AC Unit 2.pdf

Analog Communication
Angle Modulation
Introduction
Dr. Shivakumar B R
NMAM Institute of Technology
Nitte, Karkala -574110
December 10, 2022
Dr. Shivakumar B R Analog Communication December 10, 2022 1 / 126

Introduction
Modulation
What is Modulation?
Modulation is a process in which some characteristics of the carrier
signal, c(t), are varied in accordance with the information bearing
signal or modulating signal, m(t), to create a modulated signal, s(t).

Introduction
Need for Modulation
Increase the Signal Strength: The baseband signals transmitted
by the sender are not capable of direct transmission. The strength of
the message signal should be increased so that it can travel longer
distances. This is where modulation is essential. The most vital
need of modulation is to enhance the strength of the signal without
affecting the parameters of the carrier signal.
Wireless Communication System: Modulation has removed the
necessity for using wires in the communication systems. It is be-
cause modulation is widely used in transmitting signals from one
location to another with faster speed. Thus, the modulation tech-
nique has helped in enhancing wireless communication systems.

Introduction
Need for Modulation
Prevention of Message Signal from Mixing: Modulation and
its types prevent the interference of the message signal from other
signals. It is because a person sending a message signal through
the phone cannot tell such signals apart. As a result, they will
interfere with each other. However, by using carrier signals having
a high frequency, the mixing of the signals can be prevented. Thus,
modulation ensures that the signals received by the receiver are
entirely perfect.
Size of the Antenna: The signals within 20 Hz to 20 kHz fre-
quency range can travel only a few distances. To send the message
signal, the length of the antenna should be a quarter wavelength
of the used frequency. Thus, modulation is required to increase the
frequency of the message signal and to enhance its strength to reach
the receiver.

Introduction
A sinusoidal carrier wave has basically three characteristics;
Amplitude,
Frequency,
Phase.
Based upon these characteristics of the carrier wave, we obtain three
basic types of modulation.
Amplitude modulation,
Frequency modulation,
Phase modulation.

Introduction
Amplitude Modulation: The process in which the amplitude of
the sinusoidal carrier is slowly varied in accordance with the infor-
mation bearing signal.
Frequency Modulation: The process in which the frequency of
the sinusoidal carrier is slowly varied in accordance with the infor-
mation bearing signal.
Phase modulation: The process in which the phase of the sinu-
soidal carrier is slowly varied in accordance with the information
bearing signal.

Introduction
What is Angle Modulation?
It is the process in which either the frequency or phase of the si-
nusoidal carrier can be varied according to the information bearing
signal, keeping the amplitude constant.

Introduction
Let us denote the carrier signal by
s(t) =Ac cos[θi(t)] (1)
where,
Ac: is the carrier amplitude, which is held constant in agle modu-
lation.
θi(t): is the phase angle of the carrier, which is varied in accordance
with message signal m(t).

Introduction
The conventional sinusoidal signal is given by,
s(t) =Ac cos[ωct + ϕc] (2)
where,
[ωct + ϕc]: is the angle,
ϕc: is the value of θi(t) at t = 0.
The angle [ωct + ϕc] represents a straight line with a slope ωc and
interrupt ϕc, as in Fig. 1.

Introduction
Figure 1: Angle θi(t) as a function of time t.

Introduction
The plot of θi(t) for a hypothetical case happens to be tangential
to the angle [ωct + ϕc] at some instant t, as shown in Fig. 2.
Bacause [ωct + ϕc] is tangential to θi(t), the frequency of s(t) is the
slope of its angle θi(t) over the small interval ∆t of time.

Introduction
Figure 2: Angle θi(t) tangential to angle [ωct + ϕc].

Introduction
If we generalize this idea and say that if θi(t) increases linearly with
time, the average frequency over an interval from t1 to t2 or from t
to t + ∆t, is given by
f∆t(t) =
θi(t + ∆t) − θi(t)
2π∆t
(3)
The instantaneous frequency of the angle modulated wave s(t) can
be now defined as
fi(t) = lim
∆t→0
f∆t(t)
= lim
∆t→0

θi(t + ∆t) − θi(t)
2π∆t

=
1
2π
lim
∆t→0

θi(t + ∆t) − θi(t)
∆t

fi(t) =
1
2π
d[θi(t)]
dt
(4)

Introduction
Therefore, according to Eq. (1), we may interpret the angle modu-
lated signal s(t) as a rotating phasor of length Ac and angle θi(t).
The angular velocity of such a phasor is dθi(t)
dt , measured in radians
per second in accordance with (4).
For an unmodulated sinusoidal carrier, the angle θi(t) is given by
θi(t) =2πfct + ϕc (5)
and the corresponding phasor rotates with a constant angular ve-
locity equal to 2πfc.

Introduction
There are an infinite number of ways in which the angle θi(t) may
be varied in some manner with the message or baseband signal.
However, we shall consider only two commonly used methods;
1. Phase modulation,
2. Frequency modulation.

Time Domain Rep of FM and PM
Let θi(t) denote the angle of a modulated sinusoidal carrier at time
t.
θi(t) is assumed to be a function of the information-bearing signal
or message signal.
The resulting angle-modulated wave is;
where, Ac is the carrier amplitude.

Phase Modulation
Phase modulation is that form of angle modulation in which the
instantaneous angle θi(t) is varied linearly with the message signal
as given by
θi(t) =2πfct + kpm(t) (7)
where,
2πfct: angle of the unmodulated carrier,
kp: phase sensitivity of the modulator (in rad/V).
Substituting Eq. (7) in Eq. (6) we get the phase modulated signal
s(t) as;
s(t) =Ac cos[2πfct + kpm(t)] (8)

The instantaneous angular frequency fi is given by;
fi(t) =
dθi(t)
dt
=
d[2πfct + kpm(t)]
dt
fi(t) =2πfc + kp
d
dt
[m(t)]
(9)
Thus, in PM, the instantaneous frequency fi(t) varies linearly with
the derivative of the message signal, m(t).

Frequency Modulation
Frequency modulation is that form of angle modulation in which
the instantaneous frequency fi(t) is varied linearly with the message
signal m(t), as given by
fi(t) =fc + kf m(t) (10)
where,
fc: is the frequency of the unmodulated carrier,
kf : sensitivity of the modulator (in Hz/V).

