BehavioralModelingOfRandomJitter

DesignCon 2015

Behavioral Modeling of Random

Jitter with Realistic Time and

Frequency Dependence

Scott W. Wedge, Synopsys Inc.

[email protected]

Abstract

As solutions keep improving for the efficient prediction and equalization of deterministic

jitter in high-speed data channels, new limitations are emerging due to random jitter (RJ)

in sources and PLLs. Modeling RJ requires more advanced behavioral jitter models than

used in the past. In prior applications, RJ could be treated as Gaussian distributions with

contributions independently computed via convolution. But now advanced design and

analyses require RJ to be modeled with phase noise responses for data-clock and CDR

PLLs, realistic jitter spectra, and higher-order jitter models that properly model phase

jitter, period jitter, cycle-to-cycle jitter, all with appropriate jitter bandwidths. This paper

shall discuss how these higher-order random jitter models can be created, with not just

Gaussian statistics, but also with appropriate time- and frequency- dependence of the

jitter, and in such a way to be useful for accurately predicting critical jitter effects in

high-speed data channels.

Author Biography

Scott Wedge has spent over three decades designing AMS/RF circuits and systems and

developing advanced simulation & analysis software for ICs and PCBs. He is a Senior

Staff Engineer with the Design Group at Synopsys, and held prior technical and

management positions with Tanner Research, Hewlett-Packard, EEsof, and Hughes

Aircraft Company. Scott has made substantial contributions to many popular circuit

simulation & design tools including HSPICE, ADS, Tanner Tools, PUFF, and

Touchstone. Scott has authored numerous technical papers, has made contributions to

three textbooks, and holds two patents. He is a registered Professional Engineer in

California, a Senior Member of the IEEE, and a former Howard Hughes Doctoral Fellow.

Scott received his Ph.D. from the California Institute of Technology.

Introduction

Random Jitter (RJ) is often considered a simple and easy-to-account-for effect in link

simulations, where it is typically treated as an add-on step to deterministic jitter analyses.

But if the desire with RJ modeling is to accurately account for the phase noise

characteristics of clocks and PLLs, the modeling and analysis approach must take on

some additional dimensions of complexity. The RJ present in real clocks and data streams

can have rather complex time and frequency behavior. Today’s link simulation models

must capture this behavior for adequate accuracy.

Consider for a moment how we might model random jitter in a reference clock. Typically

this would be considered as a random phase drift about an ideal modulo-two-pi phase

advance related to clock frequency. Is it sufficient to know just the standard deviation of

this drift for a Gaussian distribution? For effective behavioral modeling, we may need to

consider, know, and model much more than this. An appropriate model may need to

include the proper: rate of change of drift, or period jitter; the acceleration of drift, or

cycle-to-cycle jitter; the appropriate sampling of the random phase drift; the matching of

half-cycle jitter, the phase effects on duty-cycle; the proper frequency response of the

jitter, and perhaps other effects.

Capturing and combining these various jitter elements in detail using time-domain

behavioral models is the subject of this paper. Presented here are model mathematics and

useful implementation details. Common pitfalls and errors encountered in creating

behavioral jitter models shall be discussed, and how neglecting certain aspects of jitter

can lead to poor models. Jitter model templates and specifics are presented, with

formulations that can be implemented and manipulated in Verilog-AMS and MATLAB.

Jitter in Three Dimensions

A complete model for jitter requires considering three different characteristics that

contribute to its makeup and description. Here we refer to these as the three dimensions

of jitter. Different aspects of jitter may only focus on a single dimension, yet in reality

they are all invariably dependent on each other.

The Jitter Distribution

The first dimension is the jitter distribution. Our primary concern in this paper is to

accurately model random jitter, and as is true in most situations involving random noise,

our distribution of primary interest is the Gaussian or Normal probability distribution

function given by:

2

2

2

2

1)(RJ

xx

PDF ex

In the classic formula above, ‘x’ is a sample distribution axis, and could be interpreted as,

for example, voltage when looking at voltage noise, or time offsets when looking at edge

transitions for random timing jitter. Such jitter distributions give us the range or spread of

expected or measured values over the axis of interest. The well known Gaussian

distribution is completely described by its 2 variance and mean value x that gives the

center of the distribution along the sample distribution axis. If we have RJ samples over

time given by )(tx , the mean square value of )(tx is given by 22 )( tx and the RMS

value is the standard deviation . As written above, the continuous distribution is

normalized such that integrating over its x-axis yields a value of unity, which is another

way of stating that 100% of samples are accounted for through integration. The awkward

aspect of Gaussian distributions is that theoretically they are unbounded, and samples are

spread over an infinite x-axis interval. Realistically, a high percentage of samples are

accounted for over a handful of values along the x-axis. More realistically, especially

when measuring, we usually examine samples using histograms that use discrete binning

to approximate the distribution.

