Analog & Digital Communication

1. FM Modulation:

The frequency modulation can be defined as the process of varying the frequency of the carrier
signal in accordance with instantaneous value (Amplitude) of the input modulating signal. The
carrier, modulating signal and the FM waveforms also are shown in the following figure:

The frequency of a carrier (fc) will increase as the amplitude of modulating (input) signal increases.
The carrier frequency will be maximum (fc max) when the input signal is at its peak. The carrier
deviates maximum from its normal value. The frequency of a carrier will decrease as the amplitude of
the modulating (input) signal decreases. The carrier frequency will be minimum (fc min) when the
input signal is at its lowest.

The carrier deviates minimum from its normal value. The frequency of the carrier will be at its normal
value (free running) fc when the input signal value is 0V. There is no deviation in the carrier.

➔ Frequency Deviation:
The amount of change in the carrier frequency produced, by the amplitude of the input modulating
signal, is called frequency deviation. It is denoted by Δf. The Carrier frequency swings between fmax
and fmin as the input varries in its amplitude.

The difference between fmax and fc is known as frequency deviation. Similarly, the difference
between fc and fmin also is known as frequency deviation. i.e.,

∆𝐟 = 𝐟𝐜 𝐦𝐚𝐱 − 𝐟𝐜

∆𝐟 = 𝐟𝐜 − 𝐟𝐜 𝐦𝐢𝐧

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1.1 Mathematical Representation of FM Signal:

For the message (modulating) signal m(t) and carrier signal c(t) = Accos2πfct, standard equation of
angle modulated wave is given by the expression:

s(t) = Ac cosθi(t) --------------------------------- [1]

The equation for instantaneous frequency fi in FM modulation is

fi = fc + Kf m(t)

Where, Kf = frequency sensitivity (Hz/V)

Also, we have the relation:
dθi(t)
ωi =
dt

dθi(t)
2πf =
i dt

θi(t) = 2π ∫ fi dt

Substitute the values of fi in above expression;

θi(t) = 2π ∫[fc + Kf m(t)] dt

θi(t) = 2πfct + 2πKf ∫ m(t) dt

Equation [1] ➔
s(t) = Ac cos (2πfct + 2πKf ∫ m(t) dt)

This is the generalized expression of FM modulated signal with message signal m (t).

Let, m(t) = Amcos2πfmt

s(t) = Ac cos (2πfct + 2πKf ∫ Amcos2πfmt dt)

2πKfAm
s(t) = A cos (2πf t + sin 2πf t)
c c m
2πfm

s(t) = Ac cos(2πfct + β sin 2πfmt)

.. where, β = KfAm/fm

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1.2 FM Spectrum:

Frequency modulation can be classified as Narrowband if the value of modulation index (β < 1) and
as wideband FM if (β > 1). Suitable mathematical steps required in order to find out frequency
spectrum for both these NBFM and WBFM are discussed in the following section:

A] Narrow Band FM (NBFM) ➔ The equation of frequency modulated wave is given by

s(t) = Ac cos(2πfct + β sin 2πfmt)
s(t) = Ac [cos2πfct . cos(β sin 2πfmt) − sin2πfct. sin(β sin 2πfmt)] ....... cos(A + B) = cosA. cosB − sinA. sinB

For NBFM, β < 1 and for small values of θ ➔ sinθ ≈ θ and cosθ ≈ 1
Using above approximations, cos(β sin 2πfmt) ≈ 1 and sin(β sin 2πfmt) ≈ β sin 2πfmt
s(t) = Ac [cos2πfct (1) − sin2πfct. (β sin 2πfmt) ]
s(t) = Ac [cos2πfct − β sin2πfct. sin 2πfmt ]
Again, by using trigonometry formula of ➔ sinA. sinB = 1 [cos(A − B) − cos(A + B)]
2
β
s(t) = Ac [cos2πfct − {cos2π(fc − fm)t − cos2π(fc + fm)t} ]
2

s(t)NBFM = Ac cos2πfct + cos2π(fc + fm)t − cos2π(fc − fm)t

2 2

This is the final expression for Narrow band frequency modulated signal. From the expression, it can
be observed that it consists of three components: carrier, upper sideband and lower sideband.