Recall Eq. (4), i.e., the instantaneous frequency of the angle mod-
ulated wave s(t),
fi(t) =
1
2π
d[θ(i)]
dt
2πfi(t) =
d[θ(i)]
dt
Rearranging, we get
θi(t) =
Z t
0
2πfi(t)dt (11)

Substitute Eq. (10) into (11)
θi(t) =
Z t
0
2π[fc + kf m(t)]dt
=2πfc
Z t
0
dt + 2πkf
Z t
0
m(t)dt
θi(t) =2πfct + 2πkf
Z t
0
m(t)dt
(12)
Substituting Eq. (12) in Eq. (6) we find the the frequency modu-
lated wave;
s(t) =Ac cos

2πfct + 2πkf
Z t
0
m(t)dt

(13)

Summary
For Phase modulation
s(t) =Ac cos[2πfct + kpm(t)]
For Frequency Modulation
s(t) =Ac cos

2πfct + 2πkf
Z t
0
m(t)dt


Observations
These two equations indicate that PM and FM are not only similar,
but are inseparable.
Replacing m(t) in equation for PM with
R
m(t) changes PM into
FM.
Therefore, a signal which is an FM wave corresponding to m(t) is
also the PM wave corresponding to
R
m(t).
Similarly, a PM wave corresponding to m(t) is the FM wave corre-
sponding to dm(t)
dt .
Thus in both PM and FM, the angle of the carrier signal is varied
accordingly to some measure of modulating signal m(t).
In PM, it is directly proportional to m(t), while in FM, it is pro-
portional to the integral of m(t).

Time
Domain
Rep
of
FM
and
PM
Figure 3: (a) carrier wave, (b) sinusoidal modulating
signal, (c) amplitude-modulated signal, (d)
phase-modulated signal, (e) frequency-modulated signal.
Dr.
Shivakumar
B
R
Analog
Communication
December
10,
2022
24
/
126

Single Tone FM
Single Tone Frequency Modulation
The time-domain expression for frequency modulated wave is;
s(t) =Ac cos

2πfct + kf
Z t
0
m(t)dt

(14)
This equation indicates that FM wave s(t) is a nonlinear function
of the modulating wave m(t).
Therefore, we can note frequency modulation as a nonlinear modu-
lation process.
Hence, unlike AM, the spectrum of an FM wave is not related in a
simple way to that of the modulating signal.

Single Tone FM
Consider a sinusoidal modulating signal
m(t) =Am cos(2πfmt) (15)
The instantaneous frquency of the resulting FM signal is
fi(t) =fc + kf m(t)
=fc + kf Am cos(2πfmt)
=fc + ∆f cos(2πfmt)
(16)
where,
∆f =kf Am (17)
The quantity ∆f is called the frequency deviation, representing the
maximum departure of the instantaneous frequency of the FM signal
from the carrier frequency fc.

Single Tone FM
The fundamental charactaristics of an FM signal is that the fre-
quency deviation ∆f is proportional to the amplitude of the mod-
ulating signal and is independent of the modulation frequency.
Recall Eq. (16),
fi(t) =fc + ∆f cos(2πfmt)
We know that
fi(t) =
1
2π
dθi(t)
dt
2πfi(t) =
dθi(t)
dt
(18)
Integrating on both sides of Eq. (19);
fi(t) =
1
2π
dθi(t)
dt
Z t
0
2πfi(t) =θi(t)
(19)

Single Tone FM
Rearranging;
θi(t) =2π
Z t
0
fi(t)dt
=2π
Z t
0
[fc + ∆f cos(2πfmt)]dt
=2πfc
Z t
0
dt + 2π∆f
Z t
0
cos(2πfmt)dt
=2πfc[t]t
0 + 2π∆f

sin(2πfmt)
2πfm
t
0
=2πfct +
∆f
fm
sin(2πfmt)
(20)
The ratio of frequency deviation ∆f to the modulation frequency
fm is called the modulation index of the FM signal. It is denoted
by β.
β =
∆f
fm
(21)

Single Tone FM
Substitute Eq. (21) into Eq. (19);
θi(t) =2πfct + β sin(2πfmt) (22)
From Eq. (22) we see that, in a physical sense, the parameter β rep-
resents the phase deviation of the FM signal, that is, the maximum
departure of the angle θi(t) from the angle 2πfct of the unmodu-
lated carrier.
The FM signal is therefore given by;
Substituting Eq. (22) in (23);
s(t) =Ac cos[2πfct + β sin(2πfmt)] (24)

Single Tone FM
Depending on the value of the modulation index β, we may distin-
guish two cases of frequency modulation;
1. Narrowband FM: where, β 1 radian
2. Wideband FM: where, β 1 radian

Narrowband FM
Narrowband Frequency Modulation
Consider the FM signal
We know that
cos[A + B] = cos A cos B − sin A sin B (26)
Using Eq. (26) in Eq. (25);
s(t) =Ac cos(2πfct) cos(β sin(2πfmt))
− Ac sin(2πfct) sin(β sin(2πfmt))
(27)

Narrowband FM
Assuming that the modulation index β is small compared to one
radian, we may us the following two approximations;
cos[β sin(2πfmt)] ≊1
sin[β sin(2πfmt)] ≊β sin(2πfmt)
(28)
Equ. (27) simplifies to;
s(t) ≊Ac cos(2πfct) × 1 − Ac sin(2πfct) × β sin(2πfmt)
s(t) ≊Ac cos(2πfct) − Ac sin(2πfct)β sin(2πfmt)
s(t) ≊Ac cos(2πfct) − Acβ sin(2πfct) sin(2πfmt)
(29)
Eq. (29) defines the approximate form of a narrow-band FM signal
produced by the sinusoidal modulating signal Am cos(2πfmt).

Narrowband FM
We know that;
sin A sin B =
1
2
[cos(A − B) − cos(A + B)] (30)
Using Eq. (30) in (29) we get,
s(t) ≊Ac cos(2πfct) − Acβ
1
2

cos[2π(fc − fm)] − cos[2π(fc + fm)]

s(t) ≊Ac cos(2πfct) + Acβ
1
2

cos[2π(fc + fm)] − cos[2π(fc − fm)]

(31)
The expression in Eq. (31) is similar to the amplitude modulation
signal, given by,
sAM (t) ≊Ac cos(2πfct) + Acβ
1
2

cos[2π(fc + fm)] + cos[2π(fc − fm)]

(32)

Narrowband FM
Comparing Eq. (31) and (32), we see that in the case of sonu-
soidal modulation, the basic difference between an AM signal and a
narrow-band FM signal is that the algebraic sign of the lower side
frequency in the narrow-band FM is reversed.
Thus, a narrowband FM signal requires essentially the same trans-
mission bandwidth (i.e., 2fm) as the AM signal.

Narrowband FM
Figure 4: A phasor comparision of NBFM and AM waves for sinusoidal
modulation. (a) NBFM, (b) AM.

Narrowband FM
The phasor diagram for narrowband FM signal with carrier as ref-
erence is shown in Fig. 4.a.
We see that the resultant of the two side-frequency phasors is always
at right angles to the carrier phasor.
The effect of this is to produce a resultant phasor representing the
narrowband FM signal that is approximately of the same amplitude
as the carrier phasor, but out of phase with respect to it.
Fig. (4).b indicates the phasor diagram representation for AM sig-
nal.
In this case we see that the resultant phasor representing the AM
signal has an amplitude different from that of the carrier phasor,
but always in phase with it.

