Freq reqs for comm.ppt, V. S. Reinhardt. Page 1 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is

freq reqs for comm.ppt, V. S. Reinhardt. Page 1Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source

reference is listed on each page, section, or graphic utilized.

The Calculation of Frequency Source Requirements for Digital

Communications Systems

Victor S. Reinhardt08/25/04

IEEE International Ultrasonics, Ferroelectrics, and Frequency Control 50th Anniversary Joint Conference, Montreal, August 24-28, 2004



The Calculation of Frequency Source Requirements for Digital Comm Systems

Introduction• Frequency sources (oscillators, synthesizers, etc.) are an

important part of digital communications systems

• Paper will discuss the derivation of frequency source requirements from over-all digital comm system parameters

• Will be tutorial treatment for those not familiar with digital comm theory but familiar with time & frequency theory

• Frequency source properties directly impact the performance of digital comm systems– Impact link acquisition & loss of acquisition—T&F community familiar

with synchronization issues—Will not be covered here– Impact bit error rate (BER) performance--Paper will address this

• Will utilize quadrature phase shift keyed (QPSK) systems for concrete examples – But theory applicable to other systems



Basic Digital Comm ConceptsSignals Carrying Digital Information

• At the transmitter a carrier is modulated in a regular time sequence of symbols to produce a digital communications signal or waveform

• A symbol is a temporal waveform in some modulation space representing a single digital word of information

• At the receiver the signal is sampled at discrete decision epochs to determine a modulation value of the carrier

• The modulation value is converted into a digital or data word by comparing 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 per symbol or word)

Example: Unshaped (Rectangular) Symbols in PAM

Decision Epochs

Time

ValueD

ec

isio

nT

hre

sh

old

s

Symbol3

Symbol2

Symbol 1

(1,0)

(1,1)

(0,1)

(0,0) (2-Bit)DigitalWords

Tc t3t2

t1

Carrier

Axis

Signal



Shaped Symbols

• Unshaped (rectangular) symbols are not bandwidth efficient– Sinc functions in freq domain

• Shaped symbols are sinc-like functions in time domain – Produce more bandwidth efficient

trapeziodal functions in freq domain– Do not interfere with each other at decision

epochs• The price one pays for shaping is more

stringent 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



Inter-Symbol Interference (ISI) & Eye Patterns

• Eye Pattern = Graph of the modulation value vs time at the receiver plotted modulo 1-symbol period (as in a scope trace)

• Eye opening = region with no value trajectories in it

• Inter-symbol Interference (ISI) = Contamination at decision epoch of modulation value by adjacent symbols– Ideal Decision epoch—no ISI – Clock errors cause the decision epoch to

wander off the best decision epoch increasing the ISI

– Sensitivity of ISI to clock timing = Slope of eye opening at decision epoch

• Even unshaped (square) symbols generate such eye patterns because of receiver and channel filtering necessary to limit signal BW & noise

• Shaped symbols have narrower eye widths than unshaped ones

From: Telecom Glossary 2000, American National Standard for Telecommunications, 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



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 digital states 2W): binary, quadrature, M-ary

• Shaped or unshaped• Coherent, incoherent, differential phase• Synchronous & asynchronous data

clock timing (used in hardline systems)

Binary, M-ary FSK

Freq

(0) (1)

FrequencyShift Keyed

Time

PulsePositionor Width

ModulationPWM

Time

PulseAmplitude

Shift Keyedor Modulation

Amplitude

PAM

Hybrid Modulation M-ary Quadrature Amplitude

Shift Keyed or ModulationCoherent Phase-Frequency

Shift KeyedMinimum Shift Keyed (Binary

CPFSK)

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



Bit Error Rate (BER) vs Eb/NoKey Comm System Parameter

– Rx thermal noise must limited by a filter

– For an ideal system the Rx filter’s bandwidth is equal 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 ideal system to achieve same BER as ideal

• Error correction coding (ECC) allows up to N bit 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 is incorrect

