Lecture 4 - Frequency Modulation

Principles of ElectronicCommunication

SystemsThird Edition

Louis Frenzel

Chapter 5

Fundamentals of Frequency Modulation

Topics Covered in Chapter 5

5-1: Basic Principles of Frequency Modulation

5-2: Principles of Phase Modulation5-3: Modulation Index and Sidebands5-4: Noise-Suppression Effects of FM5-5: Frequency Modulation Versus

Amplitude Modulation


A sine wave carrier can be modified for the purpose of transmitting information from one place to another by varying its frequency. This is known as frequency modulation (FM).

In FM, the carrier amplitude remains constant and the carrier frequency is changed by the modulating signal.


As the amplitude of the information signal varies, the carrier frequency shifts proportionately.

As the modulating signal amplitude increases, the carrier frequency increases.

With no modulation the carrier is at its normal center or resting frequency.


Frequency deviation (fd) is the amount of change in carrier frequency produced by the modulating signal.

The frequency deviation rate is how many times per second the carrier frequency deviates above or below its center frequency.

The frequency of the modulating signal determines the frequency deviation rate.

A type of modulation called frequency-shift keying (FSK) is used in transmission of binary data in digital cell phones and low-speed computer modems.


Figure 5-1: FM and PM signals. The carrier is drawn as a triangular wave for simplicity, but in practice it is a sine wave. (a) Carrier. (b) Modulating signal. (c) FM signal. (d) PM signal.

FREQUENCY MODULATION type of angle

modulation wherein the frequency of the carrier is varied relative to the amplitude of modulating frequency (signal)

whereVc = carrier signal peak voltage

ωc = 2Πfc = carrier signal angular frequency ωm = 2Πfm = modulating signal angular frequency t = instantaneous time mf = modulation index

General Equation of the FM Wave

νFM (t) = Vc sin (ωc t + mf

sin ωm t )

5-2: Principles of Phase Modulation

When the amount of phase shift of a constant-frequency carrier is varied in accordance with a modulating signal, the resulting output is a phase-modulation (PM) signal.

Phase modulators produce a phase shift which is a time separation between two sine waves of the same frequency.

The greater the amplitude of the modulating signal, the greater the phase shift.


The maximum frequency deviation produced by a phase modulator occurs during the time that the modulating signal is changing at its most rapid rate.


Figure 5-3: A frequency shift occurs in PM only when the modulating signal amplitude varies. (a) Modulating signal. (b) FM signal. (c) PM signal.


Relationship between the Modulating Signal and Carrier Deviation

In FM and in PM, the frequency deviation is directly proportional to the amplitude of the modulating signal.

In PM, the maximum amount of leading or lagging phase shift occurs at the peak amplitudes of the modulating signal.

In PM the carrier deviation is proportional to both the modulating frequency and the amplitude.


Figure 5-4: Frequency deviation as a function of (a) modulating signal amplitude and(b) modulating signal frequency.


Converting PM into FMIn order to make PM compatible with FM, the

deviation produced by frequency variations in the modulating signal must be compensated for.

This compensation can be accomplished by passing the intelligence signal through a low-pass RC network.

This RC low-pass filter is called a frequency-correcting network, predistorter, or 1/f filter and causes the higher modulating frequencies to be attenuated.

The FM produced by a phase modulator is called indirect FM.


Phase-Shift KeyingThe process of phase modulating a carrier

with binary data is called phase-shift keying (PSK) or binary phase-shift keying (BPSK).

The PSK signal has a constant frequency, but the phase of the signal from some reference changes as the binary modulating signal occurs.


Figure 5-6: Phase modulation of a carrier by binary data produces PSK.

5-3: Modulation Index and Sidebands

Any modulation process produces sidebands.

When a constant-frequency sine wave modulates a carrier, two side frequencies are produced.

Side frequencies are the sum and difference of the carrier and modulating frequency.

The bandwidth of an FM signal is usually much wider than that of an AM signal with the same modulating signal.


Modulation IndexThe ratio of the frequency deviation to the

modulating frequency is known as the modulation index (mf).

In most communication systems using FM, maximum limits are put on both the frequency deviation and the modulating frequency.

In standard FM broadcasting, the maximum permitted frequency deviation is 75 kHz and the maximum permitted modulating frequency is 15 kHz.

The modulation index for standard FM broadcasting is therefore 5.


Bessel Functions The equation that expresses the phase

angle in terms of the sine wave modulating signal is solved with a complex mathematical process known as Bessel functions.

Bessel coefficients are widely available and it is not necessary to memorize or calculate them.


Figure 5-8: Carrier and sideband amplitudes for different modulation indexes of FM signals based on the Bessel functions.