Narrowband FM
Recall Eq. (31), for the narrow-band FM signal;
1
2

cos[2π(fc + fm)]
− cos[2π(fc − fm)]

1
2
cos[2π(fc + fm)]
− Acβ
1
2
cos[2π(fc − fm)]
(33)
Taking Fourier transform on both sides;
S(f) ≊
Ac
2
[δ(f − fc) + δ(f + fc)]
+
Acβ
4

δ(f − (fc + fm)) + δ(f + (fc + fm))

+
Acβ
4

δ(f − (fc − fm)) + δ(f + (fc − fm))

(34)

Narrowband FM
Figure 5: Amplitude spectra of NBFM.

Narrowband FM
Observations
The amplitude spectrum shows that there are impulses at ±fc sig-
nifying the fact that the carrier term is not suppressed in the nar-
rowband FM.
Impulses are present on either side of ±fc and these impulses rep-
resent lower and upper sidebands.
The minimum transmission bandwidth = 2fm.
There is no amplitude variation in NBFM wave inspite of the fact
that both conventional AM and NBFM waves have the same spec-
tral content.

WBFM
Wideband Frequency Modulation
Consider the time-domain expression for an FM wave
Eq. (35) is of the form
s(t) =Re[s̃(t) exp(j2πfct)] (36)
where, s̃(t) is the complex envelope of the FM signal s(t), given by
s̃(t) =Ac exp[jβ sin(2πfmt)] (37)
From Eq. (37) we understand that, unlike original FM signal s(t),
the complex envelope s̃(t) is a periodic function of time with a
fundamental frequency equal to fm.

WBFM
Therefore, we may expand s̃(t) in the form of a complex Fourier
series as;
s̃(t) =
∞
X
n=−∞
Cn exp[j2πnfmt] (38)
where, the complex Fourier coefficient Cn is given by
Cn = fm
Z 1/2fm
−1/2fm
s̃(t) exp(−j2πnfmt)dt (39)
Substitute for s̃(t) from Eq. (37) into (39),
Cn =fm
Z 1/2fm
−1/2fm
Ac exp[jβ sin(2πfmt)] exp(−j2πnfmt)
=Acfm
Z 1/2fm
−1/2fm
exp[jβ sin(2πfmt) − j2πnfmt]dt
(40)

WBFM
Let us define a new variable
x =2πfmt (41)
New limits of integration will be;
When t = −
1
2fm
=⇒ x = 2πfm × −
1
2fm
= −π.
When t =
1
2fm
=⇒ x = 2πfm ×
1
2fm
= π.
dx
dt
= 2πfm =⇒ dt =
dx
2πfm

WBFM
Substituting new limits of integration and dt =
dx
2πfm
in Eq. (40);
Cn =Acfm
Z 1/2fm
−1/2fm
exp[jβ sin(2πfmt) − j2πnfmt]dt
=Acfm
Z π
−π
exp[jβ sin(x) − jnx]
dx
2πfm
=
Ac
2π
Z π
−π
exp[j(β sin(x) − nx)]dx
(42)
The integral on the RHS of Eq. (42), except for a scaling factor,
is recognized as the nth order Bessel function of the first kind and
argument β.
This function if commonly denoted by the symbol Jn(β), and is
given by;
Jn(β) =
1
2π
Z π
−π
exp[j(β sin(x) − nx)]dx (43)

WBFM
Comparing Eq. (43) and (42); we may write
Cn =AcJn(β) (44)
Substituting Eq. (44) in (38), we get the complex envelope of fre-
quency modulated signal s̃(t), in terms of Bessel function Jn(β) as
s̃(t) =Ac
∞
X
n=−∞
Jn(β) exp(j2πnfmt) (45)

WBFM
Substituting Eq. (45) in Eq. (36), we get;
s(t) =AcRe
∞
X
n=−∞
Jn(β) exp(j2πnfmt) exp(j2πfct)

=AcRe
∞
X
n=−∞
Jn(β) exp(j2π(fc + nfm)t)
(46)
Interchanging the order of summation and evaluation of the real
part in the RHS of Eq. (46), we get
s(t) =Ac
∞
X
n=−∞
Jn(β)Re

exp(j2π(fc + nfm)t)

s(t) =Ac
∞
X
n=−∞
Jn(β) cos[2π(fc + fm)t]
(47)
Eq. (47) is the desired form of the Fourier series representation of
the single-tone FM signal s(t) for an arbitrary value of β.

WBFM
To Plot Spectra
We know that
FT[Ac cos(2πfct)] =
Ac
2
[δ(f − fc) + δ(f + fc)]
Therefore,
FT[Ac cos(2π(fc + nfm)t)] =
Ac
2
[δ(f − (fc + nfm)) + δ(f + (fc + nfm))]
Taking Fourier Transform on both sides of Eq. (47) we get;
S(f) =
Ac
2
∞
X
n=−∞
Jn(β)[δ(f − (fc + nfm)) + δ(f + (fc + nfm))]
(48)
Eq. (48) is the desired form of the Fourier series representation of
the single-tone FM signal s(t) for an arbitrary value of β in Fre-
quency domain.

WBFM
Figure 6: Amplitude spectra of WBFM.

WBFM
Observations
The spectrum of the single-tone sinusoidally modulated FM signal is
composed of carrier having an amplitude
Ac
2
J0(β) and a set of side-
band frequencies spaced symmetrically on either side of the carrier
at a frequency separation of fm, 2fm, 3fm,... Thus, theoretically
FM wave has infinite bandwidth.
When β is small compared to one radian, only the Bessel coefficients
J0(β) and J1(β) have significant values. Consequently, FM signal
effectively comprises of carrier component and two side frequencies
fc − fm and fc + fm.
For β 1, J0(β), J1(β), J2(β), ..., Jn(β), where n ≊ β have signif-
icant values. Also, Jn(β) for n β have negligible values.

WBFM
Since the amplitude of the FM wave remains same as that of the
unmodulated carrier, the average power of an FM signal is same as
that of the unmodulated carrier.
For n odd, since J−n(β) = −Jn(β), odd numbered lower sideband
is reversed in phase.
In an FM wave, out of total available power A2
c/2, the power carried
by the carrier depends on the value of J0(β) and the power carried
by the sidebands on the values of Jn(β).

Indirect Method of FM Generation
Generation of FM Signals
There are essentially two basic methods for generating frequency mod-
ulated signals.
1. Indirect Method: In indirect method, the modulating signal is
first used to produce a narrowband frequency modulated signal,
and frequency multiplication is next used to increase the frequency
deviation to the desired level.
The indirect method is the preferred choice for frequency modula-
tion when the stability of carrier frequency is of major concern as
in the case of commercial radio broadcasting.
2. Direct Method: In direct method, the frequency of the carrier
is directly varied in accordance with the input modulating signal,
which is readily accomplished using a voltage-controlled oscillator.

Figure 7: Block diagram of the indirect method of generating a wideband FM
signal.

