* wivecrest Corporation 1715 Technology Dr., Suite 400 Saq Jose, CA w avecrestcorp. corn

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1 A New 'Method for Jtter Decomposton Through ts Dstrbuton Tal Fttng Mke P. L*, Jan Wlstrup+, Ross Jessen+, Denns Petrch* Abstract * wvecrest Corporaton 75 Technology Dr., Sute 400 Saq Jose, CA 95 0 mp,eng@ w avecrestcorp. corn We present a new tme-doman jtter separaton method. Such a method automatcally searches and fts the tal parts of the jtter!hstogram wth nonlnear jtter models and estmates de(ermnstc and rqdom jtter components. Bt error rate (BER) calculaton based on the determnstc and random jtter components s also dscussed and demonstrated.. ntroducton + Wavecrest Corporaton 7275 Bush lake Road Edna, MN meda, reflectons, cross-talk, electromagnetc nterference (EM), systematcal modulatons, and pattern dependency, for DJ; and thermal nose, shot nose, flck nose, random modulaton, non-statonary nterference, for RJ. n the real world applcatons, what s measured, n general, s the total jtter, namely both DJ and RJ are mxed together. To understand the root cause of the jtter, separatng and dentfyng each jtter component s essental. As the speed for mcroprocessor, memory, data bus, and transmsson meda ncreases steadly, f not exponentally, falures caused by jtter wll become more and more severe. Therefore, how to model, smulate, measure, and characterze jtter wll become even more mportant and challengng for hgh-speed sgnals n the forth comng years. Recently, we have developed a method for DJ and RJ separaton based on Blackman-Tukey alg~rthm'~]. n such a method, the DCD+S s obtaned by calculatng the mean of the tme error (measured edges versus that expected from a pattern). perodc jtter (PJ) and RJ are calculated through FFT spectrum estmaton of the varance va auto-correlaton functon of the tme jtter. Snce spectrum estmaton s requred n ths method, a tme seres of jtter measurements are needed. n the case when no jtter tme record (.e., jtter s measured as a functon of tme) s avalable, such a method may not apply. sources. n the tme doman measurements, jtter can be measured for a specfc edge transton or over a tme span of many edge transtons. n each case, many jtter samples are collected for each edge transton so that statstcal nformaton can be gathered and analyzed. A typcal example n real world practce s the jtter hstogram measured for a specfc edge transton, or jtter hstograms measured over a tme span of many edge transtons. Such a jtter hstogram reflects the mxture of DJ and RJ processes assocated wth the edge transtons. For many years, ths nformaton s avalable, but, to our knowledge, there s no theory or method establshed to decompose the total jtter hstogram nto DJ and RJ components. What was and has been used to quantfy jtter are statstcal Paper TC NTERNATONAL TEST CONFERENCE /99 $ EEE Authorzed lcensed use lmted to: UNVERSDADE TECNCA DE LSBOA. Downloaded on February 5, 2009 at 05:36 from EEE Xplore. Restrctons apply.