Figure 5-9: Plot of the Bessel function data from Fig. 5-8.


Bessel Functions The symbol ! means factorial. This tells you to

multiply all integers from 1 through the number to which the symbol is attached. (e.g. 5! Means 1 × 2 × 3 × 4 × 5 = 120)

Narrowband FM (NBFM) is any FM system in which the modulation index is less than π/2 = 1.57, or

mf < π /2.NBFM is widely used in communication. It

conserves spectrum space at the expense of the signal-to-noise ratio.


FM Signal BandwidthThe higher the modulation index in FM, the

greater the number of significant sidebands and the wider the bandwidth of the signal.

When spectrum conservation is necessary, the bandwidth of an FM signal can be restricted by putting an upper limit on the modulation index.


FM Signal Bandwidth Example:If the highest modulating frequency is 3 kHz and

the maximum deviation is 6 kHz, what is the modulation index?

mf = 6 kHz/3 kHz = 2What is the bandwidth?

BW = 2fmNWhere N is the number of significant* sidebands

BW = 2(3 kHz)(4) = 24 kHz*Significant sidebands are those that have an amplitude of greater than 1%

(.01) in the Bessel table.

5-4: Noise-Suppression Effects of FM

Noise is interference generated by lightning, motors, automotive ignition systems, and power line switching that produces transient signals.

Noise is typically narrow spikes of voltage with high frequencies.

Noise (voltage spikes) add to a signal and interfere with it.

Some noise completely obliterates signal information.


FM signals have a constant modulated carrier amplitude.

FM receivers contain limiter circuits that deliberately restrict the amplitude of the received signal.

Any amplitude variations occurring on the FM signal are effectively clipped by limiter circuits.

This amplitude clipping does not affect the information content of the FM signal, since it is contained solely within the frequency variations of the carrier.


Figure 5-11: An FM signal with noise.


Preemphasis Noise can interfere with an FM signal and

particularly with the high-frequency components of the modulating signal.

Noise is primarily sharp spikes of energy and contains a lot of harmonics and other high-frequency components.

To overcome high-frequency noise, a technique known as preemphasis is used.

A simple high-pass filter can serve as a transmitter’s pre-emphasis circuit.

Pre-emphasis provides more amplification of only high-frequency components.


Figure 5-13 Preemphasis and deemphasis. (a) Preemphasis circuit.


PreemphasisA simple low-pass filter can operate as a

deemphasis circuit in a receiver.A deemphasis circuit returns the frequency

response to its normal flat level.The combined effect of preemphasis and

deemphasis is to increase the signal-to-noise ratio for the high-frequency components during transmission so that they will be stronger and not masked by noise.


Figure 5-13 Preemphasis and deemphasis. (c) Deemphasis circuit.

5-5: Frequency Modulation Versus Amplitude Modulation

Advantages of FMFM typically offers some significant

benefits over AM.FM has superior immunity to noise, made

possible by clipper limiter circuits in the receiver.

In FM, interfering signals on the same frequency are rejected. This is known as the capture effect.

FM signals have a constant amplitude and there is no need to use linear amplifiers to increase power levels. This increases transmitter efficiency.


Disadvantages of FMFM uses considerably more frequency spectrum

space.FM has used more complex circuitry for

modulation and demodulation.In the past, the circuits used for frequency

modulation and demodulation involved were complex. With the proliferation of ICs, complex circuitry used in FM has all but disappeared. ICs are inexpensive and easy to use. FM and PM have become the most widely used modulation method in electronic communication today.


Major applications of AM and FM

Frequency Domain of Standard FM

fc - nfm

νFM(t)

fc

fc + nfm

Time Domain of Standard FM

Frequency Deviation (δ) the amount of change in carrier frequency produced

by modulating signal the maximum frequency the amount of change in

carrier frequency produced by modulating signal deviation occurs at the maximum amplitude of the

modulating signal

δ = fmax - fc

δ = fc - fmin where fc = carrier frequency

fmax = maximum frequency change of the carrier fmin = minimum frequency change of the carrier

δ

δ

δ = fmax + fc

δ = fc - fmin

CS

CS (Carrier Swing)

NOTEThe intelligence amplitude determines the amount of carrier frequency deviation. The intelligence frequency determines the rate of carrier frequency deviation.

δ α Em

δ = k Em

k =

ExampleFind the carrier and modulating frequencies , the modulation index

and the maximum deviation of the FM wave represented by the voltage equation νFM(t) = 12 sin (6 x 108t + 5 sin 1250t). What power will this FM wave dissipate in a 10Ω resistor?

