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The Calculation of Frequency SourceRequirements for Digital
Communications Systems
Victor S. Reinhardt
08/25/04
IEEE International Ultrasonics, Ferroelectrics, andFrequency Control 50th Anniversary JointConference, Montreal, August 24-28, 2004
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The Calculation of Frequency SourceRequirements for Digital Comm Systems
Introduction
Frequency sources (oscillators, synthesizers, etc.) are animportant part of digital communications systems
Paper will discuss the derivation of frequency sourcerequirements from over-all digital comm system parameters
Will be tutorial treatment for those not familiar with digital commtheory but familiar with time & frequency theory
Frequency source properties directly impact the performance ofdigital comm systems
Impact link acquisition & loss of acquisitionT&F community familiarwith synchronization issuesWill not be covered here
Impact bit error rate (BER) performance--Paper will address this
Will utilize quadrature phase shift keyed (QPSK) systems forconcrete examples
But theory applicable to other systems
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Basic Digital Comm ConceptsSignals Carrying Digital Information
At the transmitter a carrieris modulated in a regular time sequenceofsymbols to produce a digital communications signal or waveform
A symbol is a temporal waveform in some modulation spacerepresenting a single digital word of information
At the receiver the signal is sampled at discrete decision epochs todetermine a modulation value of the carrier
The modulation value is converted into a digital or data word bycomparing it to decision thresholds
The symbols occur at a symbol rate Rs=1/Tc (Tc= clock period)
The bit or data rate R = WRs (W = bits persymbol orword)
Example: Unshaped (Rectangular)Symbols in PAM
Decision Epochs
Time
Value
Decisi on
Threshold
s
Symbol3
Symbol2
Symbol1
(1,0)
(1,1)
(0,1)
(0,0)(2-Bit)Digital
Words
Tc t3t2
t1
Carrie
rAxis
Signal
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Shaped Symbols
Unshaped (rectangular) symbols are notbandwidth efficient Sinc functions in freq domain
Shaped symbols are sinc-like functionsin time domain Produce more bandwidth efficient
trapeziodal functions in freq domain
Do not interfere with each other at decisionepochs
The price one pays for shaping is morestringent timing
-3 -2 -1 0 1 2 3
Un-shaped
Shaped
tn/Tc
Symbols in Time Domain
Shaped Transmission
0 1 2 3 4 5 6
-1
1
0
Composite Signal
tn/Tc
-3 -2 -1 0 1 2 3
Un-shaped
Shaped
f/Rs
Symbols in Freq Domain
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Inter-Symbol Interference (ISI) & EyePatterns
Eye Pattern = Graph of the modulationvalue vs time at the receiver plotted
modulo 1-symbol period (as in a scopetrace)
Eye opening = region with no valuetrajectories in it
Inter-symbol Interference (ISI) =Contamination at decision epoch of
modulation value by adjacent symbols Ideal Decision epochno ISI Clock errors cause the decision epoch to
wander off the best decision epochincreasing the ISI
Sensitivity of ISI to clock timing = Slope ofeye opening at decision epoch
Even unshaped (square) symbolsgenerate such eye patterns because ofreceiver and channel filtering necessaryto limit signal BW & noise
Shaped symbols have narrower eyewidths than unshaped ones
From: Telecom Glossary 2000, AmericanNational Standard forTelecommunications, T1.523-2001,www.atis.org/tg2k/images/epdplot1.gif
Modulo Symbol Time0- +
Eye Pattern
Inter-SymbolInterference
Ideal Decision Epoch
Eye Opening
(No Trajectories)
ShapingNarrows
Eye Width
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Types of Digital Modulation
Type of carrier: RF carrier or subcarrier,baseband voltage, etc.
Parameter modulated: amplitude, phase,frequency, etc.
Modulation Order (or number of digitalstates 2W): binary, quadrature, M-ary
Shaped or unshaped
Coherent, incoherent, differential phase
Synchronous & asynchronous dataclock timing (used in hardline systems)
Binary, M-aryFSK
Freq
(0) (1)
FrequencyShift Keyed
Time
PulsePosition
or WidthModulation
PWM
Time
PulseAmplitude
Shift Keyedor Modulation
Amplitude
PAM
Hybrid ModulationM-ary Quadrature AmplitudeShift Keyed or Modulation
Coherent Phase-FrequencyShift Keyed
Minimum Shift Keyed (BinaryCPFSK)
16-QAMor 16-QASK(4-Bit word)
. .