• 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

10-4

10-5

10-6

10-7

- Ideal- Actual

Error Correction Coded BER

BERDegrad-ation

BERDegrad-ation

Eb/No - dB



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/No Prx/(NoRs + V2)

Decisionthresholds

• Thermal noise in BW Rs ( = NoRs)causes occasional bit errors• BER (uncoded) = ½*Erfc(2-0.5V/)

= ½*Erfc((Eb/No) ½)

• 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 values V(±1 ±j)/20.5 at decision epoch• No ISI (jitter) without thermal noise

Jitter Vn

Ideal QPSK System



LO Phase Jitter Requirements in RF Carrier 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 Tx clock

– a carrier rec loop at the Rx LO required for phase coherent symbols• Recovery loops track out relative Rx-Tx LO and clock jitter for

fourier frequencies < recovery loop bandwidths• This is very important in defining the appropriate jitter statistics in

terms of power spectral densities (PSD)

SymbolModulator

DataEncode

UserData

• Error Correction

• Encryption• Framing

~~

SymbolDemod-ulator

Data(Sampling)

Clock~ ~ LO

RF

Xmission

DataDecode

Recover Loops

UserData

Data(Sampling)

Clock

LO

Transmitter (Tx) Receiver (Rx)

Rx LO recovery

loop only for phase

coherent symbols

Typical RF Carrier Comm System



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 is

determined 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



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 over

many symbols (recovery loop time constant Tp 1/ Bp >> Tc)

– Must have Tp >> Tc so thermal noise does not degrade BER through recovery loop

• (Phase) Incoherent symbols– Inter-symbol phase unimportant– Ex: Freq or amplitude modulation

• Differential symbols – Data coded so change in symbol

phase carries information– Phase matters only from symbol to

symbol– No Rx carrier recovery loop needed– BER vs Eb/No worse than for

coherent systems

Incoherent (i.e., FSK, ASK)Freq

SymbolsCoherent (i.e., QPSK)

Xmitted SymbolsDifferential (i.e., DPSK)

Phase only matters over one symbol

Rx & Tx LO phase difference important over many symbols

Phase unimportant

Decoded SymbolX



• Definition of phase jitter variance for coherent systems

= 20

Rs/2 L(f) |1-Hp(f)|2df

2Bp

Rs/2 L(f) df– Hp(f) = recovery loop response

function – Assumes channel bandpass filter

width = symbol rate Rs

– L(f) = sum of SSB -PSD’s of all LO’s

• Because of the high pass cut-off from the carrier recovery loop, this standard variance exists even for flicker of frequency noise

• Rule of thumb for QPSK phase jitter– should be < 1-3 ° for < 0.1 dB

BER degradation

Calculating LO Phase Jitter for Coherent Systems

For oscillator x NThe phase jitter req must be reduced by N to compensate

for x N multiplication

L(f) (single sideband noise)

f

Sum of all LO’s

Carrier RecoveryLoop BW

Bp

Phase Jitter Integration

Region

Filter atSymbol

Rate Rs/2

RecoveryLoop tracks

out this region



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

)-d

Bc/

Hz

- 10 Hz- 100 Hz- 1 KHz- 10 KHz- 100 KHz- 1 MHz- 10 MHz- 100 MHz- 1 GHz

SymbolRate Rs

Composite SpecRs = 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, LO’s must satisfy worst case composite spec for all rates covered by that unit



LO Vibration Sensitivity and Carrier Phase Jitter

• Vibration induces phase jitter through Freq 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 PSD’s

• This can be converted to a phase PSD by adding 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 Rs to produce a phase variance–

= 0Rs/2 |Hg(f)|2 Sg(f)*(fo/f)2 |1-Hp(f)|2df

– 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

Sg(f)

fVibration Spectrum

|Hg(f)|2

fOscillator gsensitivity

StructuralResonances

fVibration Induced

Phase Noise

S(f) (fo/f)2 factor because vib generatesfrequency sidebands



Typical Vibration Levels in a Commercial Aircraft

From: PHASE NOISE PERFORMANCE OF SAPPHIRE MICROWAVE OSCILLATORS IN AIRBORNE RADAR SYSTEMS, T. Wallin, L. Josefsson, B. Lofter, GigaHertz 2003, Proceedings from the Seventh Symposium, November 4–5,