Given νFM(t) = 12 sin (6 x 108 t+ 5 sin 1250t)

Solution ωc = 6 x 108 ωm = 1250

fc = 95. 49 MHz fc = 198.94 Hz

mf = 5

Solution δ = mf fm

= (5 x 198.94)

δ = 994.72 Hz

2

P = 7.2 W

Carrier Swing

the total variation in frequency of the carrier

Modulation Index

NOTE The modulation index determines the number of significant

sidebands in an FM signal. The modulation index for FM is directly proportional to the modulating voltage and at the same time inversely proportional to the modulating signal frequency.

ExampleA modulation frequency range from 30 to 15000 Hz is permitted in an FM system, together with a maximum deviation of 50 kHz.

What will be the minimum and maximum possible values of modulation index in the system.

Givenfm = 30 – 15000 Hz δmax = 50 kHz

Solution

mf max = 1666.67 mf min = 3.33

Deviation Ratio

The ratio of the maximum permissible frequency deviation to the maximum permissible modulating frequency.

Percentage of Modulation

Wideband FM vs. Narrowband FM

Descriptions Wideband FM Narrowband FMApplications FM broadcast and

entertainmentMobile

communications

Modulation index (mf)

5 – 2500 ≈ 1

Maximum deviation (δmax)

75 kHz 5 kHz

Modulating frequency (fm)

30 Hz – 15 kHz 3 kHz (max)

ExampleFor an FM signal in the 88 – 108 MHz broadcast band with a

frequency deviation of 15 kHz, determine the percent modulation.

Givenfm = 88 – 108 Mhzδactual = 15 kHz

Solution

mf = 20%

Frequency Spectrum of the FM Wave

νFM = J0 (mf) sin ωc t + J1 (mf ) [ sin (ωc + ωm)t – sin (ωc - ωm)t ]

amplitude of the carrier

signal

sidebands with

frequencies

NOTEFor a single fm , you can produce an infinite number of

sidebands, in pairs, but most of them would be of negligible amounts.

Transmitted Power

In FM, PT is constant.

NOTEIn FM, the total transmitted

power always remains constant, but with increased depth of modulation the required bandwidth is increased.

Bandwidth Formulas for FM

a. Theoretical BW for FM

b. Approximate BW

c. NBFM

d. WBFM

BW = 2 x fm x no. of highest needed sideband

BW = 2(fm + δ) John Carson’s Formula

BW ≈ 2fm

BW ≈ 2δ

Advantages of FM over AM

1.The amplitude of the FM carrier remains constant.

2. Increase in the S/N ratio (less noise)3. It is possible to reduce noise still further by

increasing deviation and fitting with amplitude limiters.

4. Less adjacent channel interference because there are guard bands between FM stations provided by the FCC.

5. It is possible to operate several independent transmitters on the same frequency with considerably less interference.

Disadvantages of FM

1. A much wider channel is required by FM, 7 to times as large as that needed by AM.

2. FM transmitting and receiving equipment tends to be more complex, particularly for modulation and demodulation, therefore more expensive.

3. Since reception is limited to line of sight, the area of reception for FM is much smaller than AM.

Capture Effect

The inherent ability of FM to minimize the effect of undesired signals (noise), also applies to the reception of a strongest signal or stations and minimizing other signals operating on the same frequency.

Block Diagram of an FM Transmitter

RF oscillator

Exciter (the means of

providing an FM RF signal )

Power amplifier

Audioamplifier

Pre-emphasis network

Pre -emphasis

The boosting of the higher modulating frequencies at

the transmitter, in accordance with the

pre –arranged curve to improve noise immunity at FM and prevent the higher frequency component of

the transmitted intelligence being

degraded.

De -emphasis

Reducing the amplitude of the higher

modulating frequencies at the receiver to the

same amount as it was before the

pre – emphasis circuit.

NOTEThe pre –emphasis and de – emphasis networks have a time constant of 75 μs (150 μs for Europe , 25 μs for Dolby) and a cut – off frequency of 2122 Hz.

NOTEIf two modulating signals have the same initial amplitude and

one of them is pre – emphasized to twice this amplitude, whereas the other is unaffected, being at a much lower frequency , then the receiver will naturally have to de –emphasize the first signal by a factor of 2 to ensure that both signals have the same amplitude in the output of the receiver.

Forms of Interference in FM

1. Image Frequency Effect of two stations being received

simultaneously

2. Co – channel Interference true to mobile receivers; when

travelling from one transmitter toward another. This minimizes by capture effect.

3. Adjacent Channel Interference created between tuning two adjacent

stations due the imperfection of filter circuits like channel 2 and 3.

