SIGNAL MODEL BASED APPROACH TO JOINT JITTER AND NOISE DECOMPOSITION
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SIGNAL MODEL BASED APPROACH TO JOINT JITTER AND NOISE DECOMPOSITION White paper | Version 01.00 | Adrian Ispas, Julian Leyh, Andreas Maier, Bernhard Nitsch First published at DesignCon 2020
ABSTRACT
We propose a joint jitter and noise analysis framework for serial PAM transmission based
on a parametric signal model. Our approach has several benefits over state-of-the-art
methods. First, we provide additional measurements. Second, we require shorter sig-
nal lengths for the same accuracy. Finally, our method does not rely on specific symbol
sequences. In this paper, we show example measurement results as well as comparisons
with state-of-the-art methods.
AUTHORS BIOGRAPHY
Adrian Ispas received Dipl.-Ing. and Dr.-Ing. degrees in Electrical Engineering and
Information Technology from RWTH Aachen University, Aachen, Germany, in 2007 and
2013, respectively. Since 2013, he has been with Rohde & Schwarz GmbH & Co. KG,
Munich, Germany, where he is now a Senior Development Engineer in the Center of
Competence for Signal Processing.
Julian Leyh obtained his M.Sc. degree in the field of Electrical Engineering from TU
Munich in 2017 specializing in communications engineering. He then began his pro-
fessional career as a member of the Center of Competence for Signal Processing at
Rohde & Schwarz. His current focus is on time domain high-speed digital signal analysis.
Andreas Maier received an M.S. degree in Electrical Engineering and Ph.D. degree
from the Karlsruhe Institute of Technology (KIT), Germany in 2005 and 2011, respec-
tively. In 2011, he joined Rohde & Schwarz, Munich, Germany. His areas of expertise are
digital signal processing and signal integrity. He is a lecturer at the Baden-Wuerttemberg
Cooperative State University Stuttgart, Germany.
Bernhard Nitsch is the director of the Center of Competence of Signal Processing at
Rohde & Schwarz in Munich, Germany. He and his team focus on signal processing algo-
rithms and systems in the area of signal analysis, spectral analysis, power measurement,
security scanners and oscilloscopes. Rohde & Schwarz products contain software, FPGA
or ASIC based signal processing components developed in the Center of Competence.
He joined Rohde & Schwarz in 2000 and holds a PhD degree in Electrical Engineering and
Information Technology from Darmstadt University of Technology, Germany and an M.S.
degree in Electrical Engineering from Aachen University of Technology, Germany. He is
author of one book and seven technical papers and holds more than 25 patents.
ACKNOWLEDGEMENT
We want to thank Thomas Kuhwald for initiating the development of the joint jitter and
noise framework, Guido Schulze for motivating us to submit the paper to DesignCon and
finally Andrew Schaefer, Guido Schulze and Josef Wolf for their valuable feedback on
the paper.
2CONTENTS
1 Introduction....................................................................................................................................................4
2 Signal model...................................................................................................................................................5
3 Jitter and noise decomposition.....................................................................................................................6
3.1 Source and analysis domains..........................................................................................................................6
3.2 Decomposition tree..........................................................................................................................................7
4 Joint jitter and noise analysis framework....................................................................................................8
4.1 Estimation of model parameters......................................................................................................................9
4.2 Analysis domains...........................................................................................................................................10
5 Measurements..............................................................................................................................................11
5.1 Statistical values and histograms..................................................................................................................11
5.2 Duty cycle distortion and level distortion......................................................................................................11
5.3 Autocorrelation functions and power spectral densities..............................................................................11
5.4 Symbol error rate calculation........................................................................................................................11
5.5 Jitter and noise characterization at a target SER..........................................................................................12
6 Framework results........................................................................................................................................13
6.1 Example analysis...........................................................................................................................................13
6.2 Signal length influence.................................................................................................................................19
7 Comparison with competing algorithms.....................................................................................................19
8 Conclusion....................................................................................................................................................22
9 References....................................................................................................................................................23
Rohde & Schwarz White paper | Signal model based a ecomposition 3
pproach to joint jitter and noise d1 INTRODUCTION
The identification of jitter and noise sources is critical when debugging failure sources in
the transmission of high-speed serial signals. With ever increasing data rates accompa-
nied by decreasing jitter budgets and noise margins, managing jitter and noise sources is
of utmost relevance. Methods for decomposing jitter have matured considerably over the
past 20 years; however, they are mostly based on time interval error (TIE) measurements
alone [1, 2]. This TIE-centric view discards a significant portion of the information present
in the input signal and thus limits the decomposition accuracy.
The field of jitter separation was conceived in 1999 by M. Li et al. with the introduction
of the Dual-Dirac method [3]. The Dual-Dirac method was augmented and improved over
the next two decades. Originally, it was meant to isolate deterministic from random jit-
ter components based on the probability density of the input signal’s TIEs. It uses the
fact that deterministic and random jitter are statistically independent and that determin-
istic jitter is bounded in amplitude, while random jitter is generally unbounded. Three
years later, M. Li et al. reported that their Dual-Dirac method systematically overestimates
the deterministic component [4]. Despite this flaw, modelling jitter using the Dual-Dirac
model has maintained significance in commercial jitter measurement solutions due to its
simplicity [5].
Throughout the years, additional techniques were added to the original Dual-Dirac
method to separate additional jitter sources such as intersymbol interference (ISI), peri-
odic jitter (PJ) and other bounded uncorrelated (OBU) jitter. For example, PJ components
can be extracted using the autocorrelation function [6] or the power spectral density
[7, 8] of the TIEs, while the ISI part of deterministic jitter can be determined by averag-
ing periodically repeating or otherwise equal signal segments [13]. Yet another method
for estimating ISI makes use of the property that ISI can be approximately described as
a superposition of the effect of individual symbol transitions on their respective TIE [12].
Once the probability density function of one jitter component is known, any second com-
ponent can be estimated from a mix of the two by means of deconvolution approaches,
as long as the components are statistically independent of each other [9, 10, 11].
Collectively, over 40 IEEE publications and more than 50 patents can be found on the
topic of jitter analysis alone. Despite this, applications in industrial jitter measurements
commonly use a combination of the previously described methods [14, 15, 16], all based
solely on TIEs.
In this paper, we first introduce a parametric signal model for serial pulse-amplitude mod-
ulated (PAM) transmission that includes jitter and noise contributions. The key to this
model is a set of step responses, which characterizes the deterministic behavior of the
transmission system, similar to the impulse response in traditional communication sys-
tems. Based on the signal model, we propose a joint jitter and noise analysis framework
that takes into account all information present in the input signal. This framework relies
on a joint estimation of model parameters, from which we readily obtain the commonly
known jitter and noise components. Therefore, we provide a single mathematical base
yielding the well-known jitter/noise analysis results for PAM signals and thus a consistent
impairment analysis for high-speed serial transmission systems.
4known jitter/noise analysis results for PAM signals and thus a consistent impairment
analysis for high-speed
Additionally, we provideserial
deeptransmission systems.
system insight through the introduction of new mea-
surements, such as what-if signal reconstructions based on a subset of the underlying
Additionally, we provide deep system insight through the introduction of new
impairments. These reconstructions enable the visualization of eye diagrams for a selec-
measurements, such as what-if signal reconstructions based on a subset of the underlying
tion of jitter/noise
impairments. Thesecomponents, thereby
reconstructions enableallowing informed
the visualization ofdecisions about
eye diagrams for the relevance
a selection
ofofthe selected components. Similarly, we determine selective symbol error
jitter/noise components, thereby allowing informed decisions about the relevance of rate (SER)
the
and bit error rate (BER) extrapolations to allow fast calculation of (selective)
selected components. Similarly, we determine selective symbol error rate (SER) and bit peak-to-peak
jitter
errorandratenoise
(BER) amplitudes at low
extrapolations error rates.
allowing for a fast calculation of (selective) peak-to-peak
Signal Model-Based Approach to a Joint Jitter &
jitter and noise amplitudes at low error rates.