2 peak-to-peak value and 6 standard devaton, based on the entre hstogram dstrbuton that has both DJ and RJ components. The correct way to quantfy jtter s to use peak-to-peak value for DJ snce s bounded, and o standard devaton for RJ snce t s unbounded and random. t can be seen that the use of peak-to-peak value and <T standard devaton based on a total hstogram that has both DJ and RJ components s not only msleadng, but also statstcally wrong. n ths paper, we present a method/algorthm to decompose a total jtter hstogram nto DJ and RJ components through. tal fttng. n secton 2, we wll dscuss the theory of the total jtter hstogram and ts relatonshp wth DJ and RJ processes. n secton 3, we wll descrbe an algorthm that automatcally dentfes the tal parts of the dstrbuton and fts them wth the Gaussan dstrbutons, to estmate the j DJ and RJ components. n secton 4, we present our Monte Carlo smulaton results that verfy the valdty of our algorthm. n secton 5, we show some practcal applcaton examples usng our jtter software mplemented wth the algorthm and to verfy the correlaton, as well as to demonstrate the advantage of havng DJ and RJ components. n secton 6, we wll summarze our results. Note: Theoretcal analyss and computer smulatons have been performed. The tal search and fttng algorthms have been mplemented n our.current jtter analyss software product. A patent applcaton has been fled wth the U. S. patent offce. 2. Jtter Hstogram Dstrbuton and ts Relatonshp wth DJ and RJ Theoretcally, the tal part of the hstogram dstrbuton reflects the random jtter process. Physcally, random jtter s due to the random moton of partcles wthn a devce or transmsson meda. The random velocty of these partcles n an equlbrum state s proven to be best descrbed by a Gaussan dstrbuton. Therefore random jtter s naturally modeled by a Gaussan functon. Snce mult-temperature partcle dstrbuton s possble, a mut-gaussan dstrbuton functon may be needed to model certan random jtter processes. A sngle Gaussan jtter hstogram dstrbuton s defned as : The measured total jtter hstogram represents the scaledup total jtter probablty dstrbuton functon (PDF). On the other hand, the convoluton of RJ PDF wth DJ PDF gves the total PDF, f DJ and RJ processes are ndependent. n most cases, such an assumpton s vald. Therefore the tal part of the dstrbuton should be mostly determned by the random jtter, whch, n general, has a Gaussan type of dstrbuton. The random nose can be quantfed by the '0 standard devaton of Gaussan dstrbuton, whle DJ can be quantfed by the peak-topeak value. Dependng on the error probablty level, the total RJ can be a multple of the 6, deduced from the Gaussan dstrbuton. n the absence of DJ, the hstogram of the jtter should be roughly a Gaussan dstrbuton. Under ths condton, there s only one peak n the dstrbuton whch corresponds to zero DJ. The rms RJ s the o value. When both DJ and RJ come to play together, the measured jtter dstrbuton wll be broadened and no longer a Gaussan as a whole. On the other hand, both ends of the dstrbuton should stll keep the Gaussan type of tals snce DJ PDF s bounded. These tal part dstrbutons can be used to deduce the RJ number. Because of the DJ, the mean of each tal s no longer the same and mult-peaks can be present n the hstogram. The jtter dfference between far left peak value and far rght peak value wll gve rse to the DJ. Fgure s a schematc drawng of such a broadened total hstogram n the presence of both DJ and RJ. f there s no bas and statstcal samplng nose n the measurement, the two tals, whch represent the random process, should be symmetrca. Snce t s not possble to completely randomze measurements and reduce the samplng nose to zero, the (T values for the far left and far rght Gaussan tals may not be the same. The total RJ value should be the average of these two, and DJ s the dstance between two peaks of far left and far rght Gaussn tals, namely RJ =(a, +0,)/2 and. DJ=y,-y, where N,, s maxmum event count, t s the jtter, p and 6 are the Gaussan mean and standard devaton respectvely. 789 Authorzed lcensed use lmted to: UNVERSDADE TECNCA DE LSBOA. Downloaded on February 5, 2009 at 05:36 from EEE Xplore. Restrctons apply.