2003, Linköping, Sweden, Linköping 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

) –

dB

g2 /

Hz

Fourier Frequency - dBHz

With VibrationDamper

Without VibrationDamper

Sg Level0.003 g2/Hz

Double Sideband Spectrum Damper Response

-80

-60

-40

-20

0

20

-20 -10 0 10 20 30

f/fres - dB

Res

po

nse

– d

B

fres = 14.3 Hz Q = 3

Vibration levels at a crystal oscillator with and without a vibration damper



Typical LO Hg Required vs Data Rate

• Using this vib data (scaled by peak Sg without damper), one can generate the above curves of required Hg vs 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 ) : (1) Hg regs more

stringent for lower symbol rates, (2) vibration damper helps more at higher 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

10 20 30 40 50 60 70 80

Symbol Rate-dBHz

Hg

-g2/H

z

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

10 20 30 40 50 60 70 80

Symbol Rate-dBHz

Hg

-g2/H

z



Clock Jitter Requirements for Data and Sampling Clocks

• Decision epoch jitter from data clocks– Clock jitter requirement value determined by eye pattern behavior

• Sampling or aperture clock jitter in A/Ds & D/As (in digitally implemented Tx’s and Rx’s) – Jitter in aperture clock causes non-thermal SNR degradation in A/D’s

and D/A’s (creates amplitude jitter)– Reduces effective number of bits (ENOB)– Causes BER degradation

A/DAnalogInput Demod

RecoveryLoops

Typical Digital Implementation

Samplingor Aperture

Clock

SNR of N-Bit worddegraded by clock jitter

DecisionThreshold

Data Clock Jitter

Modulo time

Effective eye

Opening reduced

Symbol Period

Decision Epoch Jitter

ISI

ISI



Decision Epoch Jitter from Data Clocks

• Analysis of decision jitter similar to that of phase jitter– x

= 2 0Rs/2 Lx(f) |1-Hp(f)|2df

– x 2 Bp

Rs/2 Lx(f) df = (Tc)2

– x = /(2Rs) = clock reading error

– Lx(f) = sum of SSB x-PSD’s 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

– = 2Rsx = 2 (in radians)

– L(f) = sum of SSB -PSD’s of clocks

– = 2 0

Rs/2 L(f) |1-Hp(f)|2df

– 2 Bp

Rs/2 L(f) df

–Rule of thumb: should be < 1-3 °for < 0.1 dB BER degradation

–Same curves as LO L(f) vs Rs (for same phase 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

Jtte

r -

log

(s)

0.3% of Tc

0.9% of Tc



Effect of Sampling Clock Jitter in Digital Implementations

• Digital implementations use A/D and D/A converters to convert between analog and digital domains

• Jitter tj in aperture clock generates random amplitude noise in digitizing a signal with carrier frequency f

• Phase noise generated = 2fSWtj = V/A

• Limits SNR of digital outputto

• Can be converted to an effective number of bits (ENOB) of the converter (with assumptions about 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 Jittertj

AmplitudeJitter V

Phase Jitter = 2fSWtj

2A



SNR due to Aperture (Sampling) Clock Jitter 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

EN

OB

Sinewave Frequency - dBHz

SN

R -

dB

1 ns

0.1 ns

10 ps

1 ps

0.1 ps

Clock Jitter



Summary--Conclusions

• T&F specs for frequency sources in comm systems can be derived 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 of standard variances even in the presence of flicker of frequency noise

• The critical jitter statistics are derived from PSD’s by integrating from 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 jitter requirements because of I-Q crosstalk

• Frequency source vibration requirements are more critical for low data rate systems

Documents

Freq reqs for comm.ppt, V. S. Reinhardt. Page 1 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is