Fig. 7 shows the simplified block diagram of an indirect frequency
modulation system.
The system consists of an integrator, phase modulator, a frequency
modulator, and a crystal oscillator.
The crystal oscillator produces a carrier signal equal to Ac cos(2πf1t).
The message signal m(t) is first integrated and fed to a phase mod-
ulator.
The phase modulator itself consists of a multiplier, adder, and a
phase shifter that shifts the phase of its input, which is the crystal
oscillator signal by −π/2 radians.
This phase-shifted crystal oscillator output, equal to Ac sin(2πf1t)
is then multiplied with the integrator output.

This product signal is then fed to the adder which has its other input
taken from the crystal oscillator and is equal to Ac cos(2πf1t).
The difference between these signals is the phase modulator output
denoted by s1(t).
To minimize the distortion in the phase modulator, the maximum
phase deviation or modulation index β is kept small, thereby result-
ing in a narrowband frequency modulated signal.
The narrowband FM signal is next multiplied in frequency by means
of a frequency multiplier so as to produce the desired wideband FM
signal.

A frequency multiplier consists of a nonlinear device followed by a
band-pass filter, as shown in Fig. 8.
Figure 8: Block diagram of frequency multiplier.
The implication of the nonlinear device being memoryless is that it
has no energy-storage elements.
The input-output relation of such a non-linear device may be ex-
pressed in the general form as
s′
(t) =a1s(t) + a2s2
(t) + ... + ansn
(t) (49)
where, a1, a2, ...an are coefficients determined by the operating point
of the device, and n is the highest order of nonlinearity.

Let the NBFM wave produced at the output of the phase shifter,
which is s1(t) be given as.
s1(t) =A1 cos

2πf1t + 2πk1
Z t
0
m(t)dt

(50)
where,
f1 =⇒ frequency of the crystal oscillator,
k1 =⇒ frequency sensitivity (in Hz/V).

Let the single tone modulating signal be defined as;
m(t) = Am cos(2πfmt) (51)
Substituting Eq. (51) in Eq. (50);
s1(t) =A1 cos

2πf1t + 2πk1
Z t
0
Am cos(2πfmt)dt

=A1 cos

2πf1t + 2πk1Am
sin(2πfmt)
2πfm

=A1 cos

2πf1t +
k1Am
fm
sin(2πfmt)

s1(t) =A1 cos

2πf1t + β1 sin(2πfmt)

(52)
where,
β1 =⇒ modulation index for single-tone modulation and is kept
below 0.3 to minimize the distortion.

The output of the narrowband phase modulator is then multiplied
by a frequency multiplier, producing the desired WBFM wave hav-
ing the following time-domain description.
s(t) =Ac cos

2πnf1t + 2πnk1
Z t
0
m(t)dt

(53)
Denote =⇒ nf1 = fc and nk1 = kf in Eq. (53);
s(t) =Ac cos

2πfct + 2πkf
Z t
0
m(t)dt

(54)

For a single tone modulating signal m(t) = Am cos(2πfmt), Eq. (54)
becomes;
s(t) =Ac cos

2πfct + 2πkf
Z t
0
Am cos(2πfmt)dt

=Ac cos

2πfct + 2πkf Am
sin(2πfmt)
2πfm

=Ac cos

2πfct +
kf Am
fm
sin(2πfmt)

s(t) =Ac cos

2πfct + β sin(2πfmt)

(55)
where, β = nβ1.
Eq. (55) is the desired form of frequency modulated signal generated
using indirect frequency modulation technique.

Direct Method of FM Generation
Generation of FM Signals
There are essentially two basic methods for generating frequency mod-
ulated signals.
1. Indirect Method: In indirect method, the modulating signal is
first used to produce a narrowband frequency modulated signal,
and frequency multiplication is next used to increase the frequency
deviation to the desired level.
The indirect method is the preferred choice for frequency modula-
tion when the stability of carrier frequency is of major concern as
in the case of commercial radio broadcasting.
2. Direct Method: In direct method, the frequency of the carrier
is directly varied in accordance with the input modulating signal,
which is readily accomplished using a voltage-controlled oscillator.

In a direct FM systems, the instantaneous frequency of the carrier
wave is varied directly in accordance with the amplitude of the
message signal.
This may be achieved by using a voltage controlled oscillator (VCO).
A voltage-controlled oscillator (VCO) is an electronic oscillator whose
oscillation frequency is controlled by input voltage.
The applied input voltage determines the instantaneous oscillation
frequency fi(t).
Therefore, a VCO can be used for frequency modulation (FM) or
phase modulation (PM) by applying a modulating signal to the
control input.

VCO Construction
A VCO may be constructed using resonating elements such as a
inductor-capacitor i.e., LC combination.
The LC network can be made a highly resonant network by making
the capacitor to change its value in accordance with the modulating
signal.
We can make the capacitance of the LC network to change by using
a fixed capacitor in parallel with a voltage variable capacitor called
varactor to create a frequency selective network.
A varactor diode is a type of diode which is commonly operated in
reverse bias condition and the internal capacitance of the varactor
diode varies with respect to the applied reverse voltage.
Hence, a variable capacitance is created across the LC resonant
circuit.
The larger the reverse voltage applied to such a diode, the smaller
will be its transition capacitance.

Figure 9: Hartley oscillator.

For this discussion, let us consider a Hartley oscillator, where only
tank circuit details are shown in Fig. 9.
For this circuit, the frequency of oscillation is given by
fi(t) =
1
2π
p
(L1 + L2)C(t)
(56)
where,
C(t) =C0 + C[v(t)] (57)
In Fig. 9, inductors L1 and L2 and the capacitance C(t) form the
frequency determining network.
It should be noted that C[v(t)] is a variable capacitor, whose capac-
itance varies with v(t).

Suppose, modulating voltage v(t) be sinusoidal with a frequency fm
i.e.,
v(t) = Am cos(2πfmt) (58)
Then the total capacitance C(t) can be expressed as
C(t) =C0 + C[Am cos(2πfmt)]
C(t) =C0 + Cm cos(2πfmt)
(59)
where,
C0: is the total capacitance in the absence of modulation,
Cm: is the maximum change in the total capacitance.

Substituting Eq. (59) in (56) and solving for instantaneous fre-
quency, we get
fi(t) =
1
2π
p
(L1 + L2)[C0 + Cm cos(2πfmt)]
=
1
2π
r
(L1 + L2)C0[1 +
Cm
C0
cos(2πfmt)]
=
1
2π
p
(L1 + L2)C0
1
r
1 +
Cm
C0
cos(2πfmt)
=f0
1
r
1 +
Cm
C0
cos(2πfmt)
fi(t) =f0

1 +
Cm
C0
cos(2πfmt)
−1/2
(60)

where, f0 is the unmodulated frequency of oscillations and is given
by;
f0(t) =
1
2π
p
(L1 + L2)C0
(61)
Recall the binomial theorem;
(1 + x)−1/2
≊ 1 −
1
2
x, if |x| 1 (62)
Hence, if
Cm
C0
1, we may use binomial theorem on Eq. (60).
Therefore, Eq. (60) may be approximated as;
fi(t) =f0

1 −
Cm
2C0
cos(2πfmt)

(63)

Let −
Cm
2C0
=
∆f
f0
. Therefore, Eq. (63) reduces to,
fi(t) =f0

1 +
∆f
f0
cos(2πfmt)

=

f0 +
f0 × ∆f
f0
cos(2πfmt)

fi(t) =f0 + ∆f cos(2πfmt) (64)
Eq. (64) is the desired relationship for the instantaneous frequency
of the FM wave.
However, the FM wave thus generated is not widenband.