. .
. .
. .
. .
. .
. .
. .
I
Q
Phase Shift KeyedBPSK, QPSK, 8PSK, .., DPSK
(0,0)(0,1)
(1,0) (1,1)
Complex RF Envelope
I
Q
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Bit Error Rate (BER) vs Eb/NoKey Comm System Parameter
Rx thermal noise must limited by a filter
For an ideal system the Rx filters bandwidth isequal to the symbol rate Rs = R/W
The ideal SNR = Prx/(NoRs) = Pb/(NoR) = Eb/No
No = Thermal noise density
Pb = Prx/W = Power per bit
Eb = Pb/R = Prx/Rs = Energy per bit
BER vs Eb/No the canonical comm link
characterization
BER degradation is the extra Eb/No over idealsystem to achieve same BER as ideal
Error correction coding (ECC) allows up to Nbit errors to be corrected in a group or block of
bits--Improves BER above a certain Eb/No
The bit error rate (BER) is the probability that a received bit isincorrect
The BER is a function of the SNR at the digitalreceiver
Uncoded BER
10-3
10-4
10-5
10-6
10-7
- Ideal- Actual
Eb/No - dB
10-3
10-410-5
10-6
10-7
- Ideal- Actual
Error CorrectionCoded BER
BERDegrad-ation
BERDegrad-ation
Eb/No - dB
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BER Degradation and ISI
Causes of ISI Symbol distortion
RF carrier phase errors & jitter
Data clock errors & jitter
Simple BER degradation Models Worst case model:
BER deg = -20Log10 (1- V/V) Noise Model: Use theoretical
curve with Eb/NoPrx/(NoRs + V2)
Decisionthresholds
Thermal noise in BW Rs ( = NoRs)causes occasional bit errors BER (uncoded) = *Erfc(2-0.5 V/ )
= *Erfc((Eb/No))
ISI generates non-thermal jitter Vn When V + Vn is closer to decision
threshold higher BER with thermal noise
Net effect to increase BER for given Eb/No
Actual QPSK system
(no thermal noise)
Sampled valuesV(1 j)/20.5 atdecision epoch
No ISI (jitter)without thermalnoise
Jitter Vn
Ideal QPSK System
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LO Phase Jitter Requirements in RFCarrier Digital Comm Systems
At the transmitter (Tx) an LO and a clock are required
At the Receiver (Rx)
a clock recovery loop is always required to track the Rx clock to the Txclock
a carrier rec loop at the Rx LO required for phase coherent symbols
Recovery loops track out relative Rx-Tx LO and clock jitter forfourier frequencies < recovery loop bandwidths
This is very important in defining the appropriate jitter statistics interms of power spectral densities (PSD)
SymbolModulator
DataEncode
User
Data
ErrorCorrection
Encryption
Framing
~~
SymbolDemod-ulator
Data(Sampling)
Clock~ ~
LO
RF
Xmission
DataDecode
Recover Loops
User
Data
Data(Sampling)
ClockLO
Transmitter (Tx) Receiver (Rx)
Rx LOrecovery
loop only for
phasecoherentsymbols
Typical RF Carrier Comm System
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Carrier Phase Jitter and ISI
Phase jitter produces ISI in quadrature systems through I-Q cross-
talk Phase jitter much less of an issue in BPSK because there is no Q
channel (Just produces loss of power)
The definition of the appropriate of phase variance 2 isdetermined by the phase coherence properties of the system
PhaseJitter
Q-Symboljitter produces
cross-talkin I-Channel, etc.
