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Application Note
AN-JITTER-1
Jitter Analysis using SHF 10000
Series Bit Error Rate Testers

Introduction
This application note introduces the technique and theory of using the SHF 10000 series of bit
error rate testers for jitter analysis. The SHF 11100 A error analyzer has a precise delay line(100 fs step size) and the corresponding control software is able to measure the BER as a functionof the time position of the decision point. Applying a fitting technique to the data allows theextraction of random and deterministic jitter components. Jitter analysis is one of the mostcommonly performed tests on high-speed digital communication components.
JitterJitter is the short term variation of the signal from its ideal location in time. With increasing datarates, jitter related problems become more apparent. A simple classification of jitter is shown inFigure 1.

Total Jitter (TJ)

Deterministic Jitter (DJ)Random Jitter (RJ)

Bounded - Peak-to-Peak

Periodic Jitter (PJ)

Unbounded - RMS

Data Dependant Jitter (DDJ)

Duty Cycle Distortion (DCD)

Inter Symbol Interference (ISI)

Figure 1 - Jitter Components
Total jitter (TJ) is composed of several independent parts which can be separated into
deterministic (DJ) and random (RJ) components. The method described here allows only theextraction of these two components. A further separation of the deterministic parts is not possibleusing this method.
Random jitter has an unbounded Gaussian distribution. It is caused by noise processes (thermal,shot, flicker). Since it is Gaussian in nature, the distribution is quantified by the standard deviation( ) and mean (µ). The infinitely extended tails of the distribution are used to fit a Gaussian function
and to extract and µ.The peak-to-peak value of random jitter must always be specified for acertain bit error rate.
Deterministic jitter is the non-Gaussian distributed component of the total jitter of a signal. As
shown in Figure 1, it is composed of periodic jitter and data dependant jitter. It is characterized byits peak-to-peak value.
A more detailed treatment of jitter theory and fundamentals can be found in the literature.

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BER Scan Theory
Figure 2 shows a typical eye diagram. The ideal decision point is in the center of the eye and ismarked by the red dot. In general, the lowest bit error rate is to be expected at this point. If thedecision point is moved towards the right or left crossing point, the probability of an error increases
steadily.

Right CrossingPointLeft Crossing
Point
1t=0

1 Unit Interval (UI)

3

2t = TB

Figure 2 - Eye Diagram
To determine the jitter components, the time position of the decision point is moved from one sideof a crossing point (blue dot, point 2) to the other side (green dot, point 3). At the same time, theBER is measured for each step. As the decision point is stepped from the center of the eye
towards the crossing point, the BER increases continuously. At some point, the error analyzer willlose synchronization. The theoretical BER maximum is 0.5 for a typical pseudo-random bit
sequence with an equal number of “1”s and “0”s (equal transition density).
Figure 3 shows a typical measurement. This measurement was performed using an SHF 11100 Aerror analyzer; the decision point was stepped in 100 fs steps along the crossing point of a50 Gbps data signal from an SHF 12100 A bit pattern generator. Both instruments were connectedback-to-back with a 0.5 m long 50 GHz microwave cable assembly.

Figure 3 - Typical BER Scan
To extract the deterministic and random jitter components from the measurements, the “Dual DiracModel” or “Double Delta Model” is employed [1,3]. In this model, the probability density function
SHF reserves the right to change specifications and design without notice – AN-JITTER-1 – Rev. 1.01 – 20/DEC/2005 Page 3/11

(PDF) for the deterministic jitter component is represented by two Dirac functions separated intime by the peak to peak value DJ = µL – µR.

PDFDJ =(1)1
[δ (t − µ R ) + δ (t − µ L )]2

Random jitter is modeled as a Gaussian function with a standard deviation

R/L.

(2)

t2

PDFRJ =− 21e 2σ
2 ⋅π σThe total jitter probability density function can be determined by convoluting the PDFs of DJ andRJ.

PDFTJ = PDFDJ ⊗ PDFRJ(3)

1 ⎛⎜ 1 − 2σ R2
1 − 2σ L2=+eeσL2 2 ⋅π ⎜ σ R⎝( t −µ R )2

( t − µ L )2⎞⎟⎟⎠

Equation 3 is illustrated in Figure 4. Since this model is quite simple with respect to the complex
processes and contributions, it is necessary to select the regions for the extraction of the random
jitter parameters carefully. While the distribution is dominated by deterministic jitter in the vicinity ofthe crossing point, it becomes dominated more and more by random jitter with increasing distance.
Hence it is important to collect measurement data further away from the crossing point, i.e.
towards the tails of the distribution where it is expected to be purely Gaussian.