Stereophonic FM Multiplex System (1961)

modulation system in which two – channel system with left channel and a right channel transmitted simultaneously and independently

NOTEStereophonic uses 38 kHz subcarrier from a pilot carrier of 19

kHz to produce sum and difference of the two channels. Also, a 67 kHz subsidiary communications authorization (SCA) is added for optional transmission in FM broadcasting.

sum channel (L + R)

sub – carrier

difference channel (L - R)

Optional SCA transmission

DSBSC AM FMaudio0 15 19 23 38 53 59.5 67 74.5

Generation of FM Signals

A. Direct Method

Varying the frequency of the carrier oscillator directly

1. Reactance Modulator Reactance of the capacitive or inductive components of the tank circuit vary in direct relationship to the audio signal which causes the oscillator frequency to vary thereby directly producing AM.

The equivalent capacitance depends on the device transconductance. The capacitance can be originally adjusted to any value, within reason, by varying the components R and C

Ceq = gm RC

The expression gmRC has the correct dimensions of capacitance; R, measured in ohms, and gm measured in siemens (S).

Since

Therefore

ExampleDetermine the value of the capacitive reactance obtainable from a reactance FET whose gm is 12 millisiemens (12 mS). Assume that

the gate – to – source resistance is 1/9 of the reactance of the gate – to – drain capacitor and that frequency is 5 MHz.

Givengm = 12 mS R = 1/9 Xc Xc = 9R

Solution

n

Xc eq = 750 Ω

2. Varactor Diode Modulator

Uses a varactor (voltage variable capacitor) diode that when reversed biased exhibits a junction capacitance that varies inversely with the amount of reverse voltage.

B. Indirect Method

Armstrong Method The modulating signal is modulated using

balanced modulator then fed to a summing device together with a crystal that is shifted 90°. Wideband FM then obtain by successive frequency multiplication of the output.

Crystal oscillator

Buffer

90° Phase shifter

Combining network

Balanced Modulator

Audio equalizer

1st group of multipliers

Mixer

2nd group of multipliers

Class C power

amplifiers

Crystal oscillator

equalized audio

FM wave(very low fc and mf )

carrier only

carrier at 90°

medium fc

and low mf low fc

and mf high fc

and mf

high fc

and mf

sidebands only

modulation wherein the phase of the carrier is made proportional to the instantaneous value of the modulating signal

Phase Modulation

General Equation of the PM Wave

ѵPM(t) = Vc sin(ωc t + mp sin ωm t)

where

mp = modulation indexkp = proportionality constant

mp = kpVm

Under identical conditions: mp = mf but when fm is changed mp k

mf changed

ExampleThe equation of an angle modulated wave is ν(t) = 15 sin (3x108t + 20 sin 2000t). Calculate the maximum

deviation. Rewrite this equation if the modulating frequency is halved, but all else remains constant, assuming that the wave is:

a. frequency modulatedb. phase modulated

GivenνPM (t) = 15 sin (3x108t + 20 sin 2000t)

Solution

fm = 318.31 Hz δ = 6.37 kHz

Solutiona. frequency modulated

b. phase modulated

νFM (t) = 15 sin (3x108t + 40 sin 1000t)

νPM (t) = 15 sin (3x108t + 20 sin 1000t)

Frequency Modulation vs. Phase Modulation

Comparisons Phase Modulation Frequency Modulation

deviation (δ) δp is proportional to Vm, independent on fm

δf is proportional to Vm

modulation index mp is proportional to Vm

mf is proportional to Vm

and inversely proportional to fm

when fm is changed mp will remain constant

mf will increase as fm is reduced, vice versa

Demodulation

the process by which the modulating signal is recovered from the modulated carrier

found in receivers

Functions of a Receiver

1. selects the desired signal 2. amplifies3. demodulates / detects 4. displays

Standard AM Receivers

1. Tuned Radio Frequency 2. Superheterodyne Receiver

1st RF amplifier

2nd RF amplifier

AMdetector

a simple “logical” receiver simplicity and high sensitivity aligned at broadcast frequencies 535 – 1640 Hz

Tuned Radio Frequency (TRF) Receiver

AFamplifier

Poweramplifier

ganged

It amplifies weak signal from the antenna. It has a variable resistor that controls the RF gain and sensitivity.

It provides rectification and detection for modulated signals.

Volume – controlled amplifier that raises the power level of the audio (AF) signal to a value sufficient to drive the loudspeaker of the receiver.

RF Amplifier

Detector

AF Amplifier

Problems in TRF Receivers

NOTE RF stage provides greater gain, prevention of re –radiation of

the local oscillator, improved rejection of adjacent unwanted signals and better coupling of antenna with the receiver.

