Communication Systems, 5e

Chapter 5: Angle CW Modulatation

A. Bruce CarlsonPaul B. Crilly

2010 The McGraw-Hill Companies

Chapter 5: Angle CW Modulatation

Phase and frequency modulation Transmission bandwidth and distortion Generation and detection of FM and PM Interference

2010 The McGraw-Hill Companies

3

General Description

Angle modulation phase or frequency based

The instantaneous angle is the argument of the cosine (for complex, the exponential)

ttf2cosAts 0

ttf2t 0

tjjtf2jexpReAts 0

dt

tft

ttftfdt

t

2

1221

21

00

Narrowband Signal Demo

AM, PM, and FM Spectrum with random noise4

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-140

-120

-100

-80

-60

-40

-20

0

Frequency (Hz)

Pow

er (d

B)

AM, FM, PM Example Power Spectrum

FMPMAM

Wideband Signal Demo

AM, PM, and FM Spectrum with random noise5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-120

-100

-80

-60

-40

-20

0

Frequency (Hz)

Pow

er (d

B)

AM, FM, PM Example Power Spectrum

FMPMAM

6

PM and FM Spectrum Basis

Based on the previous analysis, we need to determine the transform of the phase components

tmt 2pPM

fMf 2pPM

t

3fFM dmt

f

fMjf 3fFM

p is the phase deviation f is the frequency deviation

7

Phase Modulation (PM)

The magnitude is constant while the phase changes in time carries the message information

The instantaneous frequency of a PM system is

tmtf2cosAts 2p0

t

tmtf2dt

ttf2 2p0

180p 1tm

t

tm2

ftf 2p0

8

Frequency Modulation (FM)

The magnitude is constant while the relative frequency changes in time carries the message information

The instantaneous frequency of a FM system is

tmftf 3f0

t

3f0 dm2tf2cosAts 1tm

t

dm2tf2

21tf

t

3f0

9

FM or PM Single Tone Modulation

The Quadrature Signal Representation

tf2sintsintf2costcosAttf2cosAts

cc

c

tf2sintf2sinsintf2costf2sincos

Atscm

cm

For Single Tone Modulation tf2sinAtm mm2 tf2cosAtm mm3

tf2sinAt mpmPM tf2sinfAt mmf

mFM

pmPM A m

fmFM f

A

10

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Tone-modulated line spectra

14.3PM

pmPM A

m

fmFM f

A

(a) FM with m fixed or PM (b) FM with Am fixed

and fm decreasing

11

Rules of Thumb

The relative amplitudes of the lines will vary with the modulation index For PM, expect that < For NB FM, expect < 1 and only J0 and J1 are

significant For WB FM, expect > 1 and numerous spectral lines

Table 5.1-2 shows selected values Keeping scaled terms of 0.01 and higher

Figure 5.2-1 shows the number of sideband pairs as a function of .

Transmission Bandwidth

From the Bessels function discussion, the optimal bandwidth for FM exponential modulation is infinite!

A practical bandwidth can be defined based on the magnitude of the spectrum that we wish to keep. 90% or 99% power for example

12

13

Carsons Rule Estimate

Carsons Rule (for >>1 and

14

Bandpass Filtering FM Systems

What happens when an FM waveform is filtered? txthty cc

fXfHfY cc

0oddn mcmc

mcmcn

0evenn mcmc

mcmcn

cc0c

fnfffnfffnfffnff

J22A

fnfffnfffnfffnff

J2A

ffffJ2AfX

For a tone modulated spectrum

15

Bandpass Filtering FM Systems (2)

And the output becomes fXfHfY cc

0oddn mcmcmcmc

mcmcmcmcn

0evenn mcmcmcmc

mcmcmcmcn

cccc0c

fnfffnfHfnfffnfHfnfffnfHfnfffnfH

J22A

fnfffnfHfnfffnfHfnfffnfHfnfffnfH

J2A

fffHfffHJ2AfX

For no distortion, the magnitude of the filter must be equal for all frequencies and the phase should be linear a perfect filter!?