The proposed framework is inherently able to perform accurate measurements even
relativelyNoise Decomposition
The proposed framework is inherently able to perform accurate measurements even for
using relatively short input signals. This is due to the significant increase in information
short input signals. This is due to the significant increase in information extracted
extracted from the signal. Furthermore, our approach has no requirements regarding spe-
from the signal. Furthermore, our approach has no requirements regarding specific input
cific inputsequences,
symbol symbol sequences, such as compliance
such as predefined predefined patterns.
compliance
On patterns. On random
the contrary, the contrary,
or
the randominput
scrambled or scrambled inputencountered
data typically data typically encountered
in real-world in real-world
scenarios is ideallyscenarios
suited to is
theide-
ally suited to the framework.
framework.
Abstract
2 SIGNAL MODEL
2 Signal Model
The proposed joint jitter I.
Theproposed andI NTRODUCTION
andnoise
noiseanalysis
analysis framework
framework is based
is based on aon a signal
signal modelmodel for
for serial
serial
PAMPAM data data II.This
transmission.
transmission. S IGNAL
This
model M
modelODEL
assumes
assumes the total
the total signal,
signal, i.e. received
i.e., the the received signal
signal
containing all
containing allcomponents,
components, totobebe
∞
Signal Model-Based ∆ [k] · h (tApproach toy a+ Joint Jitter &(1)
y(t) = − T [k] − [k], s) + (t). (1)
s sr clk h −∞ v
Noise the vector Decomposition
k=−∞
(1)
Here,s denotes Here, of the transmitted PAM symbol sequence
denotes the vector of the transmitted PAM symbol sequence and and
Signal Model-Based
h Δ(t Ts[k]
Δ[k]
−
[k] −
sr s
[k],
s[k
= s)Approach
1] 1
clk
the
= hsymbol
–
denotes the step response for the symbol
s
h T [k] −atto
(t −difference
– the
vector
t T [k]at
symbol
symbol a
[k], {s[k],
time
response for the symbol vector at time , clk
sr
,
Joint
index
difference
the
(t,
clk s) Jitter
k∆h.[k]}).
Moreover,
at
reference
ℎ ,
symbol
h
clock time
& (2)
index . ssr denotes
the reference clock time at symbol
clk
sr the step
k Noise y Decomposition
at symbol index , and the initial signal value. The disturbance sources are split into
index , and –∞ the initial signal value. The disturbance sources are split into horizontal
horizontal (time) and vertical (signal level) parts: is the total horizontal source
(time) and vertical (signal
disturbance and is
ϵ [k] NPlevel)
(h) −1 parts: h is the total
horizontal source disturbance and
Abstract
the total vertical source disturbance. Contrary to traditional
ϵ v(t
h [k] = h,P [k] + h,OBU ) is the total vertical
[k] + h,R systems, source
[k] = we useA disturbance.
stepsin(2πf Contrary to
Th,l [k]of+ traditional
anφimpulse communication sys- (3)
communication a h,l responseh,linstead h,l ) + response
h,OBU [k]to+model
h,R [k]
tems, we use a step
the total signal. This is due response to instead
the fact of
thatan animpulse
impulse response
response to
is model
neither the
able total
to signal.
ensure
l=0
This
signalis due to the under
continuity fact that an−1
non-linear impulse
(in theresponse
signal levelis neither
domain) able to ensure
horizontal signal continuity
disturbances nor
under
able tononlinear (insymbol
N
the signal I.
P (v) I NTRODUCTION
level domain) horizontal disturbances nor able to repro-
reproduce Abstract
transition
Abstract
asymmetries. The step response includes transmitter,
v (t) = v,P (t) + v,OBU
duce
channel,(t) +and
symbol v,R (t) = receiver
transition
possibly asymmetries. A
II. S IGNAL v,l sin(2πf
effects, TheM
allstep
ODEL t + φresult
response
ofv,lwhich v,l )includes
+inv,OBU (t) + v,Rchannel
transmitter,
a data-dependent (t).
signal and (4)
possibly receiver effects, all of which result in a data-dependent signal disturbance,the
disturbance, i.e., inter-symboll=0 interference. Moreover, the step response depends on i.e.
symbol vector in order toMoreover,
account for effects like symbol transition dependencies in the s in
intersymbol III.∞system.
J ITTER
interference.
InI.
N OISE D ECOMPOSITION
II NTRODUCTION
AND
I.the NTRODUCTION
the step response depends on the symbol vector
transmission
order to account for effects following, we assume the step response to depend only on the
like symbol transition dependencies in the transmission sys-
A. Source and Analysislast Domains
y(t) = and the
symbol ∆slastII.
[k]
II. · hS − the M
(t transition:
SsrIGNAL
symbol Tclk [k]
ODEL
Mstep −response
h [k], s)to+depend
y−∞ + (t). (1)
tem. In the following, we assume IGNAL ODEL onlyv on the last symbol
B. Decomposition Treeand the lastk=−∞ symbol transition:
∞
∞
y(t)
h sr =
(t − T
= 1 clk
∞ ∆
[k]
∞ [k]h··[k],
∆ss−
[k] h s)−
hsrsr (t n=T (t −−Tclk
sr[k]
Thclk h [k], −+
[k] s) y−∞{s[k],
h [k], + v (t).
∆s [k]}). (2) (1)
(1) (2)
y(t) (t∂− clk [k] − h [k], s) + y−∞ + v (t).
yR(h) (t) = k=−∞
k=−∞ ∆s [k] · n hsr (t − Tclk [k], {s[k], ∆s [k]}) · (−h,R [k])n . (5)
n! ∂t
Information Classification: General
n=1 k=−∞
We thus have up to N ⋅ (N
PAM NP (h)
PAM−1 – 1) step responses for a signal of PAM order N PAM .
h sr (t − T clk [k] − h [k], s) = h sr (t − T [k] − h [k],ϵ {s[k],
[k] ∆ϵsv[k]}).
(t ) fur- (2)
h The
(t − total
T horizontal
[k]
h [k] = h,P [k] sr+ h,OBUclk[k] + − h,R
∞
and
[k],
h [k] =s)vertical
= h sr source
(t − T disturbances
clk
Ah,lclk [k] −
sin(2πfh,l [k], {s[k],
h and
h Th,l [k] + φh,l ∆ s ) + h,OBU [k] +are
,
[k]}). respectively,
h,R [k] (2) (3)
ther decomposed as ∂
ỹR(h) (t) = − ∆s [k] · l=0hsr (t − Tclk [k], {s[k], ∆s [k]}) · h,R [k]. (6)
NP∂t(v) −1
k=−∞ NP (h)
NP
−1
(h) −1
h [k]
[k] =
v (t) =
= h,P [k]
(t)
[k] +
v,P +
+ h,OBU [k]
[k] +
v,OBU (t) +
+ h,R [k]
[k] =
v,R (t)
=
= A Av,l sin(2πf
Ah,l sin(2πf
sin(2πfh,l T
v,l t + φ
Th,l [k]
[k] +
+φ
v,l )h,l
+ v,OBU
φ )) +
(t) + +v,Rh,R
+ h,OBU [k]
(t).
[k] + [k] [k] (3) (4)
(3)
h h,P h,OBU h,R h,l h,l h,l h,l h,OBU h,R
l=0 l=0
l=0
III. N OISE D ECOMPOSITION
J ITTER
NP (v)AND
NP
−1
(v) −1
A. (t) =
Source (t)
and +
(t) Analysis (t) +
Domains (t) = A sin(2πfv,l t + φv,l ) + v,OBU (t) + v,R (t). (4)
vv (t) = v,P
v,P + v,OBU
v,OBU (t) + v,R
v,R (t) = Av,l
v,l sin(2πfv,l t + φv,l ) + v,OBU (t) + v,R (t). (3) (4)
l=0
B. Decomposition Tree l=0
III. J ITTER AND N
III. J ITTER AND OISE D
N OISE D ECOMPOSITION
ECOMPOSITION
A.