Jitter versus Time

The jitter distribution gives us the important range and likelihood of values to expect, yet

it’s only part of the jitter picture. It tells us nothing about how jitter may originate from

time-domain fluctuations, nor how it may be modeled in time. Neither does it tell us how

fast samples within the jitter distribution may move from one extreme to another. Let’s

consider an abstraction that allows us to examine jitter versus time. We can begin with

an ideal sinusoidal oscillator, and then include the effects of jitter via perturbations on the

oscillator’s phase advance [1]. Ignoring amplitude noise, we can write the instantaneous

output voltage )(tV of the oscillator according to

)(sin)( 00 ttVtV

where 0V is the nominal peak voltage amplitude, 0 is the nominal angular frequency,

and )(t is the phase deviation that represents jitter (in radians) in terms of a time-

dependent phase error. Phase deviation is a random variable that gives the instantaneous

departure from the nominal phase advance given by t0 . In the context of jitter

modeling, phase deviation is a time-dependent function that we must synthesize to result

in the jitter characteristics we desire to model a clock or oscillator or PLL.

The abstraction of jitter in terms of a sinusoidal model is particularly useful for modeling

LC and XTAL oscillators that in turn may be buffered to form clock signals. But note

that for clocks and data we can easily map this same model to square wave signals using

the )sgn(x sign function which maps zero crossings of the sinusoid to rising and falling

edges:

)(

2sinsgn1

2)(

0

0 ttT

VtVsquare

In this square wave representation, 0T is the nominal oscillation period, and the effects of

the phase deviation )(t are only noticable on the resulting waveform near the half-

period intervals where it influences the sine & sign function zero crossings.

From a measurement perspective, a phase deviation function )(t that represents random

noise will have unpredictable instantaneous values. However, its influence will be

observed in standard jitter measurements. We can see this in the phase advance diagram

of Figure 1, where the phase deviation represents random fluctuations that result in

something other than an ideal straight-line phase advance. The noise-free clock or

oscillator would have a phase advance given by t0 . Phase deviation results in a blurry

departure from this ideal shown as a grey envelope in the figure. For RJ, the phase

deviation will have a jitter distribution known to be Gaussian when sampled. It will

therefore have RMS and standard deviation values that can be measured.

For the purposes of jitter modeling, we can use phase deviation as our jitter versus time

dimension, i.e. a waveform that represents jitter information with respect to time. It can

be thought of as a time-domain function, that can be generated from known statistics. But

some thought must be given towards its construction. Consider for a moment the phase

deviation waveforms shown in Figure 2. These waveforms are formed using Gaussian

samples for each half-cycle, and considered piece-wise constant over the half-cycle

interval. This is a commonly used and simple-to-implement approach to jitter modeling.

However, a close look at the waveforms reveals that there are no restrictions in place here

to prevent the Gaussian samples to move from one extreme to another during sampling.

Figure 1. The normally straight-line phase advance of an oscillator circuit

blurred by uncertainties due to random phase fluctuations.

These fast edges will translate into high-frequency components of jitter that may not be

desireable nor physcially reasonable. Clearly other factors must be considered.

Jitter versus Frequency

The jitter distribution characterizes the range of expected jitter values, while phase

deviation gives us a time-domain function for jitter modeling. Variance values from our

expected jitter distribution can be entered into random number generators to get

distribution samples. These samples can then be used to form phase deviation functions

in time. But we still need to understand the limits on how fast our phase deviation

waveforms can be allowed to change when forming time-domain waveforms from the

discrete samples. This must be done with awareness in the frequency domain of the

expected power spectral density of the phase function.

The one-sided spectral density of the phase deviation )(t is known as the phase

instability )( fS (rad2/Hz) [1]. Note that double-sided spectral densities are also

commonly in use, and a one-sided density is defined for non-negative frequencies only.

Measurements of )( fS for random signals are accomplished using spectral estimation

approaches. Typically this involves taking many waveform time samples, and averaging

and smoothing of multiple DFT results in order to construct the power spectrum. What

has become a more common and convenient measurement for phase deviation is the

single-sideband phase noise )( fL . Phase noise and phase instability responses are

related according to

)(2

1)( fSfL

Figure 2. Gaussian distributed samples used to construct piece-wise constant

phase deviation functions. Multiple random waveforms are superimposed.

)(t

t

By convention, usually )( fL is written in dB as 10log( )( fL ), with units in dB below the

carrier in a 1-Hz bandwidth (dBc/Hz), or more strictly given as

Hz

rad 2

log10 [2].

We can therefore consider our third dimension for characterizing jitter as the phase noise

response. The phase noise response tells us the energy distribution of the phase deviation

as a function of frequency. This is critical for knowing how fast the phase deviation can

change, and how its waveforms may be bandwidth limited or shaped by circuit frequency

responses. Note that jitter spectrum is a related measurement, observed directly in time

units versus frequency [3].