Rx Q-Axis
Rx I-Axis
RMS ISI V*Sin( )
V
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RF Carrier Phase Jitter and Coherent,Incoherent, and Differential Systems
Coherent symbols Tx symbols decoded relative to
phase of Rx LO
Rx-Tx LO phase independent overmany symbols (recovery loop timeconstant Tp 1/ Bp >> Tc)
Must have Tp >> Tc so thermal noise
does not degrade BER throughrecovery loop
(Phase) Incoherent symbols Inter-symbol phase unimportant
Ex: Freq or amplitude modulation
Differential symbols
Data coded so change in symbolphase carries information
Phase matters only from symbol tosymbol
No Rx carrier recovery loop needed
BER vs Eb/No worse than for coherent
systems
Incoherent (i.e., FSK, ASK)Freq
Symbols
Coherent (i.e., QPSK)
Xmitted Symbols
Differential (i.e., DPSK)
Phase only mattersover one symbol
Rx & Tx LO phase differenceimportant over many symbols
Phase unimportant
Decoded SymbolX
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Definition of phase jitter variancefor coherent systems
2 = 2 0Rs/2 L (f)|1-Hp(f)|2df 2 2 Bp Rs/2 L (f)df Hp(f) = recovery loop response
function
Assumes channel bandpass filterwidth = symbol rate Rs
L (f)= sum of SSB -PSDs of allLOs
Because of the high pass cut-offfrom the carrier recovery loop, thisstandard variance exists even forflicker of frequency noise
Rule of thumb for QPSK phase jitter
should be < 1-3 for < 0.1 dBBER degradation
Calculating LO Phase Jitter for CoherentSystems
For oscillator x N
The phase jitter req must bereduced by N to compensate
for x N multiplication
L (f) (single sideband noise)
f
Sum of all LOs
Carrier RecoveryLoop BW
Bp
Phase JitterIntegration
Region
Filter atSymbol
Rate Rs/2
RecoveryLoop
tracksout thisregion
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Typical L (f) Requirements for QPSK LOs(vs Symbol Rate)
-140
-120
-100
-80
-60
-40
0 10 20 30 40 50 60 70 80 90Fourier Frequency-dBHz
L
(f)-dBc/Hz
- 10 Hz
- 100 Hz- 1 KHz- 10 KHz- 100 KHz- 1 MHz- 10 MHz- 100 MHz
- 1 GHz
SymbolRate Rs
Composite Spec
Rs = 10 Hz - 1 MHz
The curves above show typical L (f) requirements vs symbol rate 0.5 phase jitter allocated to particular LO Oscillator model: flicker frequency + white phase
Flicker freq and white phase each contribute equally to jitter
Carrier recovery loop BW optimized for data rate = 0.01 x Data Rate but 100 KHz (assumed hardware limit for VCO modulation rate) For multi-data-rate units, LOs must satisfy worst case composite
spec for all rates covered by that unit
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LO Vibration Sensitivity and CarrierPhase Jitter
Vibration induces phase jitter throughFreq source g-sensitivity
Hg(f) =( f/f)/ g = y/ g The vibration PSD Sg(f) generates f/f-
PSD Sy (f) directly through Hg(f)
Sy(f) = |Hg(f)|2 Sg(f)
S(f) = double-sided PSDs
This can be converted to a phase PSD byadding a (fo/f)
2 factor
S (f) = |Hg(f)|2 Sg(f)*(fo/f)2 fo = carrier frequency
As before, S (f) is integrated from Bp to Rsto produce a phase variance
2 = 0Rs/2 |Hg(f)|2 Sg(f)*(fo/f)2 |1-Hp(f)|2df 2 Bp Rs/2 |Hg(f)|2 Sg(f)*(fo/f)2 df
Because of the (fo/f)2 dependence of S (f),
there is a strong 1/Bp dependence in 2
Sg(f)
f
Vibration Spectrum
|Hg(f)|2
f
Oscillator gsensitivity
StructuralResonances
f
Vibration Induced
Phase Noise
S (f) (fo/f)2 factorbecause vibgeneratesfrequencysidebands
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Typical Vibration Levels in a CommercialAircraft
From: PHASE NOISE PERFORMANCE OF SAPPHIRE MICROWAVE OSCILLATORS IN AIRBORNE RADARSYSTEMS, T. Wallin, L. Josefsson, B. Lofter, GigaHertz 2003, Proceedings from the Seventh Symposium, November 45,
2003, Linkping, Sweden, Linkping ISSN 1650-3740 (www) , Issue: No. 8, URL: http://www.ep.liu.se/ecp/008/.