A. Source
Source and
and Analysis
Analysis Domains
Domains
∞ ∞
1 &
Rohde ∂ n | Signal model based a pproach to joint jitter and noise d ecomposition
Schwarz White paper 5
B.
B. Decomposition y
Decomposition Tree Tree
R(h) (t) = ∆ s [k] · n
hsr (t − Tclk [k], {s[k], ∆s [k]}) · (−h,R [k])n . (5)
n! n=1 k=−∞
∂tThe horizontal and vertical periodic components ϵh,P [k] and ϵv,P (t ), respectively,
are characterized by the amplitudes Ax,l , the frequencies fx,l and the phases ϕx,l for
l = 0, … , NP(x) – 1 and x = {h,v}, respectively, and Th,l [k] denotes the relevant point in time
for the horizontal periodic component l at symbol k. The terms ϵh,OBU [k] and ϵv,OBU (t ) des-
ignate other bounded uncorrelated (OBU) 1) components, and the random components
ϵh,R [k] and ϵv,R (t ) denote additive noise for the horizontal and vertical case, respectively.
At this point, we do not impose any statistical properties on the random components.
3 JITTER AND NOISE DECOMPOSITION
As introduced in section 2, various effects cause disturbances in transmitted data. The
transmitter and the channel have the most influence on the step responses that deter-
mine the data-dependent disturbance of the signal (intersymbol interference). Periodic
and OBU disturbances make up the remaining deterministic and bounded components.
Finally, there are random and unbounded disturbances such as thermal noise. Apart from
being deterministic and bounded, very little information is available about OBU com-
ponents at the receiver end. Therefore, we omit OBU components from now on. Their
influence will be visible in the extracted random components.
3.1 Source and analysis domains
With the exception of data-dependent components, all disturbances are either of hori-
zontal or vertical origin. The horizontal components constituting ϵh [k] originate at the
transmitter, whereas the vertical components in ϵv (t ) may also be added in the channel or
at the receiver end. The data-dependent components have their origin in the combination
of the data, i.e. the transmitted symbols, and the step responses that overlap to build the
signal. We thus define the source domain to be either “horizontal”, “vertical” or “data”.
Analyzing the disturbances in the time domain is referred to as jitter analysis. Timing
errors with respect to a reference clock signal are determined and analyzed. This is usu-
ally done by means of the TIE. However, the disturbances can also be analyzed in the
signal level domain, which is referred to as noise analysis. In this case, signal level errors
with respect to reference levels are determined and analyzed at symbol sampling times
given by a reference clock signal. Fig. 1 depicts the definition of the TIE and the level
error (LE). The choice between jitter analysis and noise analysis is a choice of the analysis
domain, which we accordingly define to be either “jitter” or “noise”.
Fig. 1: TIE and LE definition
Signal level
LE
Reference level
Threshold level
TIE
Sampling time
Clock time
Transition point
1)
The word “other” in OBU refers to the nonperiodicity of the disturbance and the word “uncorrelated” to the nonexistence of
any correlation between the disturbance and the symbol sequence.
6hsr (t − Tclk [k] − h [k], s) =NP
h(h) −1− Tclk [k] − h [k], {s[k], ∆s [k]}).
sr (t (2
h [k] = h,P 3.2
[k] +Decomposition
h,OBU [k] +tree
h,R [k] = Ah,l sin(2πfh,l Th,l [k] + φh,l ) + h,OBU [k] + h,R [k]
The decomposition tree of total jitter l=0 and total noise is depicted in Fig. 2. It is important to
NP (h) −1
remember that, as shown in Fig.
NP2,
(v)horizontal
−1 disturbances also have an influence in the
h [k] = h,P [k] +
noise [k]
domain
h,OBU +
and [k] =
vertical
h,R disturbancesA h,l sin(2πf
in the jitter Th,lt[k]
h,ldomain. + φh,lno
Thus ) +matter
h,OBU [k] + ah,R
whether jit-[k] (3
v (t) = v,P (t) + v,OBU (t) + v,R (t) = Av,l sin(2πf v,l + φv,l ) + v,OBU (t) + v,R (t).
ter or a noise analysis is performed,
l=0
l=0
all disturbance components need to be accounted
for, i.e. for any analysis domain, all
NP (v) −1 source
AND N OISE domains are relevant. Moreover, it is generally
notv,OBU
possible III.
to directly J
v,R (t) mapITTER
= properties Av,linsin(2πf D ECOMPOSITION
a source (4
v (t) = v,P (t) + (t) + v,ldomain
t + φv,lto
)+measurements
v,OBU (t) + in an(t).
v,R analy-
A. Source and Analysis
sis domain.Domains
Consider, e.g. thel=0 case of the horizontal random source disturbance ϵh,R [k]
only. Its effect on the signal level and thus the noise domain can be described by a Taylor
B. Decomposition Tree III. J ITTER AND N OISE D ECOMPOSITION
series of ϵh,R [k] around the mean. For the zero-mean case, we obtain
A. Source and Analysis Domains
∞ ∞
B. Decomposition Tree 1 ∂n
yR(h) (t) = ∆s [k] · n hsr (t − Tclk [k], {s[k], ∆s [k]}) · (−h,R [k])n . (4)
n=1
n! k=−∞ ∂t
∞
1 ∞
∂n
yR(h) (t) = ∆
∞s [k] · hsr (t − Tclk [k], {s[k], ∆s [k]}) · (−h,R [k])n .
n ∂
(5
n!
The first-order approximation
∂t
of equation
ỹR(h)
n=1(t) = −
k=−∞ ∆s [k] · hsr (t(4)−isTgiven by{s[k], ∆s [k]}) · h,R [k].
clk [k],
k=−∞
∂t
∞
∂
ỹR(h) (t) = − ∆s [k] · hsr (t − Tclk [k], {s[k], ∆s [k]}) · h,R [k]. (5) (6
k=−∞
∂t
We recognize that the horizontal random source disturbances are transformed to the
noise domain in a nonlinear manner that depends on the step responses and their deriva-
tives 2). Moreover, the inverse step of determining the source disturbance based on the
noise disturbance is nontrivial.
Fig. 2: Jitter and noise decomposition tree
a) Jitter analysis
Jitter analysis Unbounded
Random horizontal
jitter RJ(h)
Random jitter RJ
Random vertical
jitter RJ(v)
Total jitter TJ Bounded
Periodic horizontal
jitter PJ(h)
Periodic jitter PJ
Periodic vertical
Deterministic jitter DJ jitter PJ(v)
Data-dependent
jitter DDJ
Noise analysis Unbounded
Random horizontal
noise RN(h)
Random noise RN
Random vertical
noise RN(v)
Total noise TN Bounded
2)
For jointly stationary ϵh,R [k] and s [k], and Tclk [k] = kTs with the constant symbol period Ts, the first-order
Periodicapproximation
horizontal (5)
is cyclostationary. Moreover, for PAM signals of order NPAM > 2 and Gaussian distributed ϵh,R [k], ỹR(h) (t ) isPN(h)
noise not Gaussian dis-
tributed since the symbol difference Δ s [k] can take on values with different amplitudes.
Periodic noise PN
Periodic vertical
Rohde & Schwarz White paper | Signal
Deterministic model based a
noise DN noise PN(v)
pproach to joint jitter and noise ecomposition 7
d
Data-dependentDeterministic jitter DJ jitter PJ(v)
Data-dependent
Fig. 2: Jitter and noise decomposition tree jitter DDJ
b) Noise analysis
Noise analysis Unbounded
Random horizontal
noise RN(h)
Random noise RN
Random vertical
noise RN(v)
Total noise TN Bounded
Periodic horizontal
noise PN(h)
Periodic noise PN
Periodic vertical
Deterministic noise DN noise PN(v)
Data-dependent
noise DDN
4 JOINT JITTER AND NOISE ANALYSIS FRAMEWORK
We propose a framework that encompasses a joint jitter and noise analysis and is based
on the signal model introduced in section 2. Contrary to state-of-the-art methods, which
transition from a signal to a TIE representation as a first step, our approach is based
directly on the total signal. It thus does not discard any information present in the (total)
signal by using a condensed representation like TIEs do. The core of our framework
operates independently of the analysis domain (jitter or noise) and considers all source
domains (horizontal, vertical and data). The jitter and noise results are derived from a joint
set of estimated model parameters.
