Adaptive characterization of jitter noise in sampled high-speed signals

Abstract

We estimate the root-mean-square (RMS) value of timing jitter noise in simulated signals similar to measured high-speed sampled signals. The simulated signals are contaminated by additive noise, timing jitter noise, and time shift errors. Before estimating the RMS value of the jitter noise, we align the signals (unless there are no time shift errors) based on estimates of the relative shifts from cross-correlation analysis. We compute the mean and sample variance of the aligned signals based on repeated measurements at each time sample. We estimate the derivative of the noise-free signal based, in part, on a regression spline fit to the average of the aligned signals. Our initial estimate of the RMS value of the jitter noise depends on estimated derivatives and sample variances at time samples where the magnitude of the estimated derivative exceeds a selected threshold. This initial estimate is generally biased. Using a parametric bootstrap approach, we adaptively adjust this initial estimate of the RMS value of the jitter noise based on an estimate of this bias. We apply our method to real data collected at NIST. We study how results depend on the derivative threshold.

Full text

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 52, NO. 5, OCTOBER 2003 1537
Adaptive Characterization of Jitter Noise in
Sampled High-Speed Signals
Kevin J. Coakley, C.-M. Wang, Paul D. Hale, Senior Member, IEEE, and Tracy S. Clement, Member, IEEE
Abstract—We estimate the root-mean-square (RMS) value of
timing jitter noise in simulated signals similar to measured high-
speed sampled signals. The simulated signals are contaminated
by additive noise, timing jitter noise, and time shift errors. Before
estimating the RMS value of the jitter noise, we align the signals
(unless there are no time shift errors) based on estimates of the
relative shifts from cross-correlation analysis. We compute the
mean and sample variance ofthe aligned signals based on repeated
measurements at each time sample. We estimate the derivative
of the noise-free signal based, in part, on a regression spline fit
to the average of the aligned signals. Our initial estimate of the
RMS value of the jitter noise depends on estimated derivatives
and sample variances at time samples where the magnitude
of the estimated derivative exceeds a selected threshold. This
initial estimate is generally biased. Using a parametric bootstrap
approach, we adaptively adjust this initial estimate of the RMS
value of the jitter noise based on an estimate of this bias. We
apply our method to real data collected at NIST. We study how
results depend on the derivative threshold.
Index Terms—Adaptive, bias-correction, bootstrap, derivative
estimation, high-speed, jitter, optoelectronics, regression spline.
I. INTRODUCTION
I
systems, the target time and actual sampling time may differ
because of both systematic and random errors. The systematic
component of this difference is time base distortion (TBD)
[1]–[8]. We decompose the random timing error into a slowly
varying component called drift, and a quickly varying compo-
nent called jitter [9]–[12] (the terms “jitter” and “jitter noise”
are synonymous and should not be confused with deterministic
jitter). Within the time window during which a particular
realization of the signal is sampled, the drift error manifests
itself by shifting the waveform, while jitter manifests itself
by perturbing each of the sampling times in the window by
a random amount. Within the measured time window, each
of the jitter noise realizations is independent of all the other
jitter noise realizations. By definition, the expected value of
a realization of the jitter noise at any time is 0. Thus, the
root-mean-square (RMS) value of the jitter noise, that is the
RMS jitter, equals the standard deviation of the jitter noise.
Further, signal measurements are contaminated by additive
noise. In this work, we estimate the RMS value of timing jitter
DEALLY, one would like to sample a signal at equally
spaced time intervals. However, in high-speed measurement
Manuscript received August 22, 2002; revised May 13, 2003.
K. J. Coakley and C.-M. Wang are with the Statistical Engineering Division,
National Institute of Standards and Technology, Boulder, CO 80305 USA.
P. D. Hale and T. S. Clement are with the Optoelectronics Division, National
Institute of Standards and Technology, Boulder, CO 80305 USA.
Digital Object Identifier 10.1109/TIM.2003.817905
noise. Our work is motivated by efforts to characterize the im-
pulse response functions of high-speed sampling oscilloscopes
[13]–[16] that are contaminated by jitter noise (as well as by
TBD errors, time shift errors, and additive noise).
For the case where the noise-free signal is a sinusoid plus its
harmonics, there are methods to estimate the RMS value of the
jitter noise [8], [17]. Here, we consider the more complex case
where there is no analytical model for the signal. Hence, the
methods developed in [8], [17] do not apply to the signals con-
sidered in this work. In principle, the RMS value of the jitter
noise can be estimated from data collected over a time interval
where the noise-free signal is a linear function of time [10].