AM modulated signal can be given by:

s(t) = A cos2πf t + Acμ cos2π(f + f )t + Acμ cos2π(f − f )t

AM c c c m c m
2 2

On comparing the expressions of AM and NBFM signals, it can be concluded that the spectrum of
AM and NBFM will be same except 180 degree phase shift at lower sideband frequency component.
Note: Power and bandwidth requirement of NBFM will be same as that of AM signal.

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B] Wide Band FM (WBFM) ➔

The equation of frequency modulated wave is given by
s(t) = Ac cos(2πfct + β sin 2πfmt) . . . where, β > 1

The expression for Wide band frequency modulated signal is given by:
∞

s(t)WBFM = Ac ∑ Jn(β) cos2π(fc + nfm)t

n=−∞

where,

Jn(β) = Bessel function and

1 π j(βsinθ−nθ)
Jn(β) = ∫ e dθ
2π −π

Figure shows, plot of Bessel

Function for the different values
of β = 0, 1, 2 ….

By expanding WBFM expression for the values of n = 0, ±1, ±2 … ..

s(t)WBFM = Ac J0(β)cos2πfct + Ac J1(β)cos2π(fc + fm)t + Ac J−1(β)cos2π(fc − fm)t

+ Ac J2(β)cos2π(fc + 2fm)t + Ac J−2(β)cos2π(fc − 2fm)t +. . . . . . ..

Thus, from above expression it can be observed that there are infinite sidebands present in the
spectrum of WBFM signal. The amplitude of spectrum is nothing but value of Bessel function at that
point. Here, only five components are considered for n = 0, 1, -1, 2, -2.

Spectrum of WBFM

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From the spectrum of WBFM, it can be observed that bandwidth requirement of this signal is
theoretically INFINITE ! But, practical it is calculated using CARSON’s Rule as:

BW = 2(β + 1)fm or BW = 2(∆f + fm)

Also, total power requirement is

Ac2
Pt =
2R

2. Difference between Frequency Modulation and Amplitude Modulation:

SN Parameter Frequency Modulation (FM) Amplitude Modulation (AM)

01 Definition It is the process of varying It is the process of varying

frequency of carrier wave with amplitude of carrier wave with
respect to instantaneous value respect to instantaneous value
of modulating signal. of modulating signal.
02 Amplitude of Carrier Remains constant Is varying according to message
03 Transmitted power Remains constant Is varying according to ‘µ’
Ac2 Ac2 μ2
Pt = Pt = (1 + )
2R 2R 2
04 Modulation Index Can exceed above one [β > 1] ‘µ’ cannot exceed above 1
05 No. of Sidebands Infinite sidebands are present. Only two sidebands are present
06 Bandwidth Theoretically, infinite !
By Carson’s rule: BW = 2fm
BW = 2[1 + β]fm
07 Fidelity Better than that of AM Poor than FM
08 Broadcast frequencies 88 MHz – 108 MHz 535 KHz – 1650 KHz
09 Transmission Are Complex and costly Are Simple
Equipment
10 Noise Interference Is less [due to presence of Is more
amplitude limiters]
11 Adjacent channel Is absent [due to Guard band] Is present
interference
12 Area of reception Is small Is large

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3. Comparison between NBFM and WBFM:

SN Parameter Narrow Band FM Wide Band FM

01 Modulation Index (𝛃) β<1 β>1

02 Frequency Deviation (∆𝐟) 5 KHz 75 KHz
03 Modulation Frequency 30 Hz – 3 KHz 30 Hz – 15 KHz
04 Number of Sidebands Only 2 sidebands are present. Infinite sidebands are present.
05 Bandwidth BW = 2fm 15 times than that of NBFM
BW = 2[1 + β]fm
06 Noise Suppression Is less Is More
07 Applications Mobile Communication Entertainment & Broadcasting

4. Generation of Frequency Modulated Signals:

There are two basic methods of generating Frequency Modulated signals:

• Direct method
• Indirect method

Direct Method: This method is called as the Direct Method because we are generating a wide band
FM wave directly. In this method, Voltage Controlled Oscillator (VCO) is used to generate WBFM.
VCO produces an output signal, whose frequency is proportional to the input signal voltage. i.e.,

𝑓𝑖 ∝ 𝑚(𝑡)

𝑓𝑖 = 𝑓𝑐 + 𝑘𝑓 𝑚(𝑡)

Indirect Method: This method is called as Indirect Method because we are generating a wide band
FM wave indirectly. This means, first we will generate NBFM wave and then with the help of
frequency multipliers we will get WBFM wave.

4.1 Direct FM Generation using FET reactance modulator

One method of Generation of Frequency Modulation suggests if either the capacitance or inductance
of an LC oscillator tank is varied, frequency modulation of some form will result. If this variation can
be made directly proportional to the voltage supplied by the modulation circuits, true FM will be
obtained. There are a number of devices whose reactance can be varied by the application of
voltage. The three-terminal ones include the reactance field-effect transistor (FET).

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Basic reactance modulator: Provided that certain simple conditions are met, the impedance z, as
seen at the input terminals A–A of figure, is almost entirely reactive. The circuit shown is the basic
circuit of a FET reactance modulator, which behaves as a three-terminal reactance that may be
connected across the tank circuit of the oscillator to be frequency-modulated. It can be made
inductive or capacitive by a simple component change. The value of this reactance is proportional to
the transconductance of the device, which can be made to depend on the gate bias and its
variations. Note that an FET is used in the explanation here for simplicity only. Identical reasoning
would apply to a bipolar transistor or a vacuum tube, or indeed to any other amplifying device.

Theory of Reactance Modulators:

In order to determine z, a voltage v is applied to the terminals A–A between which the impedance is
to be measured, and the resulting current i is calculated. The applied voltage is then divided by this
current, giving the impedance seen when looking into the
terminals. In order for this impedance to be a pure reactance (it is
capacitive here), two requirements must be fulfilled. The first is
that the bias network current ib must be negligible compared to
the drain current. The impedance of the bias network must be
large enough to be ignored. The second requirement is that the
drain-to-gate impedance (Xc here) must be greater than the gate-
to-source impedance (R in this case), preferably by more than 5:1.

The following analysis may then be applied:

Rv
vg = ib R =
R − jX c
The FET drain current is
gm Rv
i = gmvg =
R − jXc

Therefore, the impedance seen at the terminals A – A is

v R − jXc
z= =
i gm R
1 jXc
z= (1 − )
gm R

If XC ≫ R in above equation, then equation will reduce to

−jXc
z=
gm R
This impedance is quite clearly a capacitive reactance, which may be written as

X = Xc = 1 = 1
eq gm R 2πf gm RC 2πf Ceq

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From above equation it is seen that under such conditions the input impedance of the device at A–A
is a pure reactance and is given by
Ceq = gm RC

This equivalent capacitance depends on the device transconductance and can therefore be
varied with bias voltage. The capacitance can be originally adjusted to any value by varying the
components R and C. The expression gmRC has the correct dimensions of capacitance; R, measured
in ohms and gm measured in siemens (s), cancel each other’s dimensions, leaving C as required.

5. Pre-emphasis and De-emphasis

A] Pre-emphasis: The noise suppression ability of FM decreases with the increase in the frequencies.
Thus, increasing the relative strength or amplitude of the high frequency components of the
message signal before modulation is termed as Pre-emphasis.

Pre-Emphasis Circuit:
At the transmitter, the modulating signal is passed through a simple network which amplifies the
high frequency, components more than the low-frequency components. The simplest form of such a
circuit is a simple high pass filter of the type shown in fig (a). Specification dictate a time constant of
75 microseconds (µs) where t = RC. Any combination of resistor and capacitor (or resistor and
inductor) giving this time constant will be satisfactory. Such a circuit has a cutoff frequency fco of
2122 Hz. This means that frequencies higher than 2122 Hz will he linearly enhanced. The output
amplitude increases with frequency at a rate of 6 dB per octave. The pre-emphasis curve is shown in
Fig (b). This pre-emphasis circuit increases the energy content of the higher-frequency signals so that
they will tend to become stronger than the high frequency noise components. This improves the
signal to noise ratio and increases intelligibility and fidelity.