Conceptually, the first step of the framework is to recover a clock that makes it possible
to decode, i.e. recover, the data symbols. Based on the recovered clock signal and the
decoded symbols, a joint estimation of model parameters is performed. Specifically, a
coarse, partly spectrum based estimator recovers rough estimates of the frequencies and
phases of the vertical periodic components as well as their number. These are then used
for a fine, least-squares estimator that jointly determines the step responses and the peri-
odic vertical components. The horizontal periodic components are similarly estimated.
These estimates, along with the recovered symbols and clock, allow synthesis of signals
with only selected disturbance components, e.g. the data-dependent signal. We then use
the total and synthesized signals to transition to the analysis domains and calculate jitter
and noise values, i.e. TIE and LE values, respectively. Fig. 3 shows a block diagram of the
framework.
8Fig. 3: Joint jitter and noise analysis framework
Clock and data recovery Estimation: step responses Signal synthesis Jitter analysis
and periodic components
Coarse estimation Fine estimation
SIGDD
SIGT EST_PV_ SYNTH_DD
CLK_REC DECODE EST_SR_PV TIEX
COARSE
TIE
SIGP(v)
SYNTH_PV
TIEP(h) + R SIGDD + P(h)
EST_PH_
EST_PH SYNTH_DD
COARSE
LEX
LE
Noise analysis
4.1 Estimation of model parameters
The fine estimator that performs a joint estimation of the model parameters is at the core
IV. J OINT
of the framework and isJ derived N OISE
as follows.
ITTER AND A NALYSIS
First, we need toFtreat the remaining horizon-
RAMEWORK
A. Estimation tal disturbances ϵ
of Model Parametersh, rem[k] as additive noise for the estimator. Their influence in the signal
level domain is quantified as in (5) by linearizing (1) with respect to ϵh, rem[k]:
∞
IV.
IV. J OINT J ITTER AND N OISE A NALYSISJ OINT J ITTER
F RAMEWORK AND N
(t) ≈ ỹh,rem (t) = − OISE A∆
ˇNALYSIS F RAMEWORK ˇ
yh,rem s [k] · h(t − Ťclk [k], {š[k], ∆s [k]}) · h,rem [k] (6)
A. Estimation of Model Parameters
Model Parameters k=−∞
∞
where š [k]Nis the decoded
P (v)−1
symbol, Δ̌ s [k] the corresponding symbol difference,
IV.
ˇ s [k] · h(t − Ťclk t,J{š[k],
h((v) šOINT
[k]=
,∆ˇ Jss [k]})
[k]
∞
ITTER
) the AND [k]N OISE
impulse ˇ A NALYSIS
response, and Τ̌ clk F
(7) [k]RAMEWORK
an estimated clock time.
rem (t) ≈ ỹh,rem (t) = −
y
∆ v,P (t) ≈ ỹP[k],
(t) ≈ ỹh,rem (t) = −(t) · A v,lˇsin(2π fv,l t + φ̌v,l ) + Av,l (∆
h,rem ˇ f,v,l 2πt +
∆s [k] · h(t − Ťclk [k], {š[k], ∆s [k]}) · h,rem [k] ∆φ,v,l ) cos(2π fˇv,l t + φ̌v,l )
(7)
A. Estimation
k=−∞ ofh,rem
Model Parameters l=0k=−∞
NP (v)−1
IV. J OINT J ITTERBy ANDfurther approximating
N OISE A NALYSIS theFvertical
RAMEWORK sinusoidal components, we obtain
∞
Estimation
(t) = ofAv,l
Model
sin(2πParameters
fˇv,l t + φ̌v,l ) + Av,l (∆
NPf,v,l
(v)−1
2πt +∆ ∞
) cos(2π fˇv,l t + φ̌v,l ) (8) ˇ
y(t) ≈ ỹ(t)
= ∆ˇ φ,v,l
[k] · h (tˇ−s [k]Ť · [k], {š[k],
l=0 v,P (t) ≈ ỹPy(v) = ≈ ỹh,remA
(t)(t)
h,rem − fˇv,l t∆
(t)v,l=sin(2π
s sr
+ φ̌clk h(t
) + −
A Ťclk∆
v,l (∆
s [k]})
[k], +∆
{š[k], ˇy−∞ + ỹ· P (v) (t)[k]
[k]})
f,v,l 2πt + ∆sφ,v,l ) cos(2π
+ v,R (t) + ỹh,rem
h,remfˇv,l t + φ̌v,l ) (7) (t).
(8)
k=−∞
v,l (7)
k=−∞
∞
l=0
∞
IV. J OINT J ITTER AND N OISE A NALYSIS F RAMEWORK
ˇ s [k] · y
∆ hh,rem Ťclk≈
sr (t −(t)
ˇ s=
{š[k],(t)
[k],ỹh,rem ∆ [k]}) ˇ sỹ[k]
− + y−∞ ∆
+ P (v)·(t)
h(t+−
v,RŤ(t)
clk [k], {š[k],
+ ỹh,rem ˇs(9)
(t). ∆ [k]})· h,rem [k] (7)
k=−∞
A. Estimation
∞
oftheModel
NP (v)−1
with frequenciesParameters fv,l = ƒ̌v,l + Δf,v,l andĥthe sr phases ϕv,l = ϕ̌ v,l + Δϕ,v,l split into coarse and
k=−∞
y(t) ≈ ỹ(t)
≈ỹP= ˇ [k] · hAsr (tparts − Ťclk flˇv,l t0,+…φ̌, v,l
= {š[k], N∆ˇ)x̂s+= 1.v,l ŷThis =2πt +A
+
ỹP+y ˇv,lỹth,rem (9)
(8)
v,P (t) (t) = ∆sremaining
(v) v,l sin(2π for[k], [k]})
P(v) – A
+ (∆yf,v,l
−∞ yields
−∞ the ∆(t)
following
(v) φ,v,l+) cos(2π
v,R (t) f +
approximation φ̌(t).
+ of the
v,l ) total
NP (v)−1 sr k=−∞+ signal:
ĥ l=0 ∞ p̂
x̂
= ŷ−∞ = A y ≈A − (10) ˇ [k] · h(t − ˇ s [k]}) · h,rem [k]
v,P (t) ≈ ỹP (v) (t) = p̂Av,l sin(2π
∞ fˇv,l tyh,rem
+ φ̌v,l(t)) + ỹh,rem
v,l (∆
(t)
f,v,l
=2πt
∞
+ ∆ ∆
φ,v,ls) cos(2π fv,l t clk
ˇ Ť +[k], φ̌v,l{š[k],
) ∆
(8)
ĥsr ˇ k=−∞
y(t) ≈l=0 ỹ(t) = ∆ˇ s [k] · hsr (t −ŷŤDD [k],
clk(t) = {š[k],
∆∆ ˇ s [k] +
s [k]}) ·+ĥsry(t−∞ −+Tcdr ỹP[k],
(v) (t)
∆ˇ s+ v,R
[k]) +(t) + ỹh,rem (t). (10)
ŷ−∞ (9)
(8)
∞ x̂ = ŷ −∞ = A y
ŷDD (t) = ∞ ˇ s [k] · ĥsr (tk=−∞
∆ − Tcdr [k], ∆ ˇ s [k]) + ŷ−∞NP (v)−1 k=−∞ p̂ (11)
NP (v) −1
y(t) ≈ ỹ(t) = k=−∞∆ ˇ s [k] · hsr (tv,P− (t)
Ťclk≈
[k], ỹP{š[k],
(v) (t) ∆ˇ= s [k]}) + y
A−∞
v,l +
sin(2πỹ fˇ (t)t + φ̌
P (v)v,l v,R
v,l )(t)
++ Av,lỹh,rem
(∆f,v,l (t).2πt + (9)∆φ,v,l ) cos(2π fˇv,l t + φ̌v
NP (v) −1 The coarse estimates
ŷ (t) and
= the
ĥ above  approximations
sin(2π f ˆ t + can
φ̂ be). used to perform a (fine) joint
k=−∞ ∞
P (v)
l=0 sr v,l v,l v,l
ˆ least-squares estimation of
· −∞the step
− responses and the vertical periodic components:
ŷP (v) (t) = Âv,l sin(2π fv,l t + φ̂v,l ). (t) =
ŷDD x̂∆ˇ= s [k] ŷ ĥsr (t
l=0 = TA +
(12)[k], ˇ
cdr y ∆s [k]) + ŷ−∞ (10)
(11)
l=0 ∞ p̂
ĥ sr
k=−∞
ˇ ˇ s [k]}) + y−∞ + ỹP (v) (t) + v,R (t) + ỹh,rem (
y(t) ≈ ỹ(t) =ŷ−∞ P N ∆ s A y sr −
[k]
−1 +· h (t T [k],−{š[k],
Ťclk[k] T ∆ (10)
x̂ = = (v) [k] (9)
(v)
Tsig,T [k] − Tsig,DD+P (v) [k] sig,T (13) sig,DD+P
k=−∞ ∞ ˆ
ŷP (v) (t)p̂= Â v,l sin(2π fv,l t + φ̂v,l ).
