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Statistical and System Transfer Statistical and System Transfer Function Based Method For Function Based Method For
Jitter and Noise Estimation In Jitter and Noise Estimation In Communication Design and TestCommunication Design and Test
Mike Li and Jan WilstrupWavecrest
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Purposes• Illustrate the shortfalls of simple parametric
based jitter/noise determination methods (Peak-to-Peak or RMS only)
• Introduce the PDF and CDF based statistical jitter/noise determination methods
• Introduce the jitter/noise transfer function and its role in serial data link
• Apply both statistical analysis and transfer function for jitter, noise, and BER in designing and testing > 1 Gb/s systems
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Outline• Overview of jitter and noise basic
• Shortfalls for parametric peak-to-peak• PDF and CDF for jitter, noise, and BER
• Overview of serial communication systems
• Methods for estimating and testing jitter, noise, and BER
• Conclusion
• Jitter/noise transfer functions
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Jitter And Noise Basics
∇v: amplitude noise
∇t: time jitter
Reference Amplitude v0
Reference Timing t0
• Timing Jitter : Any deviationfrom ideal timing
• Amplitude Noise: Any deviation from ideal amplitude
• Timing jitter occurs whenthere is an edge transition
• Amplitude noise is a constant activity
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Jitter And Noise Viewed From Signal Theory
• Jitter/noise is a statistical process• Jitter/noise has a distribution• Jitter/noise has many different components• Jitter/noise can only be completely
quantified by its PDF !!!
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Jitter/Noise Classification Scheme(Statistical Based)
Jitter/Noise
Deterministic Random
Static
PeriodicMulti-Gaussian Gaussian
BU
BU: Bounded uncorrelated
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Relationship Between Total PDF And Component PDFs: Convolution
∫∞
∞−−=
=
τττ dtRJpDJp
tDJptRJptTJp
)()(
)(*)()(
• An example for timing jitter PDF
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Why Deterministic Parametric (i.e, Peak-To-Peak or RMS) No Longer Sufficient?
• Peak-to-peak or RMS alone cannot determine a general PDF
• Peak-to-peak will be an unstable and size sample size dependent estimation in the presence of random component
• RMS will be an inaccurate estimation for a Gaussian random in the presence of deterministic component
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Peak-To-Peak Shortfalls
0 1 2 3 4 5 6 7 8 9 10
x 10 4
0.18
0.185
0.19
0.195
0.2
0.205
0.21
0.215
0.22
0.225
0.23
Total Number of Hits
Peak
-To-
Peak
in U
I
a.) Sample Size Dependency
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Peak-To-Peak Shortfallsb.) Probability distribution is un-symmetrical
0.6 0.65 0.7 0.75 0.8 0.850
50
100
150
200
250
300
350
Peak-to-Peak
Hits
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Peak-To-Peak Shortfallsc.) Peak-to-peak cannot define a unique PDF
∆t
Probability
Peak-to-Peak
PDFs
Which one ?
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A RMS Shortfall Case Study
RMS ≠ RJ sigma
σl σr
µl µr Jitter
Events
N_lmax, N_rmax
N_min
σ>
∆−∆−
= ∑=
2
1
)(1
1i
N
n
ttN
RMS
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A Deterministic And Random PDF Estimation Method: TailFit
• Total jitter PDF = DJ PDF * RJ PDF
222)(
21)( σ
µ
π
−∆−=∆
tetp
• Tail parts of distribution preserve information on RJ process.
DJ pdf
RJ pdf *
Total pdf
• RJ PDF is a Gaussian
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∆ t
prob
den
sity
J itter pdf
G aussian Tailfits
What Does The “TailFit” Offer ?
• RJ PDFS• DJ PDF (via deconvolution)• Subspace statistical parameters
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DJ PDF Via Deconvolution
-0.3 -0.2 -0.1 0 0.1 0.2 0.30
2
4
6
t (UI)
TJ P
DF
-0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.1
0
0.1
0.2
Rem
aind
er
-0.3 -0.2 -0.1 0 0.1 0.2 0.30
5
10
t (UI)
DJ
TJ PDF and DJ PDF
-0.3 -0.2 -0.1 0 0.1 0.2 0.3-2
-1
0
1
2
Erro
r
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A Jitter/Noise Spectrum Estimation Method: Autocorrelation (Patented)
)))((*2()( ntRxxcFFTfPSD ∆−=
Ideal Clock
0 1N UI
Data
∆t0 ∆tn∆t1
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What Does The “Autocorrelation” Offer ?
• RJ PSD • PJ PSD• DDJ (DJ without PJ and BUJ) PSD
f
psd(
f)
P J R J
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Jitter, Noise, And BER
ts
BER (cdf)
zero level
eye-diagram
0 1 ts
0.5 0.5
UI
opening
∆ t
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• BER: metric for overall system performance• BER: CDF from jitter and noise PDFS
Jitter, Noise, And BER Cont…
)]()()()([21)()( tdt
st
trPtdt
sttlPtCDFtBER ss ∆∆∫
∞−+∆∆∫
∞==
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A Serial Data Communication System
Transmitter Receiver
PLL
Data
Clock/ PLL
D
C
Q
Medium
CR/ PLL
Data
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Receiver Jitter Transfer Function
CR/ PLL
Jitter in +
-
Jitter out
Receiver “sees” a difference function for jitter
• A “difference” function
θi θe
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What Does A Difference Function Mean?
• Jitter is referenced to a recovered bit clock• Receiver has a jitter transfer function• “Intrinsic” jitter referenced to an ideal bit clock
is not the jitter “seen” by the receiver• BER of the system should be estimated
based on jitter “seen” by the receiver
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Jitter Transfer/Tolerance Functions For Data Recovery
Frequency (f) fc
Mag
nitu
de
(dB)
HH(f): Jitter Transfer
HT (f) Jitter Tolerance
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What Constitutes A Valid Jitter Estimating/Testing Method?
• Estimate the jitter as the receiver “sees” !!!• CR/PLL and difference functions or their
equivalents are required
CR/ PLL
Data in +
-
Measurement System
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Compliance Estimating/Testing:Total Jitter And BER Function
• Total jitter = UI – eye opening @ 10-12
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A Valid Jitter Estimating/Testing Method For Serial Links
Estimating/testing total jitter PDF via an appropriate bit clock reference
Estimating/Testing BER/CDF function to < 10-12
Estimating/Testing total jitter @ BER = 10-12 or smaller
Pass/Fail
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Jitter Transfer Function For PCI Express
Ref Clock (100 MHz) Tx
PLL 25X
PLL 25X
Rx
D
C
Q
D
C PI
• System block diagram for common clock mode
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Jitter Transfer Function For PCI Express Cont..
X(s): Ref Clock Jitter
Tx PLLH1(s) PI
H3(s)
Rx PLLH2(s)
+
-
Y(s): Jitter At The Rx
• System Transfer Function Block Diagram
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Jitter Transfer Function For PCI Express Cont..• Math Relationships
)(3))(2)(1()( sHsHsHstH •−=
)()()( sXstHsY •=
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Summary And Conclusion• Deterministic parametric methods for jitter and
noise no longer sufficient (e.g., Peak-to-Peak)• Statistical functional (PDF/CDF) methods are
required for jitter, noise, and BER estimation/test• “Tailfit” offers deterministic and random PDF/CDF
estimation/test• “Autocorrelation” offers PSD for periodic and
random components• Transfer function is required in estimating
relevant jitter, noise, and BER for the system• Statistical PDF/CDF + transfer function gives a
complete and accurate system performance estimation/testing
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