16

ABCs Linear Filter Conditions

Taking the baseband signaling tjexp

2AtxLP

tf2jexptyRety cLPc

fXffuffHfXfHfY LPccLPLPLP

Bandpass to Lowpass Filter Consideration(see Example 4.1-1) and Figure 4.1-6

ulBP ffffor,fjexpKfH

cuclccLP ffffffor,ffuffjexpKfH

17

ABCs Linear Conditions (2)

For no distortion: the gain must remain constant or flat, K, and the phase should be strictly linear (time delay only).

But

Filters usually have amplitude variation or ripple Filters usually have phase ripple

A notable exception is a symmetric digital FIR filter

ABCs Linear Distortion (1)

Assume linear amplitude and phase in the filter

18

ftftjffKK

ffuffjKfH

cc

cc

1010 2exp

exp

fXftftjffKKfY LPcc

LP

1010 2exp

dt

ttdxft2jexpf2j

Kttxft2jexpKty

1LPc0

c

1

1LPc00LP

ftjfXfjftjfj

K

ftjfXftjKfY

LPcc

LPcLP

101

100

2exp22exp2

12exp2exp

ABCs Linear Distortion (2) Taking the derivative of the lowpass input

Substitute

Combine

19

1101100

22exp2

2exp

ttxttmjftjfj

KttxftjKty

LPfcc

LPcLP

1LP1f

11

1LP

ttxttm2jdt

ttdttjexp2Aj

dtttdx

10110 2exp22

ttxftjfj

ttmKjKty LPc

c

fLP

tjexp2AtxLP

ABCs Linear Distortion (3)

20

Recognize an AM modulated baseband waveform! FM to AM conversion due to linear magnitude

filtering has occurred!

10110 2exp22

ttxftjfj

ttmKjKty LPc

c

fLP

10110 2exp ttxftjttmfK

Kty LPcc

fLP

Restating and simplifying tjexp2AtxLP

21

Limiters

Can we stop linear distortion from happening?

Use an ideal hard limiter on the signal and all filter magnitude variation is removed! A square signal with zero crossings results The distance between zero crossing provides frequency

information one interval=1/2 wavelength

Then, carefully bandpass filter the output, anda clean version of the FM signal is reacquired.

Nonlinear processing circuits

22Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

(a) Amplitude limiter or (b) frequency multiplier

23

Limited Fourier Components

Limited and Filtered

ttfVttfV

ttfVttfVtv

cc

ccLimit

772cos74552cos

54

332cos342cos4

00

00

ttfKttfVtv ccBPFLimit

2cos2cos4 0&

ttffRtAtv cin 2cos,,,

oddnforn

jtfcseries ,22cos

Amplitude Limiter and Noise Reduction

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FM signal processing using a limiter: (a) Noiseless FM signal, (b) noisy FM signal, (c) limiter output with noisy input, and (d) BPF output

25

AD8309: Log Amp/Limiter

26

Generating and Detecting FM & PM

Modulation Voltage Controlled Oscillators Narrowband Narrowband to Wideband Conversion Phase to frequency converters

Demodulation Phase Lock Loops Frequency/Phase Sloped Filters Frequency Discriminators

27

Commercial FM

Maximum frequency deviation 75 kHz For 15 kHz, D=5.0 For 53 kHz, D=1.4

Bandwidths (Carson)

For 15 kHz, Bt = 180 kHz For 53 kHz, Bt = 256 kHz

freq 20 k15k19 k38

k23

L-R

k53

L-R L+RFM Audio Baseband

max12 fDBT

28

Commercial FM

Subsidiary Communications Authority (SCA) Subbands Typically located at 67 kHz or 92 kHz Additional transmissions by licensee Mostly digital communications http://en.wikipedia.org/wiki/Subsidiary_Communications_Authority

http://www.fcc.gov/mb/audio/subcarriers/

k67 k75 k92 k100cf

29

Oscillator/VCO Design

Tuned Circuit L-C or crystal (modeled as L-C)

Voltage control of tuned circuits Varactor Diode V-C characteristics Varying C varies f Figure 5.3-1-like