ˇ s [k] · ĥsr (t − Tcdr [k], ∆ ˇ s [k]) + ŷ−∞ (12)
ins B. Analysis Domains ŷDD (t) = l=0 ∆ (11)
∞ with the estimate vector x̂ containing the Nĥ
k=−∞
sr srestimates of the step responses in the
ˇ
vector ĥ , the (v) −1 signalˇvalue estimate
NPinitial x̂ = ŷ −∞ , and
=A +
the y estimates (11) of the frequencies,
TIEX [k] ŷDD
= T(t)sig,X=[k] − Tcdr∆ [k]s [k] ·srĥsr (t −Tsig,T
Tcdr [k][k],− ∆T s [k]) + (14)
ŷ−∞ [k] (13)
TIE [k]
sig,DD+Pˆ = T [k] − T [k]
+ φ̂v,lp̂).components in the vector p̂ .
(v)
(12)
k=−∞phases,
ŷP (v) (t)and = amplitudes Âv,lofsin(2π
the verticalfv,l t periodic
X sig,X cdr
B. Analysis DomainsNP (v) −1 l=0
TIET [k] TIED
= (t)
ŷP (v) =[k] + TIERÂ [k] sin(2π fˆ t + φ̂ ). ∞ (15) (12) d ecomposition 9
v,l Rohde & Schwarzv,l v,l paper | Signal ˇ(16) ˇ sjitter
TIEDD+P (h) [k] = TIEDD [k]l=0 + TIEP (h) [k] ŷ White
(t) = TIE
Tsig,T [k] − Tsig,DD+P
DD T [k] ∆ TIE
=s model
[k] ·D
[k] ĥbased
[k]
sr (t +a
− TIE
pproach
TcdrR to[k]
[k], joint
∆ [k])and+noise
ŷ−∞ (13)
(v)
TIEDD+P (v) [k] = TIEDD [k] + TIEP (v) [k] TIE [k]
XTIEDD+P = T k=−∞
sig,X(h)
[k]
[k] −
(17)
= TTIE
cdr [k]
DD [k] + TIE P (h) [k] (14)y(t) ≈
y(t) ≈ ỹ(t)
ỹ(t) = = ∆ss[k]
∆ [k] ·· hhsrsr(t (t − − TTclk [k], {š[k],
clk[k], {š[k], ∆ ∆ss[k]})
[k]}) + + yy−∞ −∞ + + ỹỹPP(v)
(v)(t)
(t) ++ v,R
v,R(t)
(t) +
+ ỹỹh,rem (t). (9)
h,rem(t). (9)
∞ IV. J OINT J ITTER AND N OISE A NALYSIS F RAMEWORK
k=−∞
k=−∞
≈ ỹ(t) =of Model
)stimation ∆ˇ s [k] · hsr (t − Ťclk [k], {š[k], ∆ ˇ s [k]}) + y−∞ + ỹP (v) (t) + v,R (t) + ỹh,rem (t). (9)
Parameters
IV. J OINT TheJmatrix ITTERAAND + is the
A NALYSIS
Npseudoinverse
OISE
of the F RAMEWORK
observation matrix and the vector y contains
k=−∞ ĥ
ĥsrsr
A. Estimation ofIV. Model J OINTParameters
J ITTERsamples ANDofNthe OISE total A
x̂
x̂
signal.
NALYSIS
==
ŷŷ−∞Based on these
F RAMEWORK
A++yy
estimates and the recovered (or input) clock
(10)
(10)
−∞ = =A
signal ∞ Τ
cdr [k] ,
the data-dependent (DD) and the periodic vertical (P(v)) signals can be
imation of Model Parameters synthesized ˇ ĥ as
sr p̂
p̂ ˇ
yh,rem (t) ≈ ỹh,rem (t) = − ∆ s [k]· h(t −+Ťclk [k], {š[k], ∆s [k]}) · h,rem [k] (7)
x̂ = ŷ ∞
−∞ = A y (10)
yh,rem (t) ≈ ỹh,rem (t)∞= − p̂ k=−∞
∞ ∞∆ ˇ s [k] · h(t − Ťclk [k], {š[k], ∆ ˇ s [k]}) · h,rem [k] (7)
ŷŷDD ˇˇ [k] ·· ĥĥsrsr(t(t − − TTcdr ˇ
ˇ (11)
(11)
DD(t) (t)ˇ k=−∞
== ∆∆ [k] ˇ [k],
[k], ∆ ∆ss[k])
[k]) + + ŷŷ−∞
yh,rem (t) ≈ N (t) = −∞
ỹh,rem
P (v)−1 ∆s [k] · h(t −ssŤclk [k], {š[k], cdr ∆ s [k]}) · h,rem [k]
−∞
(7)
k=−∞
k=−∞
v,P (t) ≈ ỹP (v) (t) =
ˇ t + φ̌v,l ) + Av,l (∆f,v,l 2πt + ∆φ,v,l ) cos(2π fˇv,l t + φ̌v,l ) (8)
NPA sin(2π ˇfsv,l
k=−∞
ŷDD (t) =v,l
(v)−1 ∆ [k] · N ĥNPsrP
(v)−1
(t
(v) −1
− Tcdr [k], ∆ ˇ s [k]) + ŷ−∞ (11)
= fˇv,l t + φ̌Â ). ∆φ,v,l ) cos(2π fˇv,l t + φ̌v,l ) (8) (12)
l=0
v,P (t) ≈ ỹP (v)
NP(t)(v)−1= A
ŷŷPPv,l(v)
k=−∞ (v) sin(2π
(t)
(t) = ) sin(2π
Âv,lv,l
v,l + Av,l (∆
sin(2π ffˆˆv,l tt +
v,lf,v,l+2πt v,l+
φ̂φ̂v,l ). (12)
(10)
P (t) ≈ ỹP (v) (t) =∞ Av,l fˇv,l t + φ̌v,ll=0
l=0 NP (v) −1
sin(2π ) + Av,l (∆f,v,l 2πt + ∆φ,v,l ) cos(2π fˇv,l t + φ̌v,l )
l=0
(8)
(t) ≈ ỹ(t) = ˇ
ŷ
∆ [k] (t)
· h =
sr (t − Ť Â
[k], sin(2π
{š[k], ∆ˇ fˆ t
[k]}) + φ̂
+ y ). + ỹ (t) + (t) + ỹ (t). (12) (9)
Theclkhorizontal periodic
P
s (v) v,l sv,l v,l −∞
l=0
∞ components are estimated v,R
P (v)
in a two-step h,rem
approach as above. More
k=−∞
y(t) ≈ ỹ(t)
∞
= ∆ˇ s [k] · hprecisely,
l=0
sr (t − Ťclk the [k], TT sig,T [k]
sig,T
{š[k],
estimation [k]
∆ −
−based
ˇ is TTsig,DD+P
s [k]})
sig,DD+P
+ony−∞ the(v)+
(v) [k]
[k]ỹ difference
time P (v) (t) + v,R (t) + ỹh,rem (t). (9) (13)
(13)
ˇ
B.