3 pl Pr Jtter Fgure Schematc drawng of total jtter hstogram n the presence of DJ and RJ. ~ 3. Tal Search and Fttng 3.) The requrements and specfcatons dentfyng the tal part of the hstogram dstrbuton, and then to ft them wth the Gaussan functon are the key to DJ and RJ separaton wth a gven total jtter hstogram. t s not possble to tell where the tal part of the hstogram s wthout studyng each ndvdual data and ts relatonshp wth the ne'ghborng data. The easest way to dentfy a tal part s,' through the graphcal dsplayng of the hstogram and pckng up the tal part va vsual nspecton. The dsadvantage of such an approach s that t lacks repeatabltj, and t cannot be adopted for producton test. Therefore $e requrements for a search algorthm should be:.) t slcapable of fndng the true tal part quckly, accurately, and repeatedly;.) t has to be automatc (.e., no user nterventon or vsual nspecton are requred). The fttng procedure should be able to deal wth the statstcal fluctuaton and factor ths nto the fttng routnes. The tal part has the lowest event counts and statstcal &certanty can be hgh. Smple straght forward least-square ft algorthm wll not work snce the statstcal error wll propagate nto the fttng parameters. Ths n turn gves rse l,arge to errors n DJ and RJ estmaton. A more advanced non-lnear fttng algorthm s needed to meet these requrements. 3.2) Algorthm for tal dentfcaton One of the key characterstcs of Gaussan tal s ts monotondty. That means: for the left sde of the tal, t monotoncally ncreases; for the rght part of the tal, t monotoncally decreases. Due to the presence of DJ, monoton&ty wll break and local maxmums near the left and rght/ part of tals. Wthout DJ, there s only one maxmum that corresponds to the mean of the dstrbuton. A dffcult ssue that a tal search algorthm faces s the statstcal fluctuaton. n the presence of statstcal fluctuatons, the monotoncty of a real Gaussan dstrbuton s no longer true, and usng the raw fluctuated data to fnd the local maxmum ponts for both left and rght tals wll be extremely dffcult, f not mpossble. The soluton ought to be frst flter out the nose and then use the smoothed hstogram to locate the maxmum ponts. There are two ways to acheve that, n general. One s through drect tme doman averagng; another s through FFT to get the spectrum, then apply a low-pass flter, and do the FFT. n tme doman averagng one needs to determne how many data ponts to use snce ths determnes the smooth level of the curve. n the FFT/FFT approach, one has to determne the bandwdth of the flter. The number of averagng ponts and flter bandwdth may need to be adjusted, dependng on the fluctuaton nose frequency and ampltude. n other words, a rule-based artfcal ntellgent algorthm must be used to enable the smoothng algorthm to deal wth a wde range of fluctuaton ampltudes and frequences. Ths s an mportant requrement to guarantee that smoothng only washes away the unwanted fluctuaton nose, not the true feature of the jtter hstogram. Once the smoothed hstogram hs(t) s obtaned ether through the tme doman averagng or tme-frequenc y doman FFT-flterng-FFT, the maxmum locatons can be found by calculatng the frst and second order dervatves of the jtter hstogram. The only maxmum ponts of nterest are the frst maxmum from the far left and the frst maxmum from the far rght. 3.3) Algorthm for the tal fttng One should use a fttng algorthm whch weghts the data record based on the qualty of each data. The bgger the error, the less role t should play n mnmzng the dfference between model expected value and the measured value. Thus, we need to use x2 as a gauge to determne how good the ft s. The fttng functon s Gaussan and the fttng algorthm s nonlnear so t can handle both lnear and non-lnear fttng functons. For detals of x2 theory, we refer the readers to and references theren. x2 fttng s an teratng process, n contrast to lnear eqhaton solvng n the case of lnear least-squared fttng. The fnal answer s obtaned when the teraton converges. For ths reason, ntal values of the fttng parameters are needed. A prmtve way to do ths s to try dfferent ntal values and to see whether they converge to the same fnal values. f the ntal guessed values are far from the fnal actual values, t may ether take longer tme to converge, Authorzed lcensed use lmted to: UNVERSDADE TECNCA DE LSBOA. Downloaded on February 5, 2009 at 05:36 from EEE Xplore. Restrctons apply.

