Aggregation of Crosstalk in Backplanes

Transcription

1 DesignCon 2008 Aggregation of Crosstalk in Backplanes Atul Gupta, InspireSys Corporation Henry Wong, Gennum Corporation

2 Abstract As the serial data rate increases in the backplane interconnect, dielectric and skin loss, reflections due to vias, and connectors and crosstalk are becoming major obstacles. Loss and reflection can be addressed by equalization, however crosstalk becomes a major bottleneck in the backplane performance. The IEEE 802.3ap standard is a good example of specifications which take crosstalk into account. This paper will provide more insight into the crosstalk utilizing time domain analysis and attempt to answer such questions as when to consider crosstalk as a bounded signal, and when to consider it white noise and unbounded. Author(s) Biography Atul K. Gupta has recently started a design service company, InspireSys Corp., focusing on signal integrity issues in high data rate systems. Before this venture he was the Director of Advanced Development, Analog & Mixed Signal Division, for Gennum Corporation. Past experience includes equalizer, clock and data recovery (CDR) and SerDes design. Mr. Gupta holds several patents, has published papers in IEEE journals, and has presented papers and participated in panel discussions at various standards bodies. He holds a bachelor s degree in electrical engineering from the Indian Institute of Technology in Kanpur, India, and a master s degree in science from University of Calgary. Henry Wong is a Senior Systems Engineer in Advanced Development, Analog & Mixed Signal Division for Gennum Corporation, and is responsible for high-speed system design. He has more than14 years of industrial experience in digital communications, high-speed transceivers design, and signal processing. He has worked as a team leader and principal engineer on ADSL, cable, and wireless broadband (WiMAX) modems design at Nortel, Cadence, and a series of start-ups. He holds a bachelor s degree and a doctorate in electrical engineering, and has offered graduate courses and seminars in universities.

3 1. Background of backplane crosstalk aggregation After dielectric losses and reflections are addressed by equalization, crosstalk noise becomes the main bottleneck in high-speed backplane interconnect. This paper does not describe how to cancel the crosstalk, but rather describes a method to accurately predict the effect of crosstalk on the bit error rate (BER). This paper addresses this problem by quantifying the probability distribution using realistic measurements. From this, one can draw conclusions on its realistic impact on backplane system design. Crosstalk noise arises from many sources in the interconnect, due to inductive and capacitive coupling. The challenge has always been on linking BER and crosstalk. When there are multiple channels, it is even more difficult to accurately predict the BER performance. In data communications applications, the BER target is 1e-12, which means that one can only use statistical models to predict performance. To address this statistical problem, we derive rigorous statistical distributions of each crosstalk and the aggregate crosstalk. The main results are: Probability density function of individual NEXT and FEXT Probability density function of aggregate crosstalk Figure of merit to describe its difference/resemblance of a Gaussian density function Worst-case condition vs. optimistic result as a function of bit patterns Probability of occurrence for a realistic concern for system evaluation From these results, we can answer questions such as: How do we characterize complex backplane crosstalk problem using statistics? How do we simplify time-domain crosstalk measurements in lab yet generate statisticalsignificant results? How can we simulate worst-case scenarios without running hours and hours of simulations, and yet avoid the pitfall of short simulations? 2. Raw NEXT/FEXT data and statistical modeling 2.1 Raw NEXT and FEXT measurements The backplane setup used for measuring crosstalk is for a typical 10G backplane application. One model of this application can be found in Healey [2]. In this application, the total thru trace originates from the daughter card on the line card, through the backplane interconnect, and then to a switch card. Time-domain measurements of NEXT and FEXT were made in the lab by capturing step response with about 30 ps rise/fall, which is typical of a 10G SerDes. Impulse and pulse responses are then derived for post-processing. Examples of typical NEXT and FEXT measurements that relate to the IEEE KR spec can be found in Oganessyan et al [5] and Peters