) ≈ ỹ(t)B.=Analysis
Analysis ˇ Domains
∆ sDomains
k=−∞
[k] · hsr (t − Ťclk [k],[k] {š[k], (9)
Tsig,T − Tĥ∆ sr
s [k]}) + y
sig,DD+P (v) [k]
−∞ + ỹP (v) (t) + v,R (t) + ỹh,rem (t). (13) (11)
k=−∞ x̂ = ŷ−∞ ĥ =A y +
(10)
alysis Domains where Τ sig , T [k] p̂ and Τsig ,
sr
DD + P(v) [k] are the level crossing time of the total and the
x̂ TIE=XXŷ[k]
TIE [k]
−∞= = T=
Tsig,XA+[k]
sig,X y −
[k] − TTcdr cdr[k]
[k] (10) (14)
(14)
DD + P(v) ĥ signal,
sr respectively. Thus an estimate of the horizontal periodic (P(h)) contribu-
tion ∞x̂to =the ŷlevel = p̂
−∞ crossing A+ ytimes is obtained and the DD + P(h) signal can(10) be synthesized.
TIE [k]ˇ = T sig,X [k] − T cdr [k] ˇ (14)
(11)
s [k]p̂· ĥsr (t − Tcdr [k], ∆s [k]) + ŷ−∞
ŷDD (t) = X ∆∞
Summarizing, TIE
TIEare
we TT [k]
[k]able =
=to TIE
TIE D[k][k] +
synthesize
D + TIETIE
signals [k]with a subset of the underlying source(15)
R[k]
R (15)
k=−∞
ŷDD (t) = ∆ ˇ s [k] · ĥsr (t − Tcdr [k], ∆ ˇ s [k]) + ŷ−∞ (11)
Ndisturbances.
∞
P (v) −1TIE TIEDD+P DD+P(h)
These [k] what-if
(h) [k] = = TIE signal reconstructions
TIE DD[k]
DD [k] + + TIE TIEPP(h) allow us to provide new measure- (16)
(h)[k]
[k] (16)
ments ˇ k=−∞ with selectable disturbances. ˇ For example, it is possible to construct an eye
ŷDD (t) = TIE ∆ [k][k] = · TIE
ĥ (t −[k] T
ˆ + TIE
[k], ∆ [k][k]) + ŷ (15)
(11)
(12)measure- (17)
NTIE
TIE =+TIE TIE + TIE TIE (17)
ŷP (v) (t) = diagram Âwith sr
sin(2π f[k]cdr
v,l t= φ̂v,lDD).
T s D [k] R s −∞
v,l−1
DD+P
DD+P data(v) disturbances
(v) and[k]
DD [k] +
chosen (v)[k]
periodic
P
P(v) [k]disturbances only. The
k=−∞ [k]
P (v)
TIEDD+P = TIE TIE
TIE [k] + TIE [k] (16)
ŷP (v) ments
NP(t)
(v) −1
l=0
= carried Â
(h) out DD
v,l in
D the =
[k]
[k]
sin(2π
D = fˆTIE
TIE
analysis P
v,l t DD+ domains
(h)
DD φ̂[k]
[k] + TIE
+
v,l ). TIE
are PPdescribed
[k].
[k]. in the following sections. (12) (18)
(18)
TIEDD+P (v) [k] = TIE DD [k] + TIE P (v) [k] (17)
ŷP (v) (t) = 4.2 Analysis Âv,ll=0
sin(2π domains fˆv,l t + φ̂v,l ). (12)
TIE T D [k] = TIEDD
sig,T [k] − T [k] +(v)
sig,DD+P TIEP [k].
[k] (13)
(18)
Based on the total and the synthesized signals, we move away from a signal-centric view
l=0
Tsig,T LE
LE−
[k] X[k]
[k]sig,DD+P
== VVXX[k] [k] − − VVref ref[k][k] (13) (19)
(19)
nalysis Domains the T [k]
X
in order to obtain commonly(v) known jitter/noise values. Specifically, we determine TIEs
defined
T [k]as− T (13)
B. Analysis Domains sig,T sig,DD+P (v) [k]
LE X [k] = VX [k] − Vref [k] (19)
alysis Domains TIEX [k] = TLE
LE TT [k]
sig,X [k] − LE
[k] =
= LE [k] + LE
[k] +
TcdrDD[k] LERR[k]
[k] (14) (20)
(20)
(12)
TIEX [k] = Tsig,X [k] − Tcdr [k] (14)
where
LE Τsig,XLE
T [k] =
[k]Dis[k]
the+level
LERcrossing
[k] time of the signal X = {DD, DD + P(h), DD + P(v), D, T }.
(20)
TIE
ForXthe
TIE remaining
[k][k]=T TIE[k]TIEs,
− we
Tcdrimpose
[k] the following relationships: (14)
T =sig,X D [k] + TIER [k] (15)
TIEDD+P (h) [k]TIE=T [k]TIE=DDTIE [k]D+[k]TIE
+ PTIE
(h) [k]
R [k] (16) (15)
TIEDD+P
TIETDD+P
TIE [k][k] = TIE=
[k]
TIE TIE
[k]DD TIE
+ [k] + TIE[k]P (h) [k] (17) (16)
=(h) D [k] + TIE R [k] (15)
(v) DD P (v)
TIE TIEDD+P
TIE D [k]
[k] =
= (v) TIE=DD
[k]
TIE TIE
[k][k] +
TIE
+DD TIE+
[k] TIEP (v) [k]
P [k].
[k] (18) (17)
(16)
DD+P (h) DD P (h)
TIEDD+P (v) [k]TIED [k]DD
= TIE TIE
=[k] +DDTIE [k]P (v) TIEP [k].
+ [k] (17) (18) (13)
TIELE = TIE
D [k][k] VDD +VTIE
[k]−analysis,P [k]. (18)
(19) to refer-
X for=the
Similarly, [k]
noise
X ref [k] we determine LEs that are defined with respect
ence signal levels:
LEX [k] = VX [k] − Vref [k] (19)
LE
LEXT[k]
[k]=
=VLE −+
X [k][k] VrefLE
[k] [k] (19)
(20) (14)
D R
where VLE
X [k]
T [k] LED [k]
= signal
is the + LE
value (20)
R [k] at the symbol sampling times for the signal
obtained
X = {DD, DD + P(h), DD + P(v), D, T } and Vref [k] is the reference value that depends on
LE T [k] = LED [k] + LER [k] (20)
the current symbol value.
10LEX [k] = VX [k] − Vref [k] (19)
The remaining LEs are decomposed as in the TIE case, i.e. we use
LET [k] = LED [k] + LER [k] (20) (15)
TABLE I
S YNTHETIC S ETTINGS
and thus the remaining LEs can be computed.
Property Scenario 1 Scenario 2
5 MEASUREMENTS
Data rate 5 Gbps 5 Gbps
Sample rate 40 GSa/s 20 GSa/s
Number of symbols 300 k Variable
Modulation NRZ, reference levels -1 and +1 NRZ, reference levels -1 and +1
Symbol pattern Various measurements of interest can be obtained based on theRandom
Random TIEs and the LEs intro-
Rate modulationduced in yection 4.2. TheTriangular
TIE statistics
SSC can
with
TABLE I either
-5000ppmbe computed
@ 33 kHz for
Noneall values or for
Step response asymmetry Fall time > rise
specific subsets, e.g. rising or Sfalling time
edges.