4 or get stuck at a local x2 mnmum and never converge to the fnal global x2 mnmum pont. Calculatons should be carred out to estmate the ntal fttng parameters by usng the tal parts of hstogram so that the ntal fttng parameters are close to the fnal convergng values. Ths wll also make the teraton to converge rapdly and to avod stuck-n local mnmum (pvot). We would lke to emphasze that the x2 method has proven to be robust. 4. Monte Carlo Smulatons maxmums. Ths wll enhance the tal data usage and the Gaussan model wll be better constraned. Ths can correspond to the case when DJ > 2 R n the jtter analyss applcatons. Such a hstogram s very common n spreadspectrum clock devces. y=20,l7=0 4..) Hstogram wth statstcal nose To test how well the search and the fttng algorthms worked, we need smulatons. We started wth a known bmodal hstogram, whch s represented by two added Gaussan dstrbutons supermposed wth a random nose. Ths makes the overall hstogram be close to that of actual measurements. The overall hstogram s represented by the followng equaton of 2d h(t)=nle +Nre +N,ran$) (4) where N, Nr are the peak values, pl, & are means, q, CY, are standard devatons, respectvely, for two Gaussan dstrbutons. ran(t) s a random number generatng functon based on Monte Carlo method. t has a mean of zero and a standard devaton of unty. N, s the ampltude for the random number envelopes. For a Monte Carlo based random number generaton, we refer the readers A good search and fttng algorthm should return the ftted parameters that are consstent wth these pre-defned n the smulaton. A crtcal test would be: can an accurate fttng parameter be obtaned n the presence of sgnfcant statstcal fluctuatons,.e., N, s a sgnfcant porton of N or N,. Otherwse, no accurate parameters can be obtaned snce all the real world measurements are subject to statstcal fluctuaton.!lf\=2o,f$=lo pf\=8o,uf\=lo & Jtter n ps Fgure 2. Nn = 0, no statstcal fluctuaton. The second s when two Gaussan dstrbutons are not well separated, namely & - pl < q + 0,. Under such condton, the contamnaton of two dstrbutons could extend to the tal parts. As a result, one should only use the lower parts of the tals for the fttng to mnmzng contamnaton. A conservatve way s to use the tal part from the lowest event count to half of the N or N,. Ths can correspond to the case when DJ e 2 RJ n the jtter analyss applcatons. Fgures show the results correspondng to two well-separated Gaussan hstograms (.e., & - p > 0 + Or). Fgure show the results correspondng to two 4.2) Fttng results There are two scenaros we need to treat dfferently. The frst s that when two Gaussan dstrbutons are well separated,.e., when - pl > q + U,. Under such condton, the two dstrbutons are not well mxed, and the tal parts up to the pont of the frst maxmum are essentally uncontamnated. Therefore, we could use both left and rght tal data from ts lower value to the frst 79 Authorzed lcensed use lmted to: UNVERSDADE TECNCA DE LSBOA. Downloaded on February 5, 2009 at 05:36 from EEE Xplore. Restrctons apply.

5 400! Hstrogram t Tal Fttjng Jtter n ps - Fgure 2.2 Nn = 30, wth sgnfcant statstcal fluctuaton z 400 f 300 J Smulated Jner Hstrogram Jtter n ps Fgure3.2. Nn-30, wth sgnfcant statstcal fluctuaton A Practcal Case Study z 400- ', Jtter n ps Fgure 3. Nn- 0, no statstcal fluctuaton. Fgure 4. Hstogram tal search and fttng algorthm applcaton for a clock jtter, usng the Wavecrest DTS system as the measurement nstrument. Authorzed lcensed use lmted to: UNVERSDADE TECNCA DE LSBOA. Downloaded on February 5, 2009 at 05:36 from EEE Xplore. Restrctons apply.