σL

σR

µL

µR

Figure 4 – Eye Crossing point with probability density function (PDF)
As mentioned above, Equation 3 represents the probability density function of the jitter model
being used. The simplified mathematical relation between the probability density function and theBER is as follows:tBERL (t ) = ρ L ⋅ ∫ PDFTJ (t )dt

(4)

−∞
∞

BERR (t ) = ρ R ⋅ ∫ PDFTJ (t )dt(5)

tEquations 4 and 5 represent the cumulative distribution function (CDF) which is the time integral of
the probability density function. ρ L and ρ R are the average transition density, that is the ratio of
the number of logic transitions to the total number of transmitted bits. For pseudo-random bit
sequences, this ratio is 0.5.

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erfc(t ) = 1 − erf (t ) =2π∞

⋅ ∫ e − x dx2(6)tFollowing the steps given in Appendix A, we get the following expressions:tBERL (t ) = ρ L ⋅ ∫ PDFTJ (t ' )dt '(7)t ' = −∞

⎛ µ −t ⎞ρ⎟ with N L = L = 0.25
= N L ⋅ erfc⎜⎜ L⎟2⎝ 2 ⋅σ L ⎠∞BERR (t ) = ρ R ⋅ ∫ PDFTJ (t ' )dt '(8)t '=t⎛ t − µR ⎞ρ⎟ with N R = R = 0.25
= N R ⋅ erfc⎜⎜⎟2⎝ 2 ⋅σ R ⎠BER Scan Measurement
The SHF 11100 A error analyzer performs all the necessary steps to extract the random anddeterministic jitter components automatically.

Figure 5 – Jitter Analysis Display
If the jitter analysis is selected for the first time it will start with an empty diagram (Figure 5).
Several parameters corresponding to the measurement and the data fitting can be set by the user.

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Measurement Settings:
Start Pos: Determines where (time delay) the algorithm starts with the measurement. Itshould be chosen well within the low BER range of the eye. If an Auto Search is performed inadvance, a suitable time delay starting position close to the right crossing of the eye isautomatically chosen. Alternatively, a different starting point can be entered by the user.
Gate Time: Determines the maximum gate time which is employed during the wholeanalysis. To measure very low bit rates, the gate time needs to be adjusted tocorrespondingly high values, e.g. at a bit rate of 50 Gbps, a gate time of 2000 ms should beselected to take a BER of 1·10-10 into consideration, if a confidence of at least 10 errors isrequired. Note that selecting a long gate time will slow down the jitter analysis considerably inthe low BER regions.
Max. Errors: Determines when a measurement is stopped and the algorithm can proceed to
the next test point.
Fit Settings:
Trace: Several consecutive measurements can be performed. This is helpful to see theinfluence of a certain parameter in a setup or simply the pattern dependence. Thosemeasurements are stored in different arrays and displayed as different colored traces in theplot. Each trace can be enabled (check the box “visible”) or disabled individually.
Max BER: This is the maximum BER which is taken into consideration for the fitting of themeasured data. Measurement points with a BER higher than Max BER are ignored.
Min BER: This is the minimum BER which is taken into consideration for the fitting of themeasured data. Measurement points with a BER lower than Min BER are ignored.
Method: Determines whether the inverse error function or a polynomial (which approximatesthe inverse error function according to reference [2]) is used to extract the jitter contribution(Appendix C). This might be helpful when comparing results with those determined with otherinstruments.
TJ @: To calculate the total jitter a BER must be given for which the calculation should becarried out.
Plot Settings:
If an analysis has been completed, it automatically scales the plot. The scale can be changedby the user.
Min BER: Used to scale the plot. Sets the minimum displayed BER.
Max BER: Used to scale the plot. Sets the maximum displayed BER.
Min Delay: Used to scale the plot. Sets the minimum displayed time delay value.
Max Delay: Used to scale the plot. Sets the maximum displayed time delay value.
Several global parameters such as trace width and style and default fit settings can be changed in
the EA Jitter section of the Setup dialog (Figure 6).

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Figure 6 – Jitter Setup
Once everything is set, analysis can be started with the Start button. After a short time, a trace
should appear on the display similar to that shown in Figure 7.
The results are displayed in the Result section of the display, showing random, deterministic andtotal jitter (peak-to-peak). The fit settings can still be changed to include or exclude certain data
points, to change the used method or to calculate the total jitter TJ for a different BER.

Figure 7 – Jitter Measurement

SHF reserves the right to change specifications and design without notice – AN-JITTER-1 – Rev. 1.01 – 20/DEC/2005 Page 7/11

The random jitter is calculated as the average of both sigma values:

RJ =σR +σL(9)2The deterministic jitter is the difference of the mean value of the two distributions:

DJ pp = µ R − µ L(10)Total jitter is calculated using the formula:

TJ pp ( BER) = α BER ⋅ RJ RMS + DJ pp(11)

BER can be calculated using the formula:α BER = 2 ⋅ 2 ⋅erfc −1 (2 ⋅ BER )BER-03

10
10-04
10-05
10-06
10-07

BER

6.180467.438038.529789.50685
10.3987

BER-08

10
10-09
10-10
10-11
10-12

BER

11.22411.9956
12.7227
13.41214.069

BER-13

10
10-14
10-15
10-16
10-17(12)BER

14.697615.301315.8827
16.4442
16.9876

BER-18

10
10-19
10-20

BER

17.5146
18.0265
18.5247

The plot can be copied using the Clipboard button and pasted into another application; the display
can be cleared to start with a new set of measurements; the results and measurement data points
can be saved to a file using the Save button. For this, plain ASCII or Excel-XML format areavailable.