YNTHETIC
None
Similarly, the LE statistics can be computed
S ETTINGS
Periodic horizontal disturbance h,P 100 mUI pp @ 20 MHz 200 mUI pp @ 20 MHz
for certain symbol values only. The measurements are briefly presented in the following
Periodic
Propertyvertical disturbance v,P 0.1 pp @ 100
Scenario 1 MHz 0.02 pp @220 MHz
Scenario
Random subsections.
horizontal disturbance h,R 205 mUI
Data rate Gbps rms 505 mUI
Gbps rms
Random
Sample vertical
rate disturbance v,R (SNR) 3040dB GSa/s 3520dB
GSa/s
5.1 Statistical values and histograms
Number of symbols 300 k Variable
Modulation We calculate common statistical valueslevels
NRZ, reference like the minimum,
-1 and +1 maximum,
NRZ, peak-to-peak
reference levelsvalue,
-1 and +1
Symbol patternpower and standard deviation Random
V. (SD)
M EASUREMENTS Random
for both the TIEs and the LEs. Furthermore, we use
Rate modulation Triangular SSC with -5000ppm @ 33 kHz None
histogram plots to illustrate the shape of their distribution.
A. Statistical Values
Step response and Histograms Fall time > rise time
asymmetry None
Periodic horizontal disturbance h,P 100 mUI pp @ 20 MHz 200 mUI pp @ 20 MHz
B. Duty-Cycle and5.2Level
Periodic vertical DutyDistortion
cycle
disturbance v,Pdistortion and
0.1 level
pp @distortion
100 MHz 0.02 pp @ 20 MHz
Random horizontal
In thedisturbance
context of 20 mUI rms
a jitter analysis,
h,R the duty cycle distortion (DCD) is50calculated
mUI rms as
Random vertical disturbance v,R (SNR) 30 dB 35 dB
DCD = max {E{TIEDD |hsr,k }} − min {E{TIEDD |hsr,k }} (16) (2
k=0,...,Nsr −1 k=0,...,Nsr −1
V. M EASUREMENTS
A. StatisticalLD[l]
Values E {x} denotes the expectation of the random variable x and hsr,k the occurrence
= and
where
E{LE Histograms
DD |level = l + 1|} − E{LEDD |level = l}, l = 0, . . . , NPAM − 1. (2
of the k-th step response with k = 0, … , Nsr – 1. The DCD value measures the impact of
B. Duty-Cycle and Level Distortion
C. Autocorrelationtwo Functions and the
effects. First, Power Spectral
differences in the Densities
various step responses, e.g. rising and falling step
responses in the case of non-return-to-zero (NRZ) signals (NPAM = 2), and second, a mis-
D. Symbol Error Rate Calculation
match in the signal level thresholds used for determining the TIEs.
DCD = max {E{TIEDD |hsr,k }} − min {E{TIEDD |hsr,k }} (
k=0,...,Nsr −1 k=0,...,Nsr −1
The level distortion (LD)
for the eye opening l
SER(u) = Phist [lbin ] · pR (v − xhist [lbin ]) dv (2
LD[l] = E{LEDD |level = v∈e(u)
lbin l + 1|} − E{LEDD |level = l}, l = 0, . . . , NPAM − 1. (17) (
E.C. Jitter and Noise Characterization
Autocorrelation Functions and Power at a Target
Spectral SER Densities
is used in the context of noise analysis. It characterizes vertical distortion of the eye open-
D. Symbol Error ing
Rate
dueCalculation
to data-dependent disturbances.
TJ@SER = DJ + σ · c(SER) (2
5.3 Autocorrelation functions and powerδδspectral
RJ
densities
Further information can be derived from a spectral view of the TIE and LE tracks. For
F. Framework Results SER(u) = Phist [lbin ] · pR (v − xhist [lbin ]) dv (
TIEs, a spectral view is not straightforward to compute since symbol transitions do not
G. Exemplary Analysis occur between every two
lbin v∈e(u)
symbols. Usually, this is handled by linear interpolation between
E.Other
Jitterstatistics:
and Noise Characterization
given TIEs. We proposeata different
a Targetapproach
SER in order to avoid the introduction of artifacts.
First, compute the autocorrelation function of the TIE track by only considering the given
TIE values, then compute the power spectral density (PSD) based on the autocorrelation
H. Signal Length Influence TJ@SER
function, thereby obtaining = DJview
a spectral (
δδ + without any interpolation artifacts.
σRJ · c(SER)
VI. C OMPARISON AGAINST C OMPETING A LGORITHMS
F. Framework Results5.4 Symbol error rate calculation
VII. C ONCLUSION
SER plots visualize how varying a single parameter affects the number of correctly
G. Exemplary Analysis
decoded symbols by plotting ACKNOWLEDGMENT
the SER against the parameter in question. For a jitter anal-
Other statistics:ysis, the SER is plotted against the sampling time offset used to decode the symbols,
while for a noise analysis, the parameter is the decision threshold level of the considered
H. Signal Length eye. Symbol errors occur, e.g. when signal transitions take place after the corresponding
Influence
symbol has been sampled or signal levels do not cross the threshold level of the desired
symbol,VI.
respectively. The SERAisGAINST
C OMPARISON C OMPETING
equivalent to the BER forANRZ
LGORITHMS
signals.
VII. C ONCLUSION
Rohde & Schwarz White paper | Signal model based a ecomposition 11
pproach to joint jitter and noise d
ACKNOWLEDGMENTSymbol pattern Random Random
Rate modulation Triangular SSC with -5000ppm @ 33 kHz None
Step response asymmetry Fall time > rise time None
Normally,
Periodic horizontal disturbance h,P to measure
100 mUIthe pp SER,
@ 20theMHzsystem under test is200
usedmUItopp
transmit a known symbol
@ 20 MHz
Periodic vertical disturbance
sequence
v,P 0.1 pp @ 100 MHz 0.02 pp @ 20 MHz
and to evaluate the symbol errors in the received signal. Repeating this mea-
Random horizontal disturbance h,R for a 20
surement mUI rms
number 50 mUI
of values of the desired parameter rmsthe SER plots. Such a
yields
Random vertical disturbance v,R (SNR) 30 dB 35 dB
measurement setupTABLEusually
I interferes with the regular operation of the system under test
and may Stherefore
YNTHETICbe S ETTINGS
difficult to accomplish or even yield different results. Additionally,
SER measurements take a very long time for low SER values like 10–12.
Property Scenario 1 V. M EASUREMENTS Scenario 2
Data rate 5 Gbps 5 Gbps
A. rate
Sample Statistical Values and Histograms
We propose
40 GSa/s a method to calculate SER plots20based GSa/s on the jitter or noise values, i.e. the
Number of symbols TIEs300
or LEs,
k respectively, and their statistics.Variable
To calculate an SER plot, first select a histo-
B. Duty-Cycle and Level Distortion
Modulation NRZ,
gram thatreference levels
covers all -1 anddeterministic
desired +1 NRZ, reference
components levels
with bins-1representing
and +1 the desired
Symbol pattern Random Random
analysis domain and a set of probability distribution parameters describing the desired
Rate modulation Triangular SSC with -5000ppm @ 33 kHz None
Step response asymmetry DCD = random components
Fall time > rise timein the
{E{TIE
same analysis domain. For example, when the SER for all
minNone{E{TIEDD |hsr,k }} (21)
max DD |hsr,k }} −
Periodic horizontal disturbance h,P signal
100disturbances
k=0,...,N sr −1 pp @ 20plotted
mUI MHz against the sampling
k=0,...,N200 timeppoffset
sr −1mUI @ 20 is desired, we select the TIE
MHz
Periodic vertical disturbance v,P 0.1 pp @
histogram 100 MHz
covering 0.02 ppand
all deterministic components @ 20theMHz
TIE distribution covering all ran-
Random horizontal disturbance h,R dom20components.