Maxim-IC.com Application Notes AN2032: Trimless IF VCO: Part 1: Design Considerations

http://www.maxim-ic.com/appnotes.cfm/appnote_number/2032 AN688: Trimless IF VCO: Part 2: New ICs Simplify

Implementation http://www.maxim-ic.com/appnotes.cfm/an_pk/688

30

Maxim 2605-2609 Integrated IF VCO

31

Generating FM with a VCO

A voltage controlled oscillator

From the definition of FM, let

ttvkf2jexpAtVCO inc

tv2kftmftf inc3f0

This is very common for FM generation

Direct FM and VCO (radio design)

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0

0 1

Oscillator output frequency =

Oscillator tank circuit with resonant frequency of , , and ( )

v

f f

f f f L C C t

VB defines f0, (C||Cv)L defines f

33

Narrowband PM Modulators

Narrowband Phase for small

ttfAttfAts

ttfAts

cc

c

sin2sincos2cos2cos

tftAtfAts cc 2sin2cos

tf c 2cos

tx cA ts

90 tf c 2sin

Note: Narrowband FM if x(t) is integrated

Indirect FM transmitter

34Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Narrowband to Wideband Conversion ncn ttfts 2cos1

tntfnAtsts cn 2cos12

35

Generating PM and FM in MATLABPhase to Frequency Generation

Generate the instantaneous phase

nmn 2pPM

n

k3fFM kmt2n

nmcumsumf

2n 3s

fFM

njfnfjAnx

sc 2exp

36

Demodulation Concepts

1. FM to AM Conversion2. Phase-shift Discrimination3. Quadrature Phase Estimate and Discrimination4. Zero-crossing Detection5. PLL Frequency Feedback (not in Chap 5)

37

FM to AM Conversion

What happens when we take the derivative of the FM modulated waveform?

ttf2cosAts FM0

ttf2ttf2sinA

tts FM

0FM0

t

3fFM dm2t

ttf2sintm2f2Atts

FM03f0

The signal envelope is (easier with complex)

tm2f2Atenv 3f0

38

Taking a Derivative

A Differentiator in the Laplace domain WffWffor,sKsH 00

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25Differentiating Filter Absolute Value

Am

plitu

de

Frequency-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-10

-8

-6

-4

-2

0

2

4

6

8

10Differentiating Filter Phase

Pha

se

Frequency

h=firpm(44,[0 .3 .4 1],[0 .2 0 0],'differentiator')';

39

Phase Shift Discriminator

tx ty

1FM0FM0 tttf2sinttf2costmix

ttt2tf22sintttsintmix FM1FM0FM1FM

ttttttsinty FM1FMFM1FM

t

3f

tt

3f dm2dm2ty1

tm2ty 3f

40

Phase Derivative

Quadrature (or Complex) demodulation Arctan to derive instantaneous phase

This output provides the phase

Differentiation to generate FM output LPF to support reduced bandwidth

0fLO

tx

90

tx c

41

Derivative of the Arctan

Math and discrete approximation

tItQty atan

ttI

tItQ

ttQ

tItItQt

tItQ

tItQt

ty222

1

1

1

1

1

ttItQ

ttQtI

tQtItty

221

22

11nQnI

nInInQnQnQnIny

Using 1st order difference for derivative

42

Derivative Detail

The derivative of an arc-tangent function

2uarctan

2for,

dxdu

u11uarctan

dxd

2

tI

tQtu

dx

tdItItQ

dxtdQ

tI1

tItQ1

1tItQarctan

dxd

22

dx

tdItQdx

tdQtIKtQtI

dxtdItQ

dxtdQtI

tItQ

dxd

A

1arctan 22

Using a 1st order difference approximation

22 nQnI

nInQnQnInInQarctan

dxd

Demodulation Results

Using a constant magnitude input

43

22 nQnI

nInQnQnInInQarctan

dxd

nInQnQnIKA

nInQnQnInInQ

dxd

Ac

c

2arctan

44

Zero Crossing

Instantaneous measurement of frequency at each zero crossing Interpolation between zero crossings may be performed Discrete steps require low pass filtering