1. instability 2. insufficient adjacent frequency rejection 3. bandwidth radiations

RF amplifier

1st IF amplifier

2nd IF amplifier

AMdetector

AFamplifier

Crystal oscillator

Mixerfo

fs

fIF

fsi

RF AmplifierAmplifies the weak RF signal received from the antenna. Selectivity of the receiver depends on this section.

NOTE The RF section provides discrimination or selectivity against image and intermediate frequency signals, provide an efficient coupling between the antenna and the first stage of the RF amplifier.

AM Superheterodyne Receiver

ganged

Local OscillatorA crystal oscillator whose frequency “beats” with the incoming signal to produce the correct intermediate frequency.

Mixer (First Detector)Combines the incoming RF signal and the signal from the oscillator, and produces two original frequencies, their sum and differences, and harmonics. It produces desired intermediate frequency (IF).

1st IF AmplifierTuned to 455 kHz (IF of AM), amplifies it and rejects the remaining output frequencies

AM Detector (2nd Detector)It demodulates the IF signal and recovers or extracts the original audio signal.

AF AmplifierIt raises the power level of the audio signal; to a value sufficient to drive the loudspeaker of the receiver.

2nd IF AmplifierFurther amplification and selectivity of the IF signal. Most of the gain (sensitivity) of the receiver are achieved on the IF amplifiers.

FM Superheterodyne Receiver

RF amplifier and

preselector

Local oscillator

IF amplifier (10.7 MHz)

De-emphasis network

Audio amplifierMixer Limiter Detector

Demodulation

The process of shifting the spectrum back to the original baseband frequency range and reconstructing the original form. Also known as detection.

Sensitivity

Ability to amplify weak signals

Selectivity

Ability to reject unwanted signals(adjacent)

Image Rejection Ratio (α) Provided by tuned circuits to block fsi The ratio of the gain at desired frequency (fs) to the gain of image frequency signal (fsi)

whereQ = quality factor provided by

tuned circuits to block fsi fs = signal frequency fo = oscillator frequency fIF = intermediate frequencyfsi = image frequency signal

NOTE

fsi = fs + 2fIF fsi = fo + fIF

The higher the fIF, the better the image rejection

SAMPLE PROBLEMSFrequency Modulation (Angle Modulation)

Example 1

An FM modulator has a frequency deviation sensitivity of 5 kHz/V and modulating signal vm (t) = 2cos(4000pi)t. Calculate the peak frequency deviation and modulation index

10 kHz, 5

Example 2

A PM modulator has a phase deviation sensitivity of 2.5 rad/V and modulating signal vm (t) = 4cos(4000pi)t. Calculate the peak phase deviation and modulation index.

5 rad, 5

Example 3

What is the frequency swing of an FM broadcast transmitter when modulated 80%?

60 kHz

Example 4

Calculate the frequency deviation and % modulation under FCC standards for a given modulating signal that produces 100 kHz carrier swing.

+/- 50 kHz, 67%

Example 5

A system uses a deviation of 100 kHz and a modulating frequency of 15 kHz. What is the approximate bandwidth?

230 kHz

Example 6

Calculate the deviation ratio and approximate bandwidth for the worst-case modulation index for an FM broadcast-band transmitter (FCC standard).

5, 180 kHz

Example 7

A phase-locked loop has a VCO with a free running frequency of 10 MHz. As the frequency of the reference input is gradually raised from zero, the loop locks at 8 MHz and comes out of lock again at 14 MHz. Determine the lock range.

8 MHz

Example 8

A phase-locked loop has a VCO with a free running frequency of 14 MHz. As the frequency of the reference input is gradually raised from zero, the loop locks at 12 MHz and comes out of lock again at 18 MHz. Calculate the capture range.

4 MHz

Example 9

Calculate the range of frequencies will the PLL be able to capture and subsequently maintain lock if it used a VCO with a free-running frequency of 100 kHz, PLL has a 10% capture range and 20% lock range.

90 – 100 kHz, 80 – 120 kHz

Example 10

The capture range of a PLL and filter is 12%, while the lock range is 18% and the VCO has a center frequency of 20 MHz. Calculate the PLL operation with an 18.5 MHz input?

18.5 – 21.5 MHz

References

Self-Sufficient Guide to ECE by Jason M. Ampoloquio

Communications Engineering by S.S. Cuervo and K.L.N. Suello

Principles of Electronic Communication Systems by Louis Frenzel

Lecture 4 - Frequency Modulation

Documents

free running

higher modulating

standard fm

frequency

modulating

carrier frequency

carrier frequency

modulating