SHF reserves the right to change specifications and design without notice – AN-JITTER-1 – Rev. 1.01 – 20/DEC/2005 Page 8/11

Appendix A∞(13)BERR (t ) = ρ R ⋅ ∫ PDFTJ (t ' )dt 't '=t∞−

1= ρR ⋅ ∫e2 ⋅π ⋅σ Rt '=t( t ' − µ R )22σ R2dt '( t ' − µ R )2∞ −ρR=⋅ ∫ e 2σ dt '2 ⋅ π ⋅ σ R t '=t2
R

Now the following variable substitution can be used:t '− µ R2 ⋅σ Rx=(14)t ' = x ⋅ 2 ⋅σ R + µ R(15)dt '= 2 ⋅σ Rdx
dt ' = 2 ⋅ σ R ⋅ dx

(16)

Applying the substitution to the variable as well as to the integration limits yields:

2 ⋅σ R⋅
2 ⋅π ⋅σ R

BERR (t ) = ρ R ⋅=

ρR2⋅2π∞⋅x=∞x=−x∫ e dx2(17)t −µ R
2 ⋅σ R∫µe dx− x2

t− R2 ⋅σ R

The error function and complementary error function are given by:

2erf (t ) =π(18)t⋅ ∫ e − x dx
2x =0erfc(t ) = 1 − erf (t ) =∞2π⋅ ∫ e dx
− x2(19)

x =tApplying Equation 19 to Equation 17 results in:

BER R (t ) =ρR2

⋅ erfc(t − µR2 ⋅σ R(20))

SHF reserves the right to change specifications and design without notice – AN-JITTER-1 – Rev. 1.01 – 20/DEC/2005 Page 9/11

Appendix B
The approach used separates the random jitter (RJ) and deterministic jitter (DJ) components byfitting the tails of the cumulative distribution functions (Eq. 4 and 5) to the measured BER versus
time. The simplifying assumption that the BER of one edge is independent of jitter contribution
from the other edge is made.

⎛ t − µR ⎞⎟BERR (t ) = N R ⋅ erfc⎜⎜⎟⋅
2σR ⎠⎝(21)

In order to use linear regression as fitting algorithm, Equation 21 is transformed to a linear form:⎛ t − µR ⎞BERR (t )⎟= 1 − erf ⎜⎜⎟
NR⋅
2σR ⎠
⎝

(22)

⎛ t − µR ⎞BERR (t )⎟= erf ⎜⎜⎟
NR
⋅
2σR ⎠
⎝

(23)

⎛ BERR (t ) ⎞ t − µ R⎟=
erf −1 ⎜⎜1 −N R ⎟⎠2 ⋅σ R⎝

(24)

⎛ BERR (t ) ⎞tµ⎟⎟ =− R2 ⋅ erf −1 ⎜⎜1 −NR ⎠ σ R σ R⎝(25)

1−Equation 25 has the form y=m·x+n with R=1/m and µR=–n/m. Now a least squares fitting method
is used to minimize the sum of the squares of the differences between the measured data points
and the linear fit.

Appendix CTo perform the linear regression on the measurement data, Equation 25 must be used. Somemethods use an approximation of the inverse error function by a polynomial to simplify thecalculation of Equation 25. To make results comparable with those achieved with other
instruments in the Fit Settings section of the jitter analysis, inv. erf and AN1432 can be selected inthe program. If inv erf is selected, a built-in inverse error function is used to calculate Equation 25.
If AN1432 is selected then the following polynomial approximation is used [2]:⎛ BERR (t ) ⎞⎟⎟ = 1.192 − 0.6681 ⋅ log(BERR (t ) ) − 0.0162 ⋅ (log(BERR (t ) ))22 ⋅ erf −1 ⎜⎜1 −NR
⎝
⎠

(26)

SHF reserves the right to change specifications and design without notice – AN-JITTER-1 – Rev. 1.01 – 20/DEC/2005Page 10/11

References[1]

Information Technology – Fibre Channel – Methodologies for Jitter and Signal Quality
Specification – MJSQ, International Committee for Information Technology
Standardization (INCITS), June 9, 2004

[2]

Agilent Technologies Application Note AN1432 “Jitter Analysis Techniques for High DataRates”, 2003

[3]

Agilent Technologies White Paper “Jitter Analysis: The Dual-Dirac Model, RJ/DJ, and QScale”, 2005

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