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Communication Systems, 5e
Chapter 5: Angle CW Modulatation
A. Bruce CarlsonPaul B. Crilly
2010 The McGraw-Hill Companies
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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
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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
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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
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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
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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
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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
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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
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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
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10
Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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
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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 .
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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
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13
Carsons Rule Estimate
Carsons Rule (for >>1 and
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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
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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!?
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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
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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
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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
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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
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ABCs Linear Distortion (3)
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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
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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.
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Nonlinear processing circuits
22Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) Amplitude limiter or (b) frequency multiplier
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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
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Amplitude Limiter and Noise Reduction
2010 The McGraw-Hill Companies
FM signal processing using a limiter: (a) Noiseless FM signal, (b) noisy FM signal, (c) limiter output with noisy input, and (d) BPF output
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25
AD8309: Log Amp/Limiter
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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
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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
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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
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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
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Maxim 2605-2609 Integrated IF VCO
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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
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Direct FM and VCO (radio design)
2010 The McGraw-Hill Companies
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
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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
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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
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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
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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)
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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
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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')';
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Phase Shift Discriminator
tx ty
1FM0FM0 tttf2sinttf2costmix
ttt2tf22sintttsintmix FM1FM0FM1FM
ttttttsinty FM1FMFM1FM
t
3f
tt
3f dm2dm2ty1
tm2ty 3f
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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
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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
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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
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Demodulation Results
Using a constant magnitude input
43
22 nQnI
nInQnQnInInQarctan
dxd
nInQnQnIKA
nInQnQnInInQ
dxd
Ac
c
2arctan
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Zero Crossing
Instantaneous measurement of frequency at each zero crossing Interpolation between zero crossings may be performed Discrete steps require low pass filtering