A wideband FM wave can be generated by direct method using the
arrangement shown in Fig. 10.
Figure 10: Generation of WBFM wave using direct method.

Compared with the indirect method, the VCO generated FM wave
has a large frequency deviation.
Therefore, multiplication by a small factor is required to convert it
into WBFM wave.
However, this advantage is negated by a very poor frequency sta-
bility.
This is because, the basic oscillator used in Fig. 10 is not a crystal
oscillator.
Hence, arrangements must be made to get a stabilized FM wave by
direct method.
A typical schematic block diagram is shown in Fig. 11.

Figure 11: Generation of WBFM wave with frequency stabilization.

The mixer in Fig. 11 has two inputs, namely the output of the FM
generator and the output of a stable crystal oscillator.
The mixer consists of a nonlinear device followed by a BPF.
The BPF in the mixer passes only the difference frequency term.
The output of the mixer is then applied to a frequency discriminator
that will have an instantaneous amplitude which is proportional to
the instantaneous frequency of its input signal.
The output of the frequency discriminator is lowpass filtered and
applied to VCO.
If there is no variation in the carrier frequency of the wideband FM
wave, the lowpass filter output is zero.
If the carrier frequency of the WBFM wave changes, then the dis-
criminator and lowpass filter combination develops a DC voltage,
which brings back the VCO carrier frequency and hence the fre-
quency of the WBFM wave to its required value.
Note that the polarity of the DC voltage is determined by the sense
of the frequency drift of the modulator.

DEMODULATION OF FM SIGNALS
Frequency demodulation is the process that enables us to recover
the original modulating signal from a frequency-modulated signal.
The objective is to produce a transfer characteristic that is the
inverse of that of the frequency modulator, which can be realized
directly or indirectly.
The two common techniques used in FM demodulation are;
1. Direct Method - Frequency Discriminator.
2. Indirect Method - Phase Locked Loop.

Direct Method - Frequency Discriminator: The direct method
of frequency demodulation involves the use of a popular device
known as a frequency discriminator, whose instantaneous ampli-
tude is directly proportional to the instantaneous frequency of the
input FM signal.
Indirect Method - Phase Locked Loop: This technique uses a
popular device known as a phase-locked loop.

Direct Method - Frequency Discriminator
The frequency discriminator consists of a slope circuit followed by
an envelope detector.
Figure 12: FM slope detector.

An ideal slope circuit is characterized by a transfer function that is
purely imaginary, varying linearly with frequency inside a prescribed
frequency interval.
Consider the transfer function plotted in Figure 13.a, which is de-
fined by
H1(f) =











j2πa

f − fc +
BT
2

, fc −
BT
2
≤ f ≤ fc +
BT
2
j2πa

f + fc −
BT
2

, −fc −
BT
2
≤ f ≤ −fc +
BT
2
0, elsewhere
(65)
where, a is a constant.

Figure 13: (a) Frequency response of ideal slope circuit. (b) The slope
circuit’s response. (c) Frequency response of the complex low-pass filter
equivalent to ideal slope circuit complementary to that of part (a).

We wish to evaluate the response of this slope circuit, denoted by
s1(t), which is produced by an FM signal s(t) of carrier frequency
fc and transmission bandwidth BT .
It is assumed that the spectrum of s(t) is essentially zero outside
the frequency interval fc −
BT
2
≤ f ≤ fc +
BT
2
.
For evaluation of the response s1(t), it is convenient to use the pro-
cedure which involves replacing the slope circuit with an equivalent
low-pass filter and driving this filter with the complex envelope of
the input FM signal s(t).

Let H̃1(f) denote the complex transfer function of the slope circuit
defined by Figure 13.a.
This complex transfer function is related to H1(f) by
H̃1(f − fc) =2H1(f), f 0 (66)
Substituting Eq. (65) in Eq. (66), we get
H̃1(f) =



j4πa

f +
BT
2

, −
BT
2
≤ f ≤
BT
2
0, elsewhere
(67)
The frequency response of H̃1(f) is shown in Fig. 13.b.

The incoming FM signal s(t) is defined by;
s(t) =Ac cos

2πfct + 2πkf
Z t
0
m(τ)dτ

(68)
Given that the carrier frequency fc is high compared to the trans-
mission bandwidth of the FM signal s(t), the complex envelope of
s(t) is
s̃(t) =Ac exp

j2πkf
Z t
0
m(τ)dτ

(69)

Let s̃1(t) denote the complex envelope of the response of the slope
circuit defined by Figure 13.b due to s̃(t).
Then, we may express the Fourier transform of s̃1(t) as follows:
S̃1(f) =
1
2
H̃1(f)S̃(f)
=



j2πa

f +
BT
2

S̃(f), −
BT
2
≤ f ≤
BT
2
0, elsewhere
=a

j2πfS̃(f) + j2π
BT
2
S̃(f)

, −
BT
2
≤ f ≤
BT
2
(70)
where S̃(f) is the Fourier transform of s̃(t).

Since multiplication of the Fourier transform of a signal by the factor
(j2πf) is equivalent to differentiating the signal in the time domain,
we deduce from Eq. (70) that
s̃1(t) =a

ds̃(t)
dt
+ jπBT s̃(t)

(71)
Substituting Eq. (69) in Eq. (71), we get
s̃1(t) =a

d
dt

Ac exp

j2πkf
Z t
0
m(τ)dτ

+
jπBT

Ac exp

j2πkf
Z t
0
m(τ)dτ

=jπBT aAc

1 +
2kf
BT
m(t)

exp

j2πkf
Z t
0
m(τ)dτ

(72)

The desired response of the slope circuit is therefore
s1(t) =Re[s̃(t) exp(j2πfct)]
=πBT aAc

1 +
2kf
BT
m(t)

cos

2πfct + 2πkf
Z t
0
m(τ)dτ +
π
2

(73)
The signal s1(t) is a hybrid-modulated signal, in which both ampli-
tude and frequency of the carrier wave vary with the message signal
m(t).