There are often several aspects of the phase noise response that are generally anticipated

from clocks, oscillators, and PLLs. Common frequency responses can be anticipated and

incorporated into the jitter model. In some PLL approaches, we can model the individual

blocks within a PLL to anticipate its response, but it can be much more productive and

efficient to have a single model for the closed loop response that somewhat matches the

phase noise response.

Jitter Modeling in 3D

Formulating a realistic model for a clock or oscillator with random jitter requires taking

into account the three different dimensions described above. Constructing a jitter model

for clocks, oscillators, and PLLs involves knowledge of the jitter distribution, measured

as a 2 variance in the time-domain, and the phase noise measured as a function of offset

frequency. The variance and phase noise are related. The RMS phase jitter rms is a value

related to the variance, phase deviation, and phase noise according to

00

222 )(2)()(2

1lim)( dffLdffSdtt

Tt

T

TT

rms

Figure 1 depicts a closed-loop oscillator where the random fluctuations have a bounded

standard deviation given by 0/ rms . One question concerning our oscillator

model that includes random phase fluctuations: does the oscillator’s phase advance

)(tt have a deviation )(t that is bounded, or does it continue to grow in time? It

depends. In general, closed-loop oscillators and PLLs have bounded fluctations, while

with free-running (open-loop) oscillators do not.

Modeling with realistic time and frequency dependence therefore beings by examining

how we can create behavioral jitter models with a reasonable phase noise response.

Figure 3 gives three basic phase noise responses useful for jitter model creation. What

can be called the zero-order jitter model is shown as )(0 fL . This is the phase noise

response created by using the piece-wise constant phase deviation waveforms shown in

Figure 2. These phase waveforms produce an essentially flat phase noise response over

the offset range of interest. The piece-wise constant samples result in a power spectral

density that is proportional to a 22 )()(sin ff response, but with a high-frequency roll-

off close to the oscillation frequency. This phase noise response is far from realistic for

real circuitry. Although this zero-order model can be made to match the RJ off an eye

diagram, it is far from being able to approximate actual circuitry.

A 1st

Order (Open Loop) Jitter Model

We can construct a 1st order jitter model by using a simplified phase noise response based

on a free-running oscillator dominated by white FM noise [4]. This 1st order model will

have the phase noise given by

2

2

01 )(

f

fcfL .

This response increases as offset frequency drops, and will have have an unbounded

value for RMS phase jitter rms (we cannot integrate the phase noise down to near-zero

offset frequencies). However, we can solve for the resulting timing jitter according to

0

2

2

0

2

0

2

2

2

0

2

0

0

2

2

0

2

2

8)(sin8

)(sin)(8

)(

ccf

dff

fcf

dfffLT

The scalar constant parameter c from Demir [4] can therefore be thought of as setting the

white noise high-offset frequency rolloff. If we set c , the result is a first order

Figure 3. Three basic phase noise responses useful for examining and

creating realistic clock & oscillator jitter models.

model with timing jitter given by )(T which is the open-loop jitter response

discussed by McNeill [5]. In order to form this model, all we need is an RMS value for

period jitter which is equivalent to the timing jitter after an interval of one period:

0

00 )(f

ccTTTPER

The 1st order approach for jitter modeling is therefore a single-parameter open-loop

model based on period jitter which will have frequency/period deviations from one period

to the next over time. Although the frequency/period changes will be bounded, this drift

will cause the phase deviation to grow without bounded over time. It has become

commonplace to combine both the zero-order model, for random phase, and the 1st order

model, for random frequency/period, to model open-loop VCOs. This will give a

superimposed phase noise response )()( 10 fLfL . However, a higher order approach is

needed to model closed-loop oscillators and PLLs where phase drift is bounded over

time.

A 2nd

Order (Closed Loop) Jitter Model

Jitter modeling based on realistic circuitry should have the equivalent of a bounded phase

noise response that can be specified in part with an rms phase jitter value, and have a

phase noise roll-off that somewhat matches the loop bandwidth effects of a PLL. It is this

roll-off in a closed loop response that limits how fast the random phase fluctuations may

take place. A natural phase noise response useful for such modeling is the Lorentzian. It

can be written in the form:

224

0

2

2

0

2

2 )(fcf

cffL rms

Note that written in this form, we can make use of the integral identity (for 0a ):

0

22 2

df

fa

a.

which if we let cfa 2

0 , integration gives us back the rms phase jitter

0

2

2

22

2

0

22

22)(2 rms

rmsrms dffa

adffL

The autocorrelation of the phase deviation )()()( ttR is computed as

cf

rms

marms

rms

ee

dffa

mfa

dfffLR

20

2222

0

22

2

0

2

2

)cos(2

)2cos()(2)(

We therefore have a result for timing jitter given by:

22

0

2

2

2

0

2

2

2

0

2

20

20

2

12

12

)(2

)(

c

rms

cfrms

rmsT

e

e

R

We therefore have a related jitter and phase noise model in terms of the rms phase jitter,

rms , and the scalar constant parameter c which remains related to an open-loop phase

noise response. This two parameter model ( rms and c ) can be thought of as a 2nd

order

model for a closed loop oscillator where rms exists and we have an integrable phase

noise.