If this linearity assumption is violated, estimates based on the
premise of signal linearity are, in general, biased. Given the
power spectrum of the jittered signal and a parametric model
for the power spectrum of the jitter probability density function
(pdf), it is possible to estimate the model parameters that char-
acterize the power spectrum of the jitter pdf [9]. Since this ap-
proach is a nonlinear parameter estimation scheme, it is likely
to be biased since estimates that are nonlinear functions of ob-
serveddataaregenerallybiased.Formorediscussiononthebias
of nonlinear estimates, see page 40 of [18].
In thiswork, we estimate theRMS value of timing jitter noise
based on a fully empirical estimate of the time-varying variance
of the signal, and an estimate of the time-varying derivative of
the unknown noise-free signal. We obtain the estimate of the
signal derivative using a regression spline model [19] for the
signal. In a regression spline model, one can model a signal
without having a closed form analytical model for the signal.
Our work is inspired by an earlier attempt to estimate the RMS
value of jitter noise in high-speed sampled signals using a re-
gression spline method [20]. For a clear discussion of regres-
sionsplinemodelingandrelatedtechniques,wedirectreadersto
[21].In[20],repeatedmeasurementsofjitteredsignalsprovided
estimates of the sample variance of the signal for each time
sample. Based on an estimate of the derivative of the noise-free
signal provided by a regression spline model, and the sample
variance of the signal at each time sample, they estimated the
RMS value of the timing jitter noise. In their regression spline
approach, the noise-free signal of interest was approximated as
piecewise cubic polynomials. The regression spline parameters
that define each cubic polynomial in each subinterval, the RMS
value of the jitter noise, and the RMS value of the additive noise
were jointly estimated using an iterative algorithm. However,
their approach yielded biased estimates of the RMS value of the
jitter noise.
Our implementation of the regression spline method differs
from the implementation in [20]. First, we estimate the RMS
U.S. Government work not protected by U.S. copyright.

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1538IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 52, NO. 5, OCTOBER 2003
jitter from a subset of the full signal, whereas in [20] they esti-
mated RMS jitter from the full signal. To belong to this subset,
the magnitude of the estimated derivative of the signal average
must exceed a chosen threshold. We subsample because we ex-
pect that the least informative part of the signal corresponds
to time samples where the magnitude of the derivative of the
noise-free signal is least. Second and most significantly, we es-
timatethebias,thatis thesystematicerror, ofourestimate using
aparametricbootstrapmethod[22].Based onthisbiasestimate,
we adjust our estimate accordingly. Because we correct our es-
timate for bias, our procedure is adaptive.
In a Monte Carlo simulation experiment, we consider two
cases. In one case, the signals are contaminated by jitter noise
and additive noise, but not by time shift errors. In the second
case the signals are contaminated by jitter noise, additive noise,
and time shift errors. For the second case, we estimate relative
time shift errors based on cross-correlation analysis ([23] and
Appendix). Given the relative time shift estimates, we align the
signals using a Fourier method ([23] and Appendix). We es-
timate the RMS value of jitter noise from the average of the
aligned signals and the sample variance of the aligned signals.
For the simulated and real signals considered in this work,
we will show that RMS jitter can be estimated well by selecting
a threshold value so that the RMS jitter value estimate is com-
puted from the main feature in the signal which is similar to a
damped sinusoid. During the time where the main feature rises
from a local minimum to a local maximum, the signal is sam-
pled approximately seven to eight times. For the cases studied,
the RMS timing jitter value is no greater than 2
the interval between samples. In general, the accuracy of our
RMS jitter estimation method degrades as the true RMS jitter
value increases.
In Section II, we present details for calculating our jitter esti-
mate. In Section III, for simulated data, we study the perfor-
mance of our bias-corrected (and uncorrected) jitter estimate
based on the regression spline approach. We compare the per-
formance of our bias-corrected estimate to the performance of
two other estimates. One of the alternative estimates is based on
the assumption that the noise-free signal is a linear function of
timeinthethree-pointtimeneighborhoodaboutthetimesample
where the sample variance of the signal is largest. The other es-
timate is the regression spline approach presented in [20]. In
Section IV, we estimate the RMS value of jitter noise in experi-
mental signals measured at NIST.
whereis
II. ESTIMATION PROCEDURE
A. Preliminaries
Assuming that TBD errors are negligible, we can model the
th observed signal at the th time sample aswhere
(1)
is a realization of the jitter noise,
(drift) error,
is a realization of additive noise, and
unknown function of time. For each signal, the realizations of
the additivenoise,jitter noise, and time shift noise processes are
assumed to be independent. We assume that the expected value
is a random time shift
is an
of each is zero, that is,
realizations of the additive noise and jitter processes have finite
variances
and. We estimate the relative time shift errors
by an all-pairs cross-correlation method ([23] and Appendix).