However, provided that we choose
2kf
BT
m(t) 1 for all t (74)
then we may use an envelope detector to recover the amplitude vari-
ations and thus, except for a bias term, obtain the original message
signal.
The resulting envelope-detector output is therefore
|s̃(t)| =πBT aAc

1 +
2kf
BT
m(t)

(75)
The bias term πBT aAc in the right-hand side of Eq. (75) is propor-
tional to the slope a of the transfer function of the slope circuit.

This suggests that the bias may be removed by subtracting from
the envelope-detector output |s̃(t)| the output of a second enve-
lope detector preceded by the complementary slope circuit with the
transfer function H2(f) plotted in Figure 13.c.
That is, the respective complex transfer functions of the two slope
circuits are related by
H̃2(f) =H̃1(−f) (76)
Let s2(t) denote the response of the complementary slope circuit
produced by the incoming FM signal s(t).
Then, following a procedure similar to that just described, we find
that the envelope of s2(t) is
|s̃2(t)| =πBT aAc

1 −
2kf
BT
m(t)

(77)
where s̃2(t) is the complex envelope of the signal s2(t).

The difference between the two envelopes in Eqs. (75) and (77) is
s0(t) =|s̃1(t)| − |s̃2(t)|
=4πkf aAcm(t)
(78)
which is free from bias, as desired.
We may thus model the ideal frequency discriminator as a pair of
slope circuits with their complex transfer functions related by Eq.
(76), followed by envelope detectors and finally a summer, as in
Figure 14.
This scheme is called a balanced frequency discriminator.

Figure 14: Block diagram of balanced frequency discriminator.

The idealized scheme of Figure 14 can be closely realized using the
circuit shown in Figure 15.
Figure 15: Circuit diagram of balanced frequency discriminator.

The upper and lower resonant filter sections of this circuit are tuned
to frequencies above and below the unmodulated carrier frequency
fc, respectively.
In Figure 16 we have plotted the amplitude responses of these two
tuned filters, together with their total response, assuming that both
filters have a high Q-factor.
The quality factor or Q-factor of a resonant circuit is a measure of
goodness of the whole circuit.
It is formally defined as 2π times the ratio of maximum energy
stored in the circuit during one cycle to the energy dissipated per
cycle.

In the case of an RLC parallel (or series) resonant circuit, the Q-
factor is equal to the resonant frequency divided by the 3-dB band-
width of the circuit.
In the RLC parallel resonant circuits shown in Figure 15, the re-
sistance R is contributed largely by imperfections in the inductive
elements of the circuits.

Figure 16: Frequency response of balanced frequency discriminator.

The linearity of the useful portion of the total response in Figure 16,
centered at fc, is determined by the separation of the two resonant
frequencies.
As illustrated in Figure 16, a frequency separation of 3B gives sat-
isfactory results, where 2B is the 3-dB bandwidth of either filter.
However, there will be distortion in the output of this frequency
discriminator due to the following factors:
1. The spectrum of the input FM signal s(t) is not exactly zero for
frequencies outside the range −BT /2 ≤ f ≤ fc + BT /2.
2. The tuned filter outputs are not strictly band limited, and so
some distortion is introduced by the low-pass RC filters following
the diodes in the envelope detectors.
3. The tuned filter characteristics are not linear over the whole
frequency band of the input FM signal s(t).
Nevertheless, by proper design, it is possible to maintain the FM
distortion produced by these factors within tolerable limits.

FM STEREO MULTIPLEXING
Stereo multiplexing is a form of frequency-division multiplexing
(FDM) designed to transmit two separate signals via the same car-
rier.
It is widely used in FM radio broadcasting to send two different
elements of a program (e.g., two different sections of an orchestra,
a vocalist and an accompanist) so as to give a spatial dimension to
its perception by a listener at the receiving end.

The specification of standards for FM stereo transmission is influ-
enced by two factors:
1. The transmission has to operate within the allocated FM broad-
cast channels.
2. It has to be compatible with monophonic radio (single audio
channel/single speaker) receivers.
The first requirement sets the permissible frequency parameters,
including frequency deviation.
The second requirement constrains the way in which the transmitted
signal is configured.

Figure 17 shows the block diagram of the multiplexing system used
in an FM stereo transmitter.
Figure 17: Multiplexer in transmitter of FM stereo.

Let ml(t) and mr(t) denote the signals picked up by left-hand and
righthand microphones at the transmitting end of the system.
They are applied to a simple matrixer that generates the sum signal,
ml(t) + mr(t), and the difference signal, ml(t) − mr(t).
The sum signal is left unprocessed in its baseband form; it is avail-
able for monophonic reception.
The difference signal and a 38-kHz subcarrier (derived from a 19-
kHz crystal oscillator by frequency doubling) are applied to a prod-
uct modulator, thereby producing a DSB-SC modulated wave.
In addition to the sum signal and this DSB-SC modulated wave, the
multiplexed signal m(t) also includes a 19-kHz pilot (small carrier)
to provide a reference for the coherent detection of the difference
signal at the stereo receiver.

Thus the multiplexed signal is described by
m(t) =[ml(t) + mr(t)] + [ml(t) − mr(t)] cos(4πfct) + K cos(2πfct)
(79)
where fc = 19 kHz, and K is the amplitude of the pilot tone.
The multiplexed signal m(t) then frequency-modulates the main
carrier to produce the transmitted signal.
The pilot is allotted between 8 and 10 percent of the peak frequency
deviation.
The amplitude K is chosen to satisfy this requirement.

Figure 18: Demultiplexer at receiver of FM stereo.

At a stereo receiver, the multiplexed signal m(t) is recovered by
frequency demodulating the incoming FM wave.
Then m(t) is applied to the demultiplexing system shown in Figure
18.
The individual components of the multiplexed signal m(t) are sep-
arated by the use of three appropriate filters.
The recovered pilot (using a narrowband filter tuned to 19 kHz) is
frequency doubled to produce the desired 38-kHz subcarrier.
The availability of this subcarrier enables the coherent detection
of the DSB-SC modulated wave, thereby recovering the difference
signal, ml(t) − mr(t).
The baseband low-pass filter in the top path of Figure 18 is designed
to pass the sum signal, ml(t) + mr(t).
Finally, the simple matrixer reconstructs the left-hand signal ml(t)
and right-hand signal mr(t) and applies them to their respective
speakers.

PHASE-LOCKED LOOP
INDIRECT METHOD OF FM DEMODULATION:
PHASE-LOCKED LOOP
The phase-locked loop (PLL) is a negative feedback system, the
operation of which is closely linked to frequency modulation.
PLL can be used for synchronization, frequency division/multiplication
frequency modulation, and indirect frequency demodulation.
In this topic, we demonstrate the use of PLL for indirect frequency
demodulation.

PHASE-LOCKED LOOP
Basically, the phase-locked loop consists of three major components:
a multiplier, a loop filter, and a voltage-controlled oscillator (VCO)
connected together in the form of a feedback loop, as in Figure 19.
The VCO is a sinusoidal generator whose frequency is determined
by a voltage applied to it from an external source. In effect, any
frequency modulator may serve as a VCO.
Figure 19: Phase-locked loop.