-80
-70
-60-50
-40
-30
-20
-10
10 20 30 40
Sg(f)dBg2/Hz
Fourier Frequency - dBHz
WithVibrationDamper
WithoutVibrationDamper
Sg Level
0.003 g2/Hz
Double Sideband Spectrum Damper Response
-80
-60
-40
-20
0
20
-20 -10 0 10 20 30
f/fres - dB
Respon
sedB
fres = 14.3 Hz
Q = 3
Vibration levels at a crystal oscillator with and without avibration damper
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Typical LO Hg Required vs Data Rate
Using this vib data (scaled by peak Sg without damper), one cangenerate the above curves of required Hgvs symbol rate
Assumes: 0.25 allocated to vibration induced phase jitter, Bp = 0.01Rs,
fo = 10 GHz, and constant Hg vs freq
Note (because of strong Bp dependence in 2) : (1) Hg regs morestringent for lower symbol rates, (2) vibration damper helps more athigher symbol rates & can make things worse at lower rates
With Vibration DamperNo Vibration Damper
Sg=0.003 Sg=0.01 Sg=0.03 Sg=0.1
1.E-12
1.E-11
1.E-10
1.E-09
1.E-08
1.E-07
1.E-06
1020304050607080
Symbol Rate-dBHz
Hg-g
2/Hz
1.E-12
1.E-11
1.E-10
1.E-09
1.E-08
1.E-07
1.E-06
1020304050607080
Symbol Rate-dBHz
Hg-g
2/Hz
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Decision Epoch Jitter from Data Clocks
Analysis of decision jitter similar tothat of phase jitter
x
2 = 2 0
Rs/2 Lx (f)
|1-Hp
(f)|2df
x2 2 Bp Rs/2 Lx(f)df = ( Tc)2 x = /(2 Rs) = clock reading error Lx(f)
= sum of SSB x-PSDs of clocks
Rec loop: Hp(f) = response Bp = BW
Rule of thumb: should be < 0.3-0.9 %for < 0.1 dB DER deg
Data clock phase jitter = 2 Rs x = 2 (in radians) L (f)= sum of SSB -PSDs of clocks 2 = 2 0Rs/2 L (f)|1-Hp(f)|2df 2 2 Bp Rs/2 L (f)dfRule of thumb: should be < 1-3
for < 0.1 dB BER degradation
Same curves as LO L (f)vs Rs (for samephase jitter and Bp)
Clock Jitter Reqsvs Symbol Rate
-12
-11
-10
-9-8
-7
-6
-5
30 40 50 60 70 80 90Symbol Rate - dBHz
Jtter-lo
g(s)
0.3% of Tc
0.9% of Tc
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Effect of Sampling Clock Jitter in DigitalImplementations
Digital implementations use A/Dand D/A converters to convertbetween analog and digitaldomains
Jitter tj in aperture clockgenerates random amplitudenoise in digitizing a signal withcarrier frequency f
Phase noise generated = 2 fSW tj = V/A Limits SNR of digital output
to 1 Can be converted to an effective
number of bits (ENOB) of theconverter (with assumptionsabout the size of A)
From: Analog Devices, Mixed-Signal and DSP Design Techniques, Section 2, Sampled Data Systems,
http://www.analog.com/Analog_Root/static/pdf/dataConverters/MixedSignal_Sect2.pdf, p35
Modulated SinewaveInput at Frequency fSW
Time Jitter tj
AmplitudeJitter V
Phase Jitter = 2 fSW tj
2A
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SNR due to Aperture (Sampling) ClockJitter for Full Scale Sinewave Input
From: Analog Devices, Mixed-Signal and DSP Design Techniques, Section 2, Sampled Data Systems,http://www.analog.com/Analog_Root/static/pdf/dataConverters/MixedSignal_Sect2.pdf, p36
0
20
40
60
80
100
120
60 65 70 75 80 85 90
4
8
12
16
E
NOB
Sinewave Frequency - dBHz
SNR
-dB
1ns
0.1ns
10ps
1ps
0.1ps
ClockJitter
S C
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Summary--Conclusions
T&F specs for frequency sources in comm systems can bederived by understanding the relationship between BER
degradation and frequency source phase and clock jitter
Recovery loops act as high pass filters that allow the use ofstandard variances even in the presence of flicker of frequencynoise
The critical jitter statistics are derived from PSDs by integratingfrom the loop recovery BW to the symbol rate
Spurs must be included in jitter integrations (not covered in talk)
Quadrature systems have more stringent phase jitterrequirements because of I-Q crosstalk
Frequency source vibration requirements are more critical for lowdata rate systems
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