Based on the estimated relative time shift errors, we translate
eachsignalintimeusingaFouriermethod([23]andAppendix).
We denote the th time sample of the th aligned signal as
. Assuming that we have accurately aligned the signals, a
first order Taylor series argument yields an approximation for
the variance of the sampled signal
, and that the
(2)
(Here, we consider the general case where we do not have an
analytic model for the derivative of the noise-free signal at time
,. Hence, we must estimate this derivative from mea-
surements). Using (2), we approximate the jitter variance,
as
(3)
At any time sample of interest, we cannot determine the
exact value of the variance of the signal from a finite number of
signal measurements. Following the standard rule of statistical
practice, we estimate the unknown theoretical variance by
computing the empirical sample variance. As a first step in
calculating the sample variance, we compute the average of the
aligned noisy signals as follows:
(4)
The sample variance of an aligned signal at the th time sample
(5)
is an estimate of the theoretical variance
the th time sample. As the number of samples
infinity, the sample variance converges to the theoretical vari-
ance. The square root of the sample variance,
estimated standard deviation of the signal at the
sample. Throughout this paper, we refer to the estimated stan-
dard deviation as the standard error.
of the signal at
approaches
is the
th time
B. Naive Estimate
In order to exploit (3) to estimate the RMS value of the jitter
noise, we need an estimate of the derivative of the noise-free
signal and an estimate of the additive noise variance. In this
work, we estimate the additive noise variance from sample vari-
ances computed near the time boundaries where the signal is
flat.Wedefinethetimesamplewherethesamplevarianceofthe
signalislargesttobethe
thtimesample.Ifweassumethatthe
noise-free function is a linear function of time in the neighbor-
hood of this time sample, the derivative of the noise-free signal
is approximately equal to
tiveestimateisequaltotheestimatedslopedeterminedbylinear
regression analysis. Based on this estimate of the derivative and
. This deriva-

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COAKLEY et al.: ADAPTIVE CHARACTERIZATION OF JITTER NOISE1539
(3), a naive estimate of the standard deviation of the jitter noise
is
(6)
where
estimate of the additive noise variance.
Since the above naive estimate depends on data from just
three time samples, we expect it to have a large variance com-
pared to a judiciously constructed estimate that depends on data
from more than three time samples. Moreover, we expect the
naive estimate to be biased because 1) the estimated derivative
may differ from the true derivative; 2) higher order terms in the
Taylor series are neglected in (3); and 3) the (6) estimate is a
nonlinear function of the observed data and nonlinear estimates
are generally biased. For more discussion about the bias of non-
trivial nonlinear estimates, see page 40 of [18]. The regression
spline method estimate developed in this work is superior to the
(6) estimate for two main reasons. First, we incorporate infor-
mation from more than just threetime samples. Second, we cor-
rect our estimate for bias, that is, systematic error. In the next
section, we derive our first-pass (uncorrected) regression spline
method estimate of RMS jitter. Afterward, we estimate the bias
of this first pass estimate and then adjust the first pass estimate
accordingly.
is the interval between time samples, and is our
C. Regression Spline Estimate: Uncorrected
In a regression spline approach, one selects a sequence of
interior knots that partition the time interval during which the
signal is measured into contiguous subintervals. Within any of
these subintervals, the regression spline prediction of the signal
of interest is a polynomial function of time. The coefficients of
thepolynomialvaryfromsubintervaltosubinterval.In[20],and
in our work, we choose a cubic polynomial within each interval.
For the cubic case, there are K interior knots and four exterior
knots (two at each time boundary). Overall, the cubic regres-
sion spline model has
independent basis functions. The
regression spline modelparameters are determined by the
standard method of weighted least-squares where the weight at
a particular time sample is inversely proportional to the sample
variance at that time sample. In [20] as well as in our work, we
useB-splinebasisfunctionstorepresentthepolynomialsineach
subinterval. For the cubic case, the first and second derivatives
oftheregressionsplinearecontinuousattheknots.Basedonthe
regressionsplinemodelparameters,onecanestimatethederiva-
tiveofthenoise-freesignalasafunctionoftime.Thisderivative
informationalongwithempiricalestimatesofthestandarddevi-
ation of the jittered signal at each time sample, allow one to es-
timate the RMS timing jitter noise. The B-spline representation
at time , is denoted as
, and the derivative of the B-spline
representation at time
is denoted as
To compute our jitter estimate, it is convenient to define the
following quantities at the th time sample
.