PHASE-LOCKED LOOP
We assume that initially we have adjusted the VCO so that when the
control voltage (input to VCO) is zero, two conditions are satisfied:
1. The frequency of the VCO in precisely set at the unmodulated
carrier frequency fc.
2. The VCO output has a 90-degree phase-shift with respect to the
unmodulated carrier wave.

PHASE-LOCKED LOOP
Suppose that the input signal applied to the phase-locked loop is an
FM signal defined by
s(t) =Ac sin[2πfct + ϕ1(t)] (80)
where Ac is the carrier amplitude.
Let the modulating signal be denoted by m(t). Then, the angle
ϕ1(t) is related to m(t) by the integral
ϕ1(t) =2πkf
Z t
0
m(τ)dτ (81)
where kf is the frequency sensitivity of the frequency modulator.

PHASE-LOCKED LOOP
Let the VCO output in the phase-locked loop be defined by
r(t) =Av cos[2πfct + ϕ2(t)] (82)
where Av is the amplitude.
With a control voltage v(t) applied to the VCO input, the angle
ϕ2(t) is related to v(t) by the integral
ϕ2(t) =2πkv
Z t
0
v(t)dt (83)
where kv is the frequency sensitivity of the VCO.

PHASE-LOCKED LOOP
The objective of the phase-locked loop is to generate a VCO output
r(t) that has the same phase angle as the input FM signal s(t)
except for the fixed difference of 90 degrees.
The time-varying phase angle ϕ1(t) of the input FM signal s(t) may
be due to modulation by a message signal m(t) as in Eq. (81).
In such a case, we wish to recover ϕ1(t) in order to estimate the
message signal m(t).
In other applications of the phase-locked loop, the time-varying
phase angle ϕ1(t) of the incoming signal s(t) may be an unwanted
phase shift caused by fluctuations in the communication channel.
In this case, we wish to track ϕ1(t) so as to produce a signal with
the same phase angle for the purpose of coherent detection (syn-
chronous demodulation).

PHASE-LOCKED LOOP LOOP
NONLINEAR MODEL OF THE PHASE-LOCKED
LOOP
According to Figure 19, the incoming FM signal s(t) and the VCO
output r(t) are applied to the multiplier, producing two compo-
nents:
=s(t)r(t)
=kmAc sin[2πfct + ϕ1(t)]Av cos[2πfct + ϕ2(t)]
=kmAcAv sin[4πfct + ϕ1(t) + ϕ2(t)] + kmAcAv sin[ϕ1(t) − ϕ2(t)]
(84)
where km is the multiplier gain, measured in volt−1.

1. A high-frequency component, represented by the double-frequency
term
kmAcAv sin[4πfct + ϕ1(t) + ϕ2(t)] (85)
2. A low-frequency component represented by the difference-frequency
term
kmAcAv sin[ϕ1(t) − ϕ2(t)] (86)
where km is the multiplier gain, measured in volt−1.

The loop filter in the phase-lock loop is a low-pass filter, and its
response to the high-frequency component will be negligible.
The VCO also contributes to the attenuation of this component.
Therefore, discarding the high-frequency component (i.e., the double-
frequency term), the input to the loop filter is reduced to
e(t) =kmAcAv sin[ϕe(t)] (87)
where ϕe(t) is the phase error defined by
ϕe(t) =ϕ1(t) − ϕ2(t)
=ϕ1(t) − 2πkv
Z t
0
v(τ)dτ (From Eq. (83))
(88)

The loop filter operates on the input e(t) to produce an output v(t)
defined by the convolution integral
v(t) =
Z ∞
−∞
e(τ)h(t − τ)dτ (89)
where h(t) is the impulse response of the loop filter.
Substituting Eq. (89) and Eq. (87) in Eq. (88);
ϕe(t) =ϕ1(t) − 2πkv
Z ∞
−∞
Z t
0
e(τ)h(t − τ)dτ
ϕe(t) =ϕ1(t) − 2πkv
Z ∞
−∞
Z t
0
kmAcAv sin[ϕe(τ)]h(t − τ)dτ
ϕe(t) =ϕ1(t) − 2πkvkmAcAv
Z ∞
−∞
Z t
0
sin[ϕe(τ)]h(t − τ)dτ
(90)

Differentiating Eq. (90) with respect to t;
dϕe(t)
dt
=
dϕ1(t)
dt
− 2πK0
Z ∞
−∞
sin[ϕe(τ)]h(t − τ)dτ (91)
where K0 is a loop-gain parameter defined by
K0 =kmkvAcAv (92)
The amplitudes Ac and Av are both measured in volts, the multiplier
gain km in volt−1 and the frequency sensitivity kv in Hertz per volt.
Hence, it follows from Eq. (92) that K0 has the dimensions of
frequency.

Equation (91) suggests the model shown in Figure 20 for a phase-
locked loop.
Figure 20: Nonlinear model of phase-locked loop.

In this model we have also included the relationship between v(t)
and e(t) as represented by Eqs. (87) and (89).
We see that the model resembles the block diagram of Figure 19.
The multiplier at the input of the phase-locked loop is replaced
by a subtracter and a sinusoidal nonlinearity, and the VCO by an
integrator.
The sinusoidal nonlinearity in the model of Figure 19 greatly in-
creases the difficulty of analyzing the behavior of the phase-locked
loop.
It would be helpful to linearize this model to simplify th analysis,
yet give a good approximate description of the loop’s behavior in
certain modes of operation. This we do next.

PHASE-LOCKED LOOP LINEAR MODEL OF THE PHASE-LOCKED LOOP
LINEAR MODEL OF THE PHASE-LOCKED LOOP
When the phase error ϕe(t) is zero, the phase-locked loop is said to
be in phase-lock.
When ϕe(t) is at all times small compared with one radian, we may
use the approximation
sin[ϕe(t)] ≊ ϕe(t) (93)
which is accurate to within 4 percent for ϕe(t) less than 0.5 radians.
In this case, the loop is said to be near phase-lock, and the sinusoidal
nonlinearity of Figure 19 may be disregarded.
Thus, we may represent the phase-locked loop by the linearized
model shown in Figure 21.a.

Figure 21: Models for PLL (a) Linearized model. (b) Simplified model when
the loop gain is very large compared to unity.

According to this model, the phase error ϕe(t) is related to the input
phase ϕ1(t) by the linear integro-differential equation
dϕe(t)
dt
+ 2πK0
Z ∞
−∞
ϕe(τ)h(t − τ)dτ =
dϕ1(t)
dt
(94)
Transforming Eq. (94) into the frequency domain and solving for
Φe(f) the Fourier transform of ϕe(t),in terms of Φ1(f), the Fourier
transform of ϕ1(t), we get
Φe(f) =
1
1 + L(f)
Φ1(f) (95)
The function L(f) in Eq. (95) is defined by
L(f) =K0
H(f)
jf
(96)
where H(f) is the transfer function of the loop filter.
The quantity L(f) is called the open-loop transfer function of the
phase-locked loop.