6 The search and fttng algorthms dscussed n sectons 3 and 4 are mplemented n the Wavecrest Vrtual nstrument software. Usng the Wavecrest DTS 2075 system, and a clock nput to the nstrument, we measured the jtter hstogram and use the software to decompose DJ and RJ. Fgure 4 shows an example for clock jtter hstogram and DJ and RJ values deduced from the tal fttng algorthm. n ths case, the hstogram has twn-peak DJ process that s caused by a perodc modulaton. Ths s a 00 MHz clock sgnal wth a 5 MHz perodc modulaton. Thck lnes ndcate the Gaussan model ft to the tal part of the dstrbuton overlayng to the measured hstogram thn lnes. DJ value gves the modulaton jtter ampltude. Fgure 5 BER curves for the same clock sgnal Wth the DJ and RJ values, clock performance can be predcted va bt error rate error (BER) curve that s calculated through the measured total hstogram, as well RJ number. Detals on the how to calculate BER curve can be found n 3] and reference theren. Fgure 5 shows the BER curve (sometme t s also called bathtub curve). Thck lnes ndcate the actual measured BER, and thn lnes ndcate the extrapolated BER based on a RJ Gaussan PDF. BER curve s an mportant overall performance ndcator for tme crtcal Cs and systems. n seral data communcaton, total jtter s normally specfed at an error probablty level of O-'*. n Fgure 5 example, the operatonal margn s ps, and the total jtter s 26.6 ps, at probablty level. Correlaton study s carred out by comparng the RJ and DJ values wth those obtaned by usng the Tukey method, The agreement s n cases, the dfference was less than 5%. However, as we mentoned before, tal fttng method does not requre the tme span of the jtter record. Consequently, tal-fttng algorthms have a wde applcaton n the feld of jtter analyss. 6, Summary and Conclusons We have developed a general-purpose automated search and nonlnear fttng algorthm, wth specal emphass on determnstc jtter and random jtter separaton. We have shown that under sgnfcant statstcal fluctuaton, our algorthm can stll separate DJ and RJ accurately and repeatedly. Ths algorthm does not requre any user nterventon and apples n both laboratory and producton applcatons. Our algorthm can apply to ether a sngle hstogram, or a seres jtter hstogram (for determnstc and random jtter spectrum analyss). These algorthms can also be useful n other generalpurpose sgnal analyss applcatons. For example, one can use such method to analyze the phase nose spectrum, and determne what knd of nose processes are nvolved n a specfc devce, such as clock PLL or clock recovery PLL. Other examples nclude: DJ and RJ separaton for eyehstogram, and bounded uncorrelated jtter (BUJ) separaton that can be caused by crosstalk. We have also smulated the effect of samplng statstcs error usng the Monte Carlo method, for a sngle Gaussan dstrbuton. Ths s a very mportant ssue snce one can only measure the samplng statstcs and use that to represent the underlnng populaton statstcs. f the samplng statstcs are far from the populaton statstcs, one cannot get the true value for the populaton statstcs accurately, regardless of what knd of analyss tool one s usng. Ths s true for any knd of measurement. The goodness of the samplng statstcs s proportonal to the total number of measurements used to compose the hstogram. The bgger the total number of measurements, the better the samplng statstcs. For a gven total number of measurement per hstogram, f one repeats the hstogram measurement and DJ and RJ value deducton process, for a number of tmes, the DJ and RJ parameters deduced wll compose dstrbutons that are very close to Gaussan. The standard devaton for the DJ and RJ dstrbutons obtaned n ths manner s proportonal nversely to the total number of measurements n each hstogram composton. Our smulaton has shown a o error of 4.5% for DJ and 7.2% for RJ, gven 0,000 measurements per hstogram and repeatng the smulaton 00 tmes. Of course, for dfferent hstograms, or for dfferent measurement totals, these numbers can change. The pont we are makng here s that samplng statstcs s very mportant to guarantee the accuracy Of DJ and RJ* A mnmum number of measurements s needed for good samplng statstcs. That number can vary wth dfferent and accuracy 793 Authorzed lcensed use lmted to: UNVERSDADE TECNCA DE LSBOA. Downloaded on February 5, 2009 at 05:36 from EEE Xplore. Restrctons apply.

7 References [l] Y. Takasak, Dgtal transmsson desgn and jtter analyss, Artech House, nc., 99 [2] P. R. Trschtta, E. L. Varma, Jtter n dgtal transmsson systems, Artech House, nc [3] Natonal Commttee for nformaton technology standardzat7n (NCTS), workng draft for Fber channel - methodologes for jtter specfcaton, Rev 6, 998. [4] J. Wlstqup, A method of seral data jtter analyss usng one-shot tme nterval measurements, TC Proceedng, 998. [5] P. R. Belngton, D. K. Robnson, Data reducton and error analyss for the physcal scences, McGraw-Hll, nc, 992. [6] Knuth, D. E. Semnumercal Algorthm, 2& edton, Addson-Wdsley, 98.!. 794 Authorzed lcensed use lmted to: UNVERSDADE TECNCA DE LSBOA. Downloaded on February 5, 2009 at 05:36 from EEE Xplore. Restrctons apply.