Lecture notes of angle modulation, Lecture notes of Analog Communication

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Angle Modulation

Dr. Sanjay Kumar Singh

Associate Professor, Department of Communication Engineering,

School of Electronics Engineering

Frequency Modulation (FM)

Phase Modulation (PM)

FM is the type of modulation inwhich instantaneous frequency ofthe

carrier

signal

is

varied

in

accordance with the instantaneousamplitude of the message signal,keeping amplitude and phase ofthe carrier signal constant.

PM is the type of modulation inwhich instantaneous phase of thecarrier

signal

is

varied

in

accordance with the instantaneousamplitude of the message signal,keeping amplitude and frequencyof the carrier signal constant.

Both FM and PM are combinedly referred to as

Angle Modulation

and can be

obtained from each other.

Angle Modulation

Mathematical expression for Angle Modulation: Mathematically the angle modulated wave can be defined by

௖

௜

FM:

Angle Modulation

௜

or,

௜

PM:

௜

or,

௜

௜

௜

௜

௜

௜

௜

Angle Modulation

For single tone FM,

௜

௖

௙

௖

௙

௠

𝑐𝑜𝑠2π𝑓

௠

௜௠௔௫

௖

௙

௠

and,

௜௠௜௡

௖

௙

௠

𝑡

𝑓

௖

𝑓

௖

൅ ∆𝑓

𝑓

௖

െ ∆𝑓

𝑓

௜

2∆𝑓

where,

௙

௠

Modulation Index in FM/ PM:

௙

௠

௙

௠

௠

where,

௙

Freq. Sensitivity of FM Modulator

ሺHz/voltሻ

FM:

௣

௣

௠

PM:

where,

௣

Phase Sensitivity of PM Modulator

ሺRadian/voltሻ

Mathematical expression for Angle Modulation:

Angle Modulation

Mathematical expression for Angle Modulation:^ FM signal can be classified as,

Narrowband FM (NBFM)

Wideband FM (WBFM)

FM

High)

Small)

௙

Small)

ൎ 0.2 to 0.6 ൑ 1

௙

high)

Angle Modulation

Mathematical expression for NBFM:

௖

௖

௙

௖

௖

௙

௠

௠

௠

௖

௖

௙

௠

௖

௙

௠

௙

௙

is small

௖

௖

௖

௖

௙

௠

௖

௖

େୟ୰୰୧ୣ୰

௙

௖

௖

௠

୙ୗ୆

௙

௖

௖

௠

୐ୗ୆

Angle Modulation

Generation of NBFM:



𝑚 𝑡

𝐍𝐁𝐏𝐌

𝐴

௖

𝑐𝑜𝑠2𝜋𝑓

௖

𝑡

න 𝑚 𝑡

𝑑𝑡

NBFM

𝜃 𝑡

∝ න 𝑚 𝑡

𝑑𝑡

Generation of NBPM:

𝑚 𝑡

𝐍𝐁𝐅𝐌

~

𝐴

௖

𝑐𝑜𝑠2𝜋𝑓

௖

𝑡

𝑑 𝑑𝑡

𝑚 𝑡

NBPM

𝜃 𝑡

∝ 𝑚 𝑡

𝒅 𝒅𝒕

~

Alternatively,

Angle Modulation

Phasor diagram of NBFM:

𝑚

௙

𝐴

௖

2

𝑚

௙

𝐴

௖

2

2𝜋 𝑓

௖

൅ 𝑓

௠

𝑡

2𝜋 𝑓

௖

െ 𝑓

௠

𝑡

∅

𝐴

௖

O

𝐴

𝐵

𝐷

𝐸

െ

𝑚

௙

𝐴

௖

2

OA

Carrier

AB

USB

AD

LSB

AE = USB & LSB ResultantOE

NBFM

   

Comparison with AM:

2𝜋 𝑓

௖

െ 𝑓

௠

𝑡

𝐴

௖

O

𝐴

𝐵 𝐷

2𝜋 𝑓

௖

൅ 𝑓

௠

𝑡

𝐸

𝑚

௔

𝐴

௖

2

𝑚

௔

𝐴

௖

2

OA

Carrier

AB

USB

AD

LSB

AE = USB & LSB ResultantOE

AM

   

Angle Modulation

Generation of NBFM from AM:^ NBFM signal can be generated by rotating the carrier component of AM by 90

Carrier

O

𝐴

𝐸

USB LSB

O

Carrier 𝐴

𝐸

USB LSB

∅

⟹

AM

Angle Modulation

Generation of NBFM from AM:

𝟗𝟎°

Phase Shifter

~

𝑚 𝑡

𝐴

௖

𝑐𝑜𝑠2𝜋𝑓

௖

𝑡

𝐴

௖

𝑠𝑖𝑛2𝜋𝑓

௖

𝑡

൅

൅

NBFM

AM െ DSB/SC

OE ൌ

OA

ଶ

൅ AE

ଶ

ൌ

𝐴

௖

ଶ

൅ 𝑚

௔

𝐴

௖

ଶ

ൌ 𝐴

௖

1 ൅ 𝑚

௔

ଶ

∅ ൌ tanି

ଵ

𝑚

௔

𝐴

௖

𝐴

௖

⟹ ∅ ൌ tanି

ଵ

𝑚

௔

and,

Angle Modulation

௖

௖

௙

௠

For single tone modulation,

௠

௠

௙

ᇱ

௖

௖

௙

௠

௖

ᇱ

or,

୛୆୊୑

௖

௖

ᇱ

௙

ᇱ

௠

௖

௖

௙

NBFM signal is represented by,

Generation of WBFM:Indirect Method or Armstrong Method:

Angle Modulation

Generation of WBFM:Direct Method:^ Uses Hartley oscillator with timevarying circuit in feedback.Frequency of oscillation of Hartleyoscillator can be given as

௢

ଵ

ଶ

௢

ଵ

ଶ

௢

௢

ଵ ଶ

⁄

௢

ଵ

ଶ

Angle Modulation

Spectrum of WBFM:

௖

௖

௙

௠

௖

௝ଶగ௙

೎

௧

௝௠

೑

௦௜௡ଶగ௙

೘

௧

௖

௝ଶగ௙

೎

௧

ฬ

∗ ∵ 𝑠𝑖𝑛2𝜋𝑓

௠

𝑡

is a periodic signal

∴ 𝑠̂ 𝑡

is also periodic

.

Now, expressing

by the exponential Fourier series,

௡

௡ୀஶ ௡ୀିஶ

௝ଶగ௡௙

೘

௧

௡

⁄ି்ଶ

ଶ ⁄

௘

ೕమഏ೙೑೘೟

௡

௠

௝ ௠

೑

௦௜௡ଶగ௙

೘

௧ିଶగ௡௙

೘

௧

ଵ ଶ௙

೘

⁄ି

ଵ ଶ௙

೘

⁄

Angle Modulation

Spectrum of WBFM:

As

ଵ

ଶ௙

೘

Put,

௠

௠

௠

As

ଵ

ଶ௙

೘

௡

௠

௠

௝ ௠

೑

௦௜௡௫ି௡௫

గି

గ

௡

௝ ௠

೑

௦௜௡௫ି௡௫

గି

గ

௡

௡

௙

®

Bessel Function

Integer

Real quantity

Standard Bessel function oforder 𝑛 and argument 𝑚

௙

directly read from

tables.

For exp:

1 2𝜋

න

𝑒

௝ ଷ௦௜௡௫ିଶ௫

గି గ

𝑑𝑥

ൌ 𝐽

ଶ

3