Suppose that for all values of f inside the baseband we make the
magnitude of L(f) very large compared with unity.
Then from Eq. (95) we find that Φe(f) approaches zero. That is,
the phase of the VCO becomes asymptotically equal to the phase
of the incoming signal.
Under this condition, phase-lock is established, and the objective of
the phase-locked loop is thereby satisfied.
From Figure 21.a we see that V (f), the Fourier transform of the
phase-locked loop output v(t), is related to Φe(f) by
V (f) =
K0
kv
H(f)Φe(f) (97)
Substituting for H(f) from Eq. (96) we may write,
V (f) =
jf
kv
L(f)Φe(f) (98)

Therefore, substituting Eq. (95) in (98), we get
V (f) =
(jf/kv)L(f)
1 + L(f)
Φ1(f) (99)
Again, when we make |L(f)| 1 for the frequency band of inter-
est, we may approximate Eq. (99) as follows:
V (f) ≊
(jf)
kv
Φ1(f)
≊
(jf)
kv
Φ1(f) ×
2π
2π
≊
(jf2π)
2πkv
Φ1(f)
(100)

The corresponding time-domain relation is
v(t) ≊
1
2πkv
dϕ1(t)
dt
(101)
Thus, provided that the magnitude of the open-loop transfer func-
tion L(f) is very large for all frequencies of interest, the phase-locked
loop may be modeled as a differentiator with its output scaled by
the factor 1/2πkv as in Figure 21.b.
The simplified model of Figure 21 provides an indirect method of
using the phase-locked loop as a frequency demodulator.
When the input is an FM signal as in Eq. (80), the angle ϕ1(t) is
related to the message signal m(t) as in Eq. (81).

Therefore, substituting Eq. (81) in (101), we find that the resulting
output signal of the phaselocked loop is approximately
v(t) ≊
kf
kv
m(t) (102)
Equation (102) states that when the loop operates in its phase-
locked mode, the output v(t) of the phase-locked loop is approxi-
mately the same, except for the scale factor kf /kv, as the original
message signal m(t); frequency demodulation of the incoming FM
signal s(t) is thereby accomplished.

NONLINEAR EFFECTS IN FM SYSTEMS
Nonlinearities, in one form or another, are present in all electrical
networks.
There are two basic forms of nonlinearity to consider:
1. The nonlinearity is said to be strong when it is introduced in-
tentionally and in a controlled manner for some specific application.
Examples of strong nonlinearity include square-law modulators, lim-
iters, and frequency multipliers.
2. The nonlinearity is said to be weak when a linear performance is
desired, but nonlinearities of a parasitic nature arise due to imper-
fections. The effect of such weak nonlinearities is to limit the useful
signal levels in a system and thereby become an important design
consideration.
In this section we examine the effects of weak nonlinearities on fre-
quency modulation.

Consider a communications channel, the transfer characteristic of
which is defined by the nonlinear input-output relation
v0(t) =a1vi(t) + a2v2
i (t) + a3v3
i (t) (103)
where vi(t) and v0(t) are the input and output signals, respectively.
The channel described in Eq. (103) is said to be memoryless in that
the output signal v0(t) is an instantaneous function of the input
signal vi(t).
We wish to determine the effect of transmitting a frequency-modulated
wave through such a channel.

The FM signal is defined by
vi(t) =Ac cos[2πfct + ϕ(t)] (104)
where
ϕ(t) =2πkf
Z t
0
m(τ)dτ (105)
Substituting Eq. (104) in (103);
v0(t) =a1Ac cos[2πfct + ϕ(t)] + a2A2
c cos2
[2πfct + ϕ(t)]
+ a3A3
c cos3
[2πfct + ϕ(t)]
(106)
Recall
cos2
(θ) =
1 + cos(2θ)
2
cos3
(θ) =
1
4
cos(3θ) −
3
4
cos(θ)
(107)

Therefore,
v0(t) =
1
2
a2A2
c +

a1Ac +
3
4
a3A3
c

cos[2πfct + ϕ(t)]
+
1
2
a2A2
c cos[4πfct + 2ϕ(t)]
+
1
4
a3A3
c cos[6πfct + 3ϕ(t)]
(108)
Thus the channel output consists of a dc component and three
frequency-modulated signals with carrier frequencies fc, 2fc,and
3fc; the sinusoidal components are contributed by the linear, second-
order, and third-order terms of Eq. (103), respectively.
To extract the desired FM signal from the channel output v0(t), that
is, the particular component with carrier frequency fc, it is necessary
to separate the FM signal with this carrier frequency from the one
with the closest carrier frequency: 2fc.

Let ∆f denote the frequency deviation of the incoming FM signal
vi(t), and W denote the highest frequency component of the message
signal m(t).
Then, applying Carson’s rule and noting that the frequency devia-
tion about the second harmonic of the carrier frequency is doubled,
we find that the necessary condition for separating the desired FM
signal with the carrier frequency fc from that with the carrier fre-
quency 2fc is
2fc − (2∆f + W) fc + ∆f + W (109)
or
fc 3∆f + 2W (110)

Thus, by using a band-pass filter of mid-band frequency fc and
bandwidth 2∆f +2W (Carson‘s rule), the channel output is reduced
to
v′
0(t) =

a1Ac +
3
4
a3A3
c

cos[2πfct + ϕ(t)] (111)
We see therefore that the effect of passing an FM signal through
a channel with amplitude nonlinearities, followed by appropriate
filtering, is simply to modify its amplitude.
That is, unlike amplitude modulation, frequency modulation is not
affected by distortion produced by transmission through a channel
with amplitude nonlinearities.
It is for this reason that we find frequency modulation widely used
in microwave radio and satellite communication systems.
It permits the use of highly nonlinear amplifiers and power trans-
mitters, which are particularly important to producing a maximum
power output at radio frequencies.

An FM system is extremely sensitive to phase nonlinearities, how-
ever, as we would intuitively expect.
A common type of phase nonlinearity that is encountered in mi-
crowave radio systems is known as AM-to-PM conversion.
This is the result of the phase characteristic of repeaters or am-
plifiers used in the system being dependent on the instantaneous
amplitude of the input signal.
In practice, AM-to-PM conversion is characterized by a constant K,
which is measured in degrees per dB and may be interpreted as the
peak phase change at the output for a 1-dB change in envelope at
the input.

When an FM wave is transmitted through a microwave radio link, it
picks up spurious amplitude variations due to noise and interference
during the course of transmission, and when such an FM wave is
passed through a repeater with AM-to-PM conversion, the output
will contain unwanted phase modulation and resultant distortion.
It is therefore important to keep the AM-to-PM conversion at a low
level. For example, for a good microwave repeater, the AM-to-PM
conversion constant K is less than 2 degrees per dB.

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