(7)
and
(8)
Given (3), we expect the ratio of
mate of the jitter variance at all . Intuitively, we expect more
information in (
,) data at time samples where the magni-
tude of the derivative is relatively large. In our studies, the ratio
had a very large variance at time samples where
the magnitude of the signal derivative was very small. Thus, the
averageof allthe
valueswould be a poorestimate ofthejitter
variance. To reduce the influence of noisy (
estimate, we take two actions. First, we design our estimate so
that it depends on (
,) values at time samples where the
magnitude of the estimated derivative is greater than a selected
threshold. Second, we estimate the jitter noise variance as the
ratio of the pooled
data and the pooled
natural way to reduce the influence of highly variable, i.e. non-
informative, (
,) values on the estimate. For a discussion of
data pooling in other statistical estimation problems, see [24],
[25]. Finally, we require that our variance estimate be nonneg-
ative. Thus, our (nonnegative) estimate of the variance of the
jitter noise is
andto be a rough esti-
,) pairs on our
data. Pooling is a
(9)
where
if
otherwise
(10)
and
ofthejitternoiseis
as
By lowering the threshold, we incorporate more of the mea-
sureddataintoourestimate.However,ifthethresholdistoolow,
prediction error may increase if we incorporate too much noisy
data with little or no additional information content. Since the
optimal choice of the threshold is not obvious, we study how
the choice of threshold affects results in a Monte Carlo simu-
lation experiment. In general, for any choice of threshold, we
expect that the above estimate is biased since it is a nonlinear
function of the observed data (nonlinear estimates are generally
biased [18]). Next, we describe how to correct our estimate for
this bias.
is an adjustable threshold. Our estimate of the RMS value
(above,wedenotethemaximumof 0and
).
D. Regression Spline Estimate: Corrected
We estimate the bias of our estimate using a parametric boot-
strap procedure [22]. The parametric bootstrap procedure is a
Monte Carlo resampling scheme for simulating synthetic data
based on the observed data. Based on the distribution of the (9)
jitter estimates computed from the synthetic data, one can esti-
mate the standard deviation of the (9) jitter estimate computed
from the observed data. Further, one can estimate the bias of the
estimate and correct it accordingly. In the bootstrap simulation
model, the noise-free signal is equated to the regression spline
model estimate of the average of the aligned observed signals
.Liketheobserveddata,thesyntheticsignalsarecorrupted

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1540IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 52, NO. 5, OCTOBER 2003
TABLE I
STATISTICAL PROPERTIES OF ESTIMATES OF THE RMS VALUE OF THE JITTER NOISE. FOR EACH DERIVATIVE THRESHOLD VALUE ??? AND TRUE RMS JITTER
NOISE VALUE ?? ?, WE DISPLAY THE ESTIMATED BIAS (BIAS), STANDARD ERROR (se), AND RMS PREDICTION ERROR (rms) OF THE UNCORRECTED (9) AND
BIAS-CORRECTED (14) ESTIMATE COMPUTED FROM 100 RUNS OF A SIMULATION EXPERIMENT. IN EACH RUN, RMS JITTER IS ESTIMATED FROM 100 NOISY
SIGNALS. FOR EACH RUN, WE ESTIMATE THE STANDARD DEVIATION OF THE ESTIMATE BY A BOOTSTRAP RESAMPLING SCHEME. WE LIST THE MEAN (COMPUTED
FROM ALL 100 RUNS) VALUE OF THE BOOTSTRAP ESTIMATE (15) OF THE STANDARD DEVIATION OF THE ESTIMATE AS ??
QUANTITIES BY THE INTERVAL BETWEEN TIME SAMPLES ??. IN THIS STUDY, THERE ARE NO TIME SHIFT ERRORS. FOR EACH SIMULATION RUN, THERE
ARE 100 SIGNALS. WE LIST THE NAIVE ESTIMATE (6) FOR COMPARISON
. WE NORMALIZE ALL STATISTICAL
by time shift errors, additive noise and jitter noise. In the simu-
lation,thetimeshiftparametersareequatedtotheestimatedrel-
ative time shift parameters computed from the observed data. In
thebootstrapprocedure, weassumethatjitterand additivenoise
are Gaussian random variables with expected values equal to 0
and variances equal to those estimated from the primary “ob-
served” data. The number of signals in each bootstrap set is the
same as the number of observed signals. Throughout this work,
we simulate
bootstrap replications of the observed data.
Since the observed data consist of 100 repeat measurements of
the noisy signal, each bootstrap replication consists of 100 real-
izations of a noisy signal.
