m(ς)dς.
Similarly, a PM wave corresponding to m(t) is the FM wave
corresponding to
d
dt
m(t).
Therefore, by looking only at an angle-modulated signal, there
is no way of telling whether it is FM or PM.
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15. Angle Modulation
Relation Between FM and PM
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16. Example 0
Sketch FM and PM waves for the modulating signalm(t)shownin
Figure. The constantskfandkpare 10
5
and
ω
2
,respectively,and
fc= 100 MHz.
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18. Answer to Example 0
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19. Properties of Angle-Modulated Signals
1.Constancy of Transmit Power
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20. Properties of Angle-Modulated Signals
2.Nonlinearity of Modulation Process
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21. Properties of Angle-Modulated Signals
3.Visualization Di!culty of the message waveform
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22. Figure:Angle modulation for sinusoidal message signal. (a) Message
signal. (b) Unmodulated carrier. (c) Output of phase modulator with
m(t). (d) Output of frequency modulator withm(t).
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23. Example 1
The message signalm(t)=acos(2εfmt)isusedtoeither
frequency modulate or phase modulate the carrierAccos 2εfct.
Find the modulated signal in each case.
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24. Angle Modulation
We can extend the definition of the modulation index for a
general nonsinusoidal signalm(t)as
φp=kpmax[|m(t)|]
φf=
kfmax[|m(t)|]
W
whereWdenotes the bandwidth of the message signalm(t).
In terms of the maximum phase and frequency deviation” ϱ
and” f,wehave
φp=”ϱmax
φf=
”fmax
W
Depending on the value of modulation index,φ,wemay
distinguish two cases of frequency modulation:
Narrowband Angle Mod. :φ<1
Wideband Angle Mod. :φ↑1
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25. Example 2
An FM transmitter has a frequency deviation constant of 15 Hz/V.
Assuming a message signal ofm(t) = 9 cos(40εt), write the
expression for modulated FM signal and determine the maximum
frequency deviation and modulation index.
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26. Narrowband Angle Modulation
Recall the angle-modulated signal in the form as,
u(t)=Accos(2εfct+ϱ(t))
=Accos(2εfct) cos(ϱ(t))↓Acsin(2εfct)sin(ϱ(t))
If the deviation constantskpandkfand the message signal
m(t)aresuchthatforallt
ϱ(t)↔1
Then, we can use an approximation
u(t)↗Accos(2εfct)↓Acϱ(t)sin(2εfct)
where we have used the approximations cosϱ(t)↗1and
sinϱ(t)↗ϱ(t)forϱ(t)↔1.
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27. Narrowband FM Analysis
in FM, for single tone modulation, we found that
ϱ(t)=2εkf
%
t
→↑
m(ϖ)dϖ=
kfa
fm
sin(2εfmt)=φsin(2εfmt)
Then, we can write the approximation
u(t)↗Accos(2εfct)↓Acϱ(t)sin(2εfct)as,
uFM(t)↗Ac[cos(2εfct)↓φsin(2εfct)sin(2εfmt)]
or
uNBFM(t)↗Accos(2εfct)+
φAc
2
[cos(2ε(fc+fm)t)↓cos(2ε(fc↓fm)t)]
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28. Conv. AM vs. Narrowband FM
The narrowband FM signal is very similar to a conventional AM signal.
The only di!erence is that the sign of the lower side frequency in the
narrowband FM is reversed.
The bandwidth of this signal is similar to the bandwidth of a conventional AM
signal, which is twice the bandwidth of the message signal.
BW=2W
Seldom used in practice for communication purposes
Used as an intermediate stage for generation of wideband angle-modulated
signals
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29. Conv. AM vs. Narrowband FM
Figure:Comparison of AM and narrowband angle modulation. (a) Phasor
diagrams. (b) amplitude spectrum. (c) phase spectrum.
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30. Spectral Characteristics of Angle-Modulated Signals
Angle Modulation by a Sinusoidal Signal
Consider the case where the message signal is a sinusoidal signal (to be more
precise, sine in PM and cosine in FM)
u(t)=Accos(2ωfct+εsin 2ωfmt)
whereεis the mod index that can be eitherεporεf,andinPMsin2ωfmtis
substituted by cos 2ωfmt.
Using Euler’s relation, the modulated signal can be written as
u(t)=Re(Ace
j2ωfct
e
jεsin 2ωfmt
)
Since sin 2ωfmtis periodic with periodTm=
1
fm
,thesameistrueforthe
complex exponential signale
jεsin 2ωfmt
.SoitcanbeexpandedinaFourier
Series representation. The FS coe”cients:
xn=fm
!1
fm
0
e
jεsin 2ωfmt
e
→jn2ωfmt
dt;u=2ωfmt
=
1
2ω
!
2ω
0
e
j(εsinu→nu)
du
This integral:Bessel function of the first kind of order nand denoted byJn(ε).
We have the Fourier series for the complex exponential
e
jεsin 2ωfmt
=
↑
"
n=→↑
Jn(ε)e
j2ωnfmt
.
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31. Spectral Characteristics of Angle-Modulated Signals
By substituting inu(t)
u(t)=Re
#
$Ac
↑
"
n=→↑
Jn(ε)e
j2ωnfmt
e
j2ωfct
%
&
=
↑
"
n=→↑
AcJn(ε) cos [2ω(fc+nfm)t]
The spectrum ofu(t)isobtainedbytakingtheFouriertransformsofbothsides
U(f)=
Ac
2
↑
"
n=→↑
Jn(ε)[ϑ(f→fc→nfm)+ϑ(f+fc+nfm)]
Even in this simple case (a sinusoid modulating signal), the angle modulated signal
contains all frequencies of the formfc+nfmforn=0,±1,±2,...
The actual bandwidth of the modulated signal is infinite!
However, the amplitude of the sinusoidal components of frequenciesfc±nfmfor large
nis very small. Hence, we can define a finite e!ective bandwidth for the modulated
signal.
Bc=2(ε+1)fm
Contains 98% of the signal power, in general.
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32. Spectral Characteristics of Angle-Modulated Signals
Figure:Bessel functions for various values of n.
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33. Properties of Bessel Function
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34. Spectrum of Angle-modulated Signal
Figure:Spectrum of an angle-modulated signal. (a) amplitude spectrum.
(b) phase spectrum. They are single-sided spectrums. We don not show
the negative frequencies.
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35. Example 3
Let the carrier and the message signal be given by
c(t) = 10 cos 2εfct,m(t) = cos 20εt
Assume that the message is used to frequency modulate the carrier
withkf= 50.
a.Find the expression for the modulated signal
b.Determine how many harmonics should be selected to contain
99% of the modulated signal power
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36. Table
Values of Selected Bessel Functions
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37. Summary
Angle modulation conveys information by changing the angle
(frequency or phase) of the carrier
High quality audio and relatively immune to amplitude
variations (fading, noise, interference, etc.)
Requires wider bandwidths than other modulation types.
Frequency deviation is the excursion of the carrier from the
nominal center frequency
Modulation index (deviation ratio) is max frequency deviation
divided by max modulating frequency
Narrowband and wideband based on modulation index
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38. Next Week
Angle modulation (continued) and 4th Quiz
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