The Lorentzian model for phase noise has other interesting and useful properties. It has

the low-offset frequency plateau given by

cfL rms

2

0

2

2

2 )0(

We can rewrite the response in terms of loop bandwidth or time constants according to

cffff

ffL L

L

Lrms 2

022

2

2 )(

or

LL

L

Lrms fL

211

12)(

22

2

2

If we reconsider the one-sided spectral density of the phase deviation, i.e. the phase

instability )( fS , it can now be written

22

2

1

14)(

L

LrmsS

We can model this spectral density response by passing a white noise source with RMS

value given by

Lrmsin 2 through an RC-type filter network with transfer function

ssH

L

1

1)(

Since

22

22

2

)()(

1

1)(

sHS

sH

in

L

In this manner, filtering can be used to match the spectral density of our phase deviation

function to that of a desired phase noise response.

A Strategy for Clock RJ Modeling

We can combine the elements described previously to create a strategy for creating

behavioral models for clock jitter that have realistic time and frequency dependence.

1. Begin with a random number generator capable of creating Gaussian distributions

with a given standard deviation. For example, in Verilog-A this could be done

with the $rdist_normal() function.

2. These random samples will be used to represent samples for a phase deviation

function )(t . The standard deviation of the Gaussian distribution will be related

to the RMS phase jitter rms value for the phase fluctuations.

3. An interpolation approach for connecting the phase deviation samples must be

determined to construct a )(t waveform. The selection here may or may not be

important in order to match measured jitter with modeled jitter. This can be

evaluated using a phase trajectory diagram with careful attention given to how

phase errors map to zero-crossings. The phase interpolation function may be used

to establish the fundamental power spectal density (PSD) response of the )(t

waveform.

4. The phase deviation function )(t is next passed through a filter in order to

match its filtered PSD with that of the expected phase noise response. Typically

this is low-pass filtering, used to match, for example, PLL loop time constant or

bandwidth parameters.

5. The oscillator total phase )(tt is sampled adquately to construct sinusoidal

or square wave signals with the desired time resolution to reveal jitter.

The simplest approach recommended here is a 2nd

order model, specified with RMS

phase jitter rms and a loop time constant parameter L . The loop time constant can be

derived from a specified RMS period jitter PER value (units of seconds) according to:

2

22

0

0

21ln

rms

PER

L

T

This time constant will then result in matching both desired RMS period jitter and phase

jitter values. It is likely to have situations where appropriate values for the loop time

constant L or RMS period jitter PER may be unknown. In this case, it is reasonable to

employ the approach used commonly is serial data communications based on the Golden

PLL and the 1667 rule:

0

0 2

1667

2

1667

2

1T

ffL

L

Conclusions

This paper has presented an approach for capturing realistic time and frequency effects

for behavioral jitter modeling of clocks, oscillators, and PLLs. The approach is based on

considering not only jitter distribution functions, but also realistic oscillator phase noise

responses, and using this knowledge to formulate time-domain functions for phase

deviation that model jitter accurately. The approach uses phase filtering functions, and is

well described for a 2nd

order model that matches a Lorentzian phase noise response.

However, the filtering approach can be extended to model even higher-order and more

complex oscillator & PLL noise behaviors.

References

[1] IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and

Time Metrology – Random Instabilities, IEEE Std. 1139-2008.

[2] S.W. Wedge, “Phase noise and jitter translations for signal integrity”, Proc.

DesignCon, Santa Clara, 2011.

[3] G.D. Le Cheminant, “Solving jitter problems in high-speed digital transmission

systems”, Test & Measurement, Nov. 2010.

[4] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying

theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I, vol. 47,

pp. 655-674, May 2000.

[5] J.A. McNeill, “Jitter in ring oscillators”, IEEE J. Solid-State Cir., vol. 32, no. 6, pp.

870-879, June 1997.

[6] A. Hajimiri, S. Limotyrakis, and T.H. Lee, “Jitter and phase noise in ring oscillators,”

IEEE J. Solid-State Circuits, vol. 34, no. 6, pp. 790-804, June 1999.

[7] Jitter Analysis Techniques for High Data Rates, Application Note 1432, Agilent

Technologies, Feb. 2003.

[8] Characterization of Clocks and Oscillators, NIST Technical Note 1337, National

Institute of Standards and Technology.