More formally, the th bootstrap replication of the observed
signal at the th time sample is
, where
(11)
(12)
and
estimate of the average of the aligned signals,
spacing betweens time samples,
ative time shift of the th signal with respect to the first signal
(Appendix and [23]),
is a simulated jitter noise realization,
is a constant. Above,is the regression spline model
is the nominal
is our estimate of the rel-
and
ulation model, we assume that time base distortion is 0 at all
times [If time base distortion is not 0, we would add a term
equal to the estimated TBD to the right hand side of (12). Given
that TBD is nonzero, the regression spline model
be fit to the unequally-spaced time series ( ,
where
the TBD. For more details on this approach, see [15], [26]].
The realizations of the jitter noise and additive noise are mutu-
allyindependentrealizationsofGaussianrandomvariableswith
standarddeviationsequaltothecorrespondingvaluescomputed
from the observed signals (
and
the simulated jitter noise and additive noise realizations are 0.
For each bootstrap replication of the observed data, we es-
timate relative time shift errors and align the signals using the
same algorithms used for the observed data. We estimate a new
set of regression spline model parameters, a new RMS additive
noise value, and a new RMS jitter noise value
estimate of the bias of our jitter estimate is
is a simulated additivenoise realization. Inthe (12) sim-
) would
) where
is the estimate of
). The expected values of
. The bootstrap
(13)

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COAKLEY et al.: ADAPTIVE CHARACTERIZATION OF JITTER NOISE1541
TABLE II
STATISTICAL PROPERTIES OF ESTIMATES OF THE RMS VALUE OF THE JITTER NOISE. FOR EACH VALUE OF THE DERIVATIVE THRESHOLD ??? AND TRUE RMS
JITTER NOISE VALUE ?? ?, WE DISPLAY THE ESTIMATED BIAS (BIAS), STANDARD ERROR (se), AND RMS PREDICTION ERROR (rms) OF THE UNCORRECTED (9)
AND BIAS-CORRECTED (14) ESTIMATE COMPUTED FROM 100 RUNS OF A SIMULATION EXPERIMENT. FOR EACH RUN, WE ESTIMATE THE STANDARD DEVIATION
BY A BOOTSTRAP RESAMPLING SCHEME. WE LIST THE MEAN VALUE OF THE BOOTSTRAP ESTIMATE (15) OF THE STANDARD DEVIATION OF THE ESTIMATE AS
??
. WE NORMALIZE ALL STATISTICAL QUANTITIES BY THE INTERVAL BETWEEN TIME SAMPLES ??. FOR EACH SIMULATION RUN, THERE ARE 100 SIGNALS.
WE LIST THE NAIVE ESTIMATE (6) FOR COMPARISON. UNLIKE FOR THE TABLE I CASE, THE SIGNALS ARE MISALIGNED DUE TO TIME SHIFT ERRORS. WE
ESTIMATE RELATIVE TIME SHIFT ERRORS BY A CROSS-CORRELATION METHOD, AND ALIGN THE SIGNALS BY A FOURIER METHOD
where
bootstrap replication. Our bias-corrected estimate of RMS jitter
noise is
is the estimate of RMS jitter computed from the th
(14)
Thebootstrapestimate ofthestandarddeviationoftheuncor-
rected jitter estimate is
(15)
III. SIMULATION STUDY
A. Results
WequantifytheperformanceofourestimateinaMonteCarlo
experiment. In the simulation study, we equate the noise-free
signal to the regression spline estimate of the average of aligned
experimental signals collected in an experiment at NIST. Each
simulatedsignalissampledat2048times.Theregressionspline
modelhas1028knots. Betweenknots,theinterpolating polyno-
mial is cubic. In each of many runs of the simulation, we gen-
erate 100 synthetic signals that are contaminated by jitter noise,
additive noise, and possibly time shift errors. In the first case
(Table I), there are no time shift errors. For this case, we assume
the signals are aligned, that is, we do not estimate relative time
shift errors. In the second case (Table II), the signals are mis-
aligned due to time shift errors. For this case, we estimate the
relative time shifts and align the signals (Table II, Appendix).
In case 2, we model the time shift errors as realizations of a
Gaussian AR(1) process [27] where the autocorrelation at lag
1 is 0.5 and the standard deviation is 0.5
we display the average of 100 aligned noisy signals, the mag-
nitude of the estimated derivative of the signal average based
on the regression spline model, and the standard error of the
aligned signals as a function of time. In Tables I and II, we list
the standard error of the bias estimate in parentheses. For in-
stance,0.0003(10)signifiesthatthebiasestimateandassociated
standard error are 0.0003 and 0.0010.
In the simulation experiments, the true value of the RMS
valueoftheadditivenoise,
of the additive noise is estimated from the first 50 samples after
. In Figs. 1 and 2,
.TheRMSvalue