The pre-emphasis circuit also has a upper break frequency fu where signal enhancement flattens out.
This upper break frequency is computed by the expression:

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R2
fu = R1 +
2πR1R2C

It is usually set to very high value beyond the audio frequency range, typically it is 30 KHz.

B] De-emphasis: To reproduce the signal at the receiver, at its normal level; a de-emphasis circuit is
used. This is simple RC circuit with time constant of 75 microseconds (µs). It is shown along with its
response curve in the figure below. The de-emphasis circuit has a cutoff frequency of 2122 Hz. The
signals above this frequency is attenuated at the rate 6dB per octave. Therefore, pre-emphasis at the
transmitter is exactly offset by de-emphasis at the receiver, giving normal frequency response.

Note:
✓ To increase the SNR at higher modulation frequencies, a high pass circuit called pre-
emphasis, is used at the transmitter.
✓ Another circuit called de-emphasis, the inverse process of pre-emphasis is used at the
receiver, which is a low pass circuit.
✓ The pre-emphasis and de-emphasis circuits are widely used in FM transmitter and receiver to
effectively increase the output SNR.

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1. Discuss the need of Pre-Emphasis and De-Emphasis circuit in the FM with circuit diagrams and waveforms.
2. Compare the NBFM and WBFM. Explain the Bandwidth requirement in WBFM and give its formula.
3. Explain the signal generation using FET reactance Modulator with circuit diagram.
4. Compare AM and FM.
5. Draw and explain the block diagram of FM receiver and explain its characteristics.
6. With the help of neat Sketch, explain ratio detector in detail and how it demodulates the FM signal?
7. Explain Balanced Slope Detector with neat circuit diagram.
8. A 100 MHz carrier is frequency modulated by a sinusoidal signal of amplitude 20 V and frequency 100 KHz.
The frequency sensitivity of modulator is 25 KHz/Volt.
I) Determine Frequency Deviation, Modulation Index & B.W.
II) Repeat above calculation when amplitude of message signal is doubled.
9. FM signal is s(t) = 10 cos (2π10^6t + 8sin4π10^3t)
I. Determine Modulation index, Frequency Deviation and power
II. Repeat above calculation when message frequency is doubled.
10. A frequency modulated Signal is represented by the voltage equation e FM = 10 sin (6*108 t + 5 sin 1250 t)
Calculate: I) carrier frequency II) Modulating frequency III) Maximum deviation
IV) what power will this FM wave dissipates in 20Ω resistor.
11. Determine the value of the capacity reactance obtainable from a reactance FET whose gm is 10 milli siemens
(10 mS). Assume that the gate-to-source resistance is one – sixth of the reactance of the gate–to–drain capacitor
and that the frequency is 8 MHz.

12. A 92.3 MHz carrier is frequency modulated by a sinusoidal signal of amplitude 20 V and frequency 10 KHz.
The frequency sensitivity of modulator is 15 KHz/Volt.
I) Determine Frequency Deviation, Modulation Index & B.W.
13. A 101.7 MHz carrier is frequency modulated by a sinusoidal signal of amplitude 20 V and frequency 50 KHz.
The frequency sensitivity of modulator is 5 KHz/Volt.
I) Determine Frequency Deviation, Modulation Index & B.W.
14. In an FM system, the audio frequency is 2KHz, and the audio voltage is 5V, with a deviation of 5 kHz. If the
audio signal voltage increases to 12.2V, determine the new deviation. Additionally, if the audio signal voltage
further increases to 18V while the audio frequency drops to 100 Hz, find the deviation and the modulation
index.
15. Explain the role of amplitude limiter in FM. Why limiter is not required in AM?
16. What do you understand by frequency modulation? Obtain mathematical expression for representing frequency
modulated wave and draw its spectrum as well as waveform.

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