mUI rms By choosing histograms and 50 mUI rms
distributions that cover only part of the
Random vertical disturbance v,R (SNR) 30 dB 35 dB
LD[l] = E{LEDD |levelsignal
measured =l+ 1|} − E{LEwhat-if
perturbations, DD |level
SER= l}, can
plots l =be0,generated
. . . , NPAM
that 1. the user(22)
− allow
to gain insight into how the SER would change if the user were to resolve certain signal
C. Autocorrelation Functions and issues.
integrity PowerThe Spectral
information Densities
stored in the histogram is, for every bin l bin , the prob-
V. M EASUREMENTS
ability of hitting that bin P hist [l bin] and the position of that bin x hist [l bin]. Assuming a
D. Symbol Error Rate Calculation
Statistical Values and Histogramsprobability distribution pR of the random components, we get the SER
Duty-Cycle and Level Distortion
SER(u) = Phist [lbin ] · pR (v − xhist [lbin ]) dv (23)
(18)
lbin v∈e(u)
E. Jitter DCD max {E{TIEDDat|hsr,k
= Characterization
and Noise where e (u) = {xa
k=0,...,Nsr −1
}} − SER
Target
: position
min {E{TIEDD |hsr,k }}
x generates
k=0,...,N
(21)
sr −1 a decoding error for parameter u } holds informa-
tion about the error conditions. Note that while random disturbances are often assumed
to be Gaussian distributed, this may not be the case in certain scenarios due to the non-
LD[l] = E{LEDD |levellinear
=l+ 1|}TJ@SER
transformation = DJ
− E{LEfrom
DD |level
δδ +=
source l},
σtoRJ · lc(SER)
= 0,domain
analysis . . . , N(see − 1. 3.2).
PAM section (22) (24)
Autocorrelation Functions
F. Framework Resultsand Power Spectral
5.5 Jitter Densities
and noise characterization at a target SER
Symbol
G.Error Rate Calculation
Exemplary Analysis A number of secondary measurements are derived from the SER calculation. One of
these is TJ@SER, the total jitter at a target SER. This value represents the jitter budget
Other statistics:
has to anticipate when trying to reach a target SER. It can be iden-
that a system designer
SER(u) =tified as
P the[lwidth
hist bin ] · of the SER
R p (vvs.−sampling
x [l time
hist bin ]) dvoffset plot at the desired SER.
(23)
H. Signal Length Influence lbin v∈e(u)
Another example is the Dual-Dirac value DJδδ from the Dual-Dirac model. The Dual-Dirac
VI.model
Jitter and Noise Characterization C OMPARISON
at a assumes thatAthe
Target SER GAINST C OMPETING
deterministic jitter canAonly
LGORITHMS
take on two discrete values and that
the relationship VII. C ONCLUSION
ACKNOWLEDGMENT
TJ@SER = DJδδ + σRJ · c(SER) (24) (19)
Framework Results holds, where the value c (SER) depends solely on the target SER. Sufficiently low SERs
Exemplary Analysis need to be ensured for the above relationship to hold, otherwise the (Gaussian) random
jitter does not dominate the total jitter.
Other statistics:
For noise analysis, we compute the EH@SER value, which gives the eye height that is
Signal Length Influence achieved at a target SER.
VI. C OMPARISON AGAINST C OMPETING A LGORITHMS
VII. C ONCLUSION
ACKNOWLEDGMENT
126 FRAMEWORK RESULTS
In this section, we present measurement results from the joint jitter and noise analy-
sis framework presented in section 4. To make the vast amount of measurements and
comparisons more accessible, we present a single scenario in section 6.1 and use a
majority of the available results to track down a signal integrity issue and understand its
implications. In section 6.2, we show how the length of the input signal affects the mea-
surements and thus the framework’s performance. The scenarios used in this subsection
are summarized in Table 1. We use a second order phase-locked loop (PLL) with a band-
width factor of 500 and a damping factor of 0.7 for these scenarios.
Table 1: Settings for synthetic scenarios
Property Scenario 1 Scenario 2
Data rate 5 Gbps 5 Gbps
Sample rate 40 Gsample/s 20 Gsample/s
Number of symbols 300k variable
Modulation NRZ, reference levels: –1 and +1 NRZ, reference levels: –1 and +1
Symbol pattern random random
triangular SSC with –5000 ppm
Rate modulation none
at 33 kHz
Step response asymmetry fall time > rise time none
Periodic horizontal disturbance ϵh, P 100 mUI pp at 20 MHz 200 mUI pp at 20 MHz
Periodic vertical disturbance ϵv, P 0.1 pp at 100 MHz 0.02 pp at 20 MHz
Random horizontal disturbance ϵh, R 20 mUI RMS 50 mUI RMS
Random vertical disturbance ϵv, R
30 dB 35 dB
(SNR)
6.1 Example analysis
Throughout this section, we use scenario 1 from Table 1, i.e. a synthetic 5 Gbps NRZ data
signal with random symbols to show the framework’s capabilities.
Fig. 4 shows the data rate over time as recovered by the clock and data recovery (CDR).
At the very start of the signal, the CDR’s locking behavior can be observed. After the
lock time (indicated by the red line), the CDR follows a triangular modulation profile of
–5000 ppm at 33 kHz. Such a modulation profile is commonly used for spread spectrum
clocking (SSC) in, e.g. USB 3.0. Note that CDR misconfiguration happens quite frequently
and is easily caught at this stage.
Fig. 4: Symbol rate from CDR
1.005
Rate [nominal rate]
1.000
0.995 Symbol rate
Symbol rate
CDR lock time
CDR lock time
0.990
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
Symbol • (105)
Rohde & Schwarz White paper | Signal model based a ecomposition 13
pproach to joint jitter and noise dThe recovered clock can be used to produce eye diagrams not only of the total signal,
but also of synthesized signals covering only a subset of the disturbances in the total
signal. This allows the user to graphically assess the most severe causes of signal degra-
dation and to acquire an expectation of what could be achieved in a given scenario once
a certain issue is fixed. The variety of available synthetic eye diagrams improves upon
the possibilities offered by state-of-the-art methods. Fig. 5 shows some of these eye dia-
grams: a) the total signal, b) the data-dependent signal, c) the data-dependent signal
with periodic horizontal disturbances and d) the signal with all deterministic components.
We can see that the data-dependent component is responsible for most of the jitter and
noise. Successively adding the periodic disturbances yields signals that approach the jit-
ter and noise of the total signal.
Fig. 5: Total and synthetic eye diagrams
a) T b) DD
c) DD+P(h) d) DD+P(h)+P(v)
Deeper understanding of the individual jitter and noise components is obtained from TIE
and LE histograms as in Fig. 6. These show the distribution of all timing and level errors
for various components. Some effects can be seen more easily using TIE histograms
for rising and falling edges and LE histograms for symbols 0 and 1. Subplot c) in Fig. 6
shows that falling edges cause significantly more data-dependent jitter than their rising
counterparts. Periodic jitter in e) adds a moderate amount of additional jitter. In the noise
domain, data-dependent components dominate over periodic components. The random
components in g) and h) show a close to Gaussian distribution.
14Fig. 6: TIE and LE histograms
a) TIET b) LET
Frequency
Frequency
Rising Symbol 0
2000 Falling 4000 Symbol 1
1000 200
0 0
–0.2 –0.1 0 0.1 0.2 –0.4 –0.2 0 0.2 0.4
TIE [UI] LE
Frequency c) TIEDD d) LEDD
Frequency
8000
Rising Symbol 0
8000
Falling 6 Symbol 1
6000
4
4000
2
2000
0 0
–0.2 –0.1 0 0.1 0.2 –0.4 –0.2 0 0.2 0.4
TIE [UI] LE
e) TIEP f) LEP
Frequency
Frequency
4000
2000 Rising Symbol 0
Falling Symbol 1
3000
1000 2000
1000
0 0
–0.2 –0.1 0 0.1 0.2 –0.4 –0.2 0 0.2 0.4
TIE [UI] LE
g) TIER h) LER
Frequency
Frequency
Rising 6000 Symbol 0
3000 Falling Symbol 1
4000
2000
1000 2000
0 0
–0.2 –0.1 0 0.1 0.2 –0.4 –0.2 0 0.2 0.4
TIE [UI] LE
Rohde & Schwarz White paper | Signal model based a ecomposition 15
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