ECE 546 – Jose Schutt-Aine 1 Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering...

ECE 546 – Jose Schutt-Aine 1

Spring 2014

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]

ECE 546 Lecture - 20

Jitter


• Timing uncertainties in digital transmission systems

• Utmost importance because timing uncertainties cause bit errors

• There are different types of jitter

Jitter Definition

Jitter is difference in time of when somethingwas ideally to occur and when it actually did occur.


• Jitter is a signal timing deviation referenced to a recovered clock from the recovered bit stream

• Measured in Unit Intervals and captured visually with eye diagrams

• Two types of jitter– Deterministic (non Gaussian)– Random

• The total jitter (TJ) is the sum of the random (RJ) and deterministic jitter(DJ)

Jitter Characteristics


Types of Jitter• Deterministic Jitter (DDJ)

Data-Dependent Jitter (DDJ)Periodic Jitter (PJ)Bounded Uncorrelated Jitter

(BUJ)

• Random Jitter (RJ)Gaussian Jitterf-a Higher-Order Jitter


Bandwidth LimitationsCause intersymbol interference

(ISI)ISI occurs if time required by signal

to completely charge is longer than bit interval

Amount of ISI is function of channel and data content of signal

Jitter Effects

Oscillator Phase NoisePresent in reference clocks or high-

speed clocksIn PLL based clocks, phase noise

can be amplified


Jitter Statistics

Most common way to look at jitter is in statistical domain

Because one can observe jitter histograms directly on oscilloscopes

No instruments to measure jitter time waveform or frequency spectrum directly

Jitter Histograms and Probability Density Functions (PDF)Built directly from time waveforms Frequency information is lostPeak-to-peak value depends on observation

time


Jitter Classification


Gaussian Random Jitter• Random jitter can be described by a

Gaussian distribution with the following probability density function

2221

( )2

x

RJPDF x e

x

: independent value

: RMS value

: mean of distribution (zero by definition)

Note: the PDF of a Gaussian process is unbounded, i.e, its PDF is not zero unless the jitter Dt approaches infinity


Gaussian Jitter PDF

Can be used to estimate the probability when the deviation of the random jitter variable Dt is within a multiple of its s value.

0.6826P t

2 0.9545P t

3 0.9973P t


Cummulative Density Function

Cummulative density function (CDF) is defined as:

( ) ( )t

CDF t PDF x dx

CDF(t) tells us the probability that the transition occurred earlier than t. For random jitter, we get:

1 1( )

2 2 2RJ

xCDF x erf

erf is the error function


PDF and CDF of Random Jitter

PDF CDF


• Crosstalk– Noisy neighboring signals

• Interference• Reflections

– Imperfect terminations– Discontinuities (e.g. multidrop

buses, stubs)• Simultaneous switching noise (SSN)

– Noisy reference plane or power rail– Shift in threshold voltages

Causes of Deterministic Jitter


Data-Dependent Jitter• Most commonly encountered DJ type• Dominant limiting factor for link

channels• Due to memory of lossy electrical or

optical system• Bit transition of current bit depends on

the transition times of the previous bits


Data-Dependent Jitter• DDJ depends on the impulse response

of the system that generates the pattern

• DDJ depends on the input pattern• DDJ is a distribution with its sample

size equal to the number of transitions of the data patent

• Duty cycle distortion (DCD) occurs for clock patterns of repeating bits


Data-Dependent Jitter

• Since channel does not have zero-rise time step response or infinite bandwidth, jitter is to be expected

• Settling time gives good indication of DDJ


DDJ Estimation for RC Network

/( ) 1 toU t e

Assume an RC time constant of =RC. The step response for an RC circuit is given by:

The DDJ time displacement at the 50% voltage level is:

ln 1 50% 0.6931DDJt

2

1( )

1H

3

0.2757dBf

3

0.191DDJ

dB

tf

In the frequency domain the transfer function is:

The 3dB bandwidth is:

The DDJ displacement

is:


Model for DDJ

1

( )N

DDJ DDJDDJ i i

i

f t P t D

DDJiP is the probability for the DDJ value of

DDJiD

The generic form for DDJ PDF is:

DDJiP

1

1N

DDJi

i

P

satisfies the condition


Periodic Jitter

Periodic jitter is a repeating jitter signal at a certain period or frequency. It is described by:

cos ot A t o

: angular frequency: initial phase

From a signal perspective, it is the same as any periodic signal in terms of frequency and phase, but its amplitude is jitter in units of timing.


Define the overall phase by:

ot

Phase has a uniform distribution if it is observed over a few periods. Its PDF is given by

10 2

2f for

The inverse function of t is:

1cos /t A

Periodic Jitter


Using the rule for PDF of related variables, the PDF for PJ t is given by

1cos ( / )( )PJ

d t Ad tf t f f

dt dt

2

1,

1 /PJf t A t A

t A

Which can be approximated by

1

2PJf t t A t A

Periodic Jitter

After substitution and differentiation, we get


Periodic Jitter

PDF for single sinusoidal

2

1,

1 /PJf t A t A

t A


Periodic JitterThere are 3 common waveforms for the theoretical analysis of periodic jitter

1 1( )

2 2 2 2PJ rect

m mPDF x

Rectangle Periodic Jitter

1( ) 2

0PJ triang

mfor x

PDF x motherwise

Triangle Periodic Jitter


Periodic Jitter

2

1

22

( ) / 2

0

PJ line

mfor x

PDF x m xm

otherwise

Sinusoidal Periodic Jitter


Rectangular Periodic Jitter

PDFCDF


Triangular Periodic Jitter

PDFCDF


Sinusoidal Periodic Jitter

PDFCDF


PDF of Two PJsA single PJ does not depend on its

initial phase if it is observed over many periods.

In the case of 2 PJs, the relative phase is important

When two PJs with the same magnitude, frequency and phase are added together, they form another PJ with twice the amplitude

When two PJs with the same magnitude, frequency and opposite phase are added together, their sum is zero.

The sum of two PJs can have totally different shapes depending on their phase relationships

Periodic Jitter


Phase Noise & Phase Jitter• Phase noise in clock oscillators

Phase offset term that continually changes timing of signal

( ) ( )S t P t t signal waveformwith phase noise

undistortedsignal

phase noise

Example:

9( ) sin 10 10 2P t t

91( ) sin 2 10 2

4t t

9 9( ) sin 10 10 2 0.25sin(2 10 2 )S t t t


Phase Noise

clean signal

noisy signal

2 GHz phase noise


• Phase jitter in digital systemsVariability in timing of transition in

digital systems is called phase jitterPhase jitter is digital equivalent of

phase noiseAlways defined relative to the ideal

position of the transitions

Phase Jitter

n n nt T For a jittered digital signal

nt

nT

is the actual time of the nth transition

nis the ideal timing value of the nth transition

is the time offset of the transition phase jitter term

Example: 10 Gbits/s Tn has bit intervals of 100 ps. Transitions take place at 0, 100, 200 ps


Phase Jitter

clean signal

noisy signal


• Phase jitter causes bit periods to contract and expand

• Actual bit periods are given by the time difference between 2 consecutive transitions

1 1 1n n n n n n nP t t T T

Ideal bit period:

1n n nTB T T

Period jitter:

n n nPerJ TB P

1 1 1 1n n n n n n n n nPerJ T T T T

Cycle-to-Cycle Jitter


Cycle-to-cycle jitter:

1n n nCCJit P P

1n n nCCJit PerJ PerJ

Cycle-to-Cycle Jitter


Bounded Uncorrelated Jitter

2

22

2

0

BUJ

t

BUJBUJ

PJ BUJ

BUJ

pe for t A

f t

for t A

BUJ is primarily due to crosstalk

The PDF for BUJ is given by


Mix of Random and Periodic Jitters

*2 2rect

m mRJ PJ RJ t d

Obtain convolution of 2 PDFs

2 2

2 2

/2 /2

2 21

2 2

t m t m

e e

Gaussian RJ and Rectangle PJ

Result is the sum of 2 Gaussian distributions with equal RMS value offset by the PJ peak-to-peak value . It is called the DUAL DIRAC DISTRIBUTION


• ProblemIn tests, we have measured jitter

histograms and need to extract the individual jitter components

Ideally, we could use deconvolution into components. However without prior knowledge of deterministic jitter, it is not possible

Use dual Dirac distribution model which would yield the worst case deterministic jitter

Jitter Mixing


Total Jitter Time Waveform

The total jitter waveform is the sum of individual components

TJ(t) = PJ(t) + RJ(t)


Jitter Statistics

TJ(x) = PJ(x) * RJ(x)

The total jitter PDF is the convolution of individual components


• Transfer of Level Noise into the Time DomainNoise on digital data signals causes

jitter because it offsets the threshold crossing point in time

• Bandwidth LimitationsPrimarily caused by intersymbol

interference

• Oscillator Phase NoisePhase noise present in reference clocks

especially in systems based on PLL

Jitter Mechanisms


Jitter MechanismsTransfer of noise into time

domainBandwidth limitation in

channelsOscillator phase noiseNoise

pk pk tH L

VNJ t

V V

NoiseVtt

HV

LV

rise time

pk-pk noise amplitude

Hi signal level

Lo signal level

Jitter Mechanisms


Jitter Mechanisms

Transfer of voltage noise into time domain – Linear model

Random noise caused by thermal effects


Jitter Mechanisms

Transfer of voltage noise into time domain – First order model

Periodic noise: switching power, crosstalk, etc…


Jitter Mechanisms

Multiple threshold crossing of a signal with high-frequency level noise


Bandwidth Limitations

0001111 data pattern


0101111 data pattern

Bandwidth Limitations


Q-Scale Transformation

1 1( )

2 2 2

xCDF x erf

Use CDF

Q-scale is defined such that the Gaussian distribution mapped onto the Q-scale is a straight line

1( ) 2 2 ( ) 1x

Q x erf CDF x

A Gaussian CDF is a straight line in the Q scale with slope 1/ . s DJ is given by distance d



PDFCDF

Gaussian RJ

= 0.5



PDFCDF

Gaussian RJ

= 0.25


Q-Scale - Generalization

PDFCDF

1( ) 2 2 ( ) 1x

Q x erf CDF x

Mixed Gaussian RJ and PJ

= 0.1


PDFCDF

Q-Scale - Generalization 1( ) 2 2 ( ) 1

xQ x erf CDF x

Mixed Gaussian RJ and PJ

= 0.25


Dual Dirac Model

Mixed Gaussian RJ and Triangular PJ


Jitter Classification


Measuring Jitter


Eye Diagrams• Eye diagrams are a time domain display of digital data

triggered on a particular cycle of the clock. Each period is repeated and superimposed. Each possible bit sequence should be generated so that a complete eye diagram can be made


Eye Diagram


High-Speed Oscilloscope

8-bit flash ADCs provide 256 discrete levels along vertical axis


Interleaving Architecture


High-Speed Scope Digitizers

• SiGe-Based Technologies Fastest ADCs run at 3.125 Gsamples/s Typically 8-16 digitizers

• CMOS Designs ADCs sample at lower rate 80 digitizers or more


Timing Diagram


Once waveform samples have been reassembled into a representation of the waveform, they are stored to digital memory

The maximum number of samples is the record length

Record length are typically in excess of 100 million samples

Sampling Procedure


Frequency Interleaving


A signal of bandwidth B that has been sampled at regular intervals T can be exactly recovered if the sampling rate satisfies

12*NF B

T

NF

T

B

: Nyquist rate

: sampling interval

: bandwidth

Nyquist Criterion


High-Speed Oscilloscopes

• Oscilloscopes use DSP techniques to: Extend their analog bandwidth Flatten their amplitude

• Practice has benefits

• However, limitations should be understood


Scope Channel Equalization


Edge Triggering

TOFF is recorded with high resolution but is subject to noise


Trigger jitter is the amount of effective timing instability between the trigger path and the signal capture path

Trigger Jitter

In eye diagram construction, multiple waveform acquisitions are overlayed. Trigger jitter is then an externally introduced noise that cannot be distinguished from the true jitter

Typical value: ~ 1 ps RMS


Trigger Jitter


Much of the timing instability in an oscilloscope is a combination of phase noise in the instrument’s time base and aperture jitter in the track-and-hold circuits

Sample Jitter

They exhibit a Gaussian probability distribution

Interleaving errors from the digitizers are another large source of errors. They are deterministic and are manifested as deterministic jitter can be calibrated out


Oscillator Phase Noise


• Gaussian Errors Phase noise Aperture jitter in track-and-hold circuits

• Deterministic Errors Interleaving mismatches Can be calibrated out

Sample Jitter


An eye diagram is a time-folded representation of a signal that carries digital information

Eye Diagram

Eye is horizontally centered on the ideal sampling instant


• Unit interval (UI) of a bit sequence is typically independent of the waveform sampling interval of the measurement instrument. Waveform sampling interval must be no more than

one half the unit interval to avoid aliasing Rule of thumb for eye diagrams is to sample 5 to 10

times the bit rate For 2.5 Gb/s, the sampling rate should be 20

GSamples/s

Eye Diagram

Large eye openings ensure that the receiving device can reliably decide between high and low logic states even when the decision threshold fluctuates or the decision time instant varies.


Eye Diagram Construction

Eye diagram construction in real-time oscilloscope is based on hardware clock recovery and trigger circuitry




1. Capture of the Waveform Record

2. Determine the Edge Times




3. Determine the Bit Labels


4. Clock Recovery




5. Slice Overlay

6. Display


Eye Diagram Measurements


Eye Diagram Measurements


Reference Levels


Eye HeightEye Height is the measuremnt of the eye height in volts

3 3PTop PTop PBase PBaseEye Height

PTop

PBasePBasePTop

: mean value of eye top

: standard deviation of eye top

: mean value of eye base

: standard deviation of eye base


Eye WidthEye Width is the measuremnt of the eye width in seconds

2 2 1 13 3TCross TCross TCross TCrossEyeWidth

1Crossing Percent 100%PCross PBase

PTop PBase

Crossing percent measurement is the eye crossing point expressed as a percentage of the eye height


Jitter MeasurementsJitter peak-to-peak is the peak-to-peak value for the edge jitter in the current horizontal units

Jitter pp max 1 min 1TCross TCross

Jitter root mean square is the RMS value of the edge jitter in the current horizontal units

1Jitter RMS TCross

Jitter 6s represents the same measurement reporting the 6TCross1 value


Noise MeasurementsNoise peak-to-peak is the peak-to-peak value of the noise at the top or base of the signal as specified by the user

max min ,Noise pp

max min

PTop PTop or

PBase PBase

Noise root mean square is the RMS value of the noise at the top or base of the signal

Noise RMS orPTop

PBase


Noise MeasurementsSignal-to-noise ratio is the ratio of the signal amplitude to the noise at either the top or the base of the signal

S/N RatioPTop PBase

PTop PBase

Duty cycle distortion is the peak-to-peak time variation of the first eye crossing measured at the mid-voltage reference as a percent of the eye period

p-p

2 1

TDCDDCD 100%

TCross TCross


Eye Quality Factor

Quality factor is the ratio of the eye size to noise

Quality Factor

PTop PBase

PTop PBase


Eye Diagram Specifications

PCI Express 2.0 eye diagram specification for full and deemphasized signals


Margin Testing

Eye diagram with low margin


Pseudorandomsequencegenerator

Transmitter Receiver

Scope

Trig Vert

Clk

Data

Fiber

Eye Pattern Analysis


Typical Eye Diagrams

Eye Diagram


BERT Scan

The sample delay BERT scan curve is a direct measurement of the jitter cummulative density function (CDF).


( ) (1 ( 0.5))leftBER t TD CDF t

( ) ( ( 0.5))rightBER t TD CDF t

( ) ( ) ( )left rightBER t BER t BER t

( ) (1 ( ( 0.5) ( ( 0.5))BER t TD CDF t CDF t

BERT Scan


Bathtub Curve

Linear Scale Logarithmic Scale


• ProblemIn tests, we have measured jitter

histograms and need to extract the individual jitter components

Ideally, we could use deconvolution into components. However without prior knowledge of deterministic jitter, it is not possible

Use dual Dirac distribution model which would yield the worst case deterministic jitter

Jitter Mixing


Random Jitter Extraction• Spectrum Analysis

Extract random jitter by using the assumption that it has a piecewise linear spectrum

Impulses are attributed to DJ

Noise floor is due to RJ


Extracting Random Jitter

Total jitter

Random jitter

Time domain

Statistical domain

Spectral domain


Jitter Spectrum

A longer FFT yields a spectrum with greater frequency resolution and lower noise floor.

Time record: 10N Time record: N


Random Jitter Extraction• Tail-Fit

Extract random jitter under the assumption that its probability density function follows a Gaussian distribution

Make use of the Dual-Dirac Model


Dual Dirac Model

• Equal Amplitudes Two unknown variables Linear Problem Explicit solution

- gap between 2 impulses- for Gaussian distribution

Unknowns


Dual Dirac Model

• Unequal Amplitudes Three unknown variables Nonlinear Problem No explicit solution

- gap between 2 impulses- for Gaussian distribution- ratio of 2 impulse amplitudes

Unknowns


Dual Dirac Model

*2 2rect

m mRJ PJ RJ t d

Obtain convolution of 2 PDFs

2 2

2 2

/2 /2

2 21

2 2

t m t m

e e

Assume Gaussian RJ and Rectangle PJ

Result is the sum of 2 Gaussian distributions with equal RMS value offset by the PJ peak-to-peak value . It is called the DUAL DIRAC DISTRIBUTION


DDJ and DC D• DDJ and DCD are correlated to the data

pattern

For N bits, transmitted at rate FR, the jitter components due to DDJ and DCD will appear in the spectrum at multiple of FR/N

FR=1.0625 Gbits/s

N=40 bits


Pattern Correlation


Pattern Correlation

The phase errors from all occurences of each M-bit patterns are averaged together to estimate the phase error due to that M-bit pattern


Extracting DDJ

Spectral domain Eye

DDJ Dominant

RJ Dominant

DDJ & RJ


Periodic Jitter

PJ PJ subcomponent

Time domain

Statistical domain

Spectral domain


Clock jitter is the single most important degrader of clock performance

Clock JitterIn a computer system, the clock is used to provide timing or synchronization for the system.

In a communication system, the clock is used to specify when a data switch or bit transaction should be transmitted and received

In a synchronized system, a central global clock is distributed to its subsystem


Definition• Most of the definitions of data jitter

(DJ, Rj,…) apply to clock jitter

• ISI does not apply to clock jitter


Clock jitter analysis is subject to fewer sampling constraints compared to data signal jitter; therefore, more direct and versatile methods are possible for clock jitter analysis.

Clock Jitter


Synchronized System

- Initial clock pulse causes A to latch data from input and launch it into channel- Second clock causes B to latch the incoming data


Timing Parameters


The minimum conditions are that both setup time and hold time margin should be larger than 0

0 _ _ _c jitt c skew d pd suT T T T T

_ _ _hd d pd c skew c jittT T T T

Timing Conditions

These give a quantitative description of how clock jitter and clock skew affect the performance of the synchronized system in which a common or global clock for both driver and receiver is used


Skew Impact

• Tc_jitter=0, Tc_skew>0The minimum clock period

increases. The maximum hold time increases hold time condition easier to meet

• Tc_jitter=0, Tc_skew<0The minimum clock period

decreases. The maximum hold time decreases hold time condition harder to meet (race condition)


Jitter Impact

• Tc_skew=0, Tc_jitter>0 (longer cycle)The minimum clock period

increases. The maximum hold time decreases hold time condition harder to meet

• Tc_skew=0, Tc_jitter<0 (shorter cycle)The minimum clock period

decreases. The maximum hold time increases hold time condition easier to meet


1. Positive jitter over one clock period makes both clock period and hold time hard to meet

2. A longer cycle does more harm to system performance

3. When both skew and jitter are present, system performance can be any of the four scenarios just discussed

System Performance


Asynchronized System

The skew of a synchronized system becomes hard to manage when the data rate increases(~1 Gb/s). At multiple Gb/s data rates, an asynchronized system is commonly used.


• Synchronized SystemGlobal clock is used to update and

determine bits

• Asynchronized SystemOnly data is sentClock is embedded in dataClock recovery unit (CRU) recovers

clock at receiver

Clock Types


Asysnchronized Link

_ _ _clk tot clk tx clk rxDJ DJ DJ

_ _ _

2 2 2

clk tot clk tx clk rx

Low-frequency jitter from the transmitter clock can be tracked or attenuated by the clock recovery function if it has a high enough corner frequency. A low phase noise oscillator within a PLL clock recovery also provides smaller random jitter generations.


Phase Jitter

n n nt t T

nt : timing for nth edge for jittery clock

nT : timing for nth edge for ideal clock

oT : ideal clock period

n oT nT


Phase Jitter

Phase jitter captures the instance timing deviation from the ideal for each transition. Jitter measured with phase jitter is absolute and accumulates over time.

2nn

o

t

T

In frequency domain


Period JitterPeriod jitter is defined as the period deviation from the ideal period.

1pn n n ot t t T

1pn n nt t t

using previous relations

in terms of phase units

'1n n n

Period jitter and phase jitter are not independent we can derive one from the other.


Phase, Period and CTC Jitter


Phase Jitter in Time Domain

If the phase varies, the waveform V(t) shifts back and forth along the time axis and this creates phase jitter


Phase Jitter in Spectral Domain

Phase noise appears as sidebands centered around the carrier frequency


Phase Jitter

( )( ) n

o

P fL f

P f

: phase noise power (in watts) ( )nP f

: carrier’s power (in watts) oP

: phase noise bandwidth (in hertz) f

1( ) ( )

2L f S f

Phase noise magnitude is specified relative to the carrier’s power on a per-hertz basis

: PSD of phase noise( )S f

10

( )( ) 10log

2

S fL f

or


Phase Noise to Phase Jitter

From the phase noise PSD, random jitter and deterministic jitter can be identified

Need: convert phase noise measured in the frequency domain to phase jitter for PLLs, clocks and oscillators


Probe Further

• Course web site http://emlab.uiuc.edu/ece546/appnotes

• May 09 issue of IEEE Transactions on Advanced Packaging

• D. Derickson and M. Muller, “Digital Communications Test and Measurement”, Prentice Hall, 2007.

• Mike Peng Li, “Jitter, Noise and Signal Integrity at High-Speed”, Prentice Hall, 2008.

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ECE 546 – Jose Schutt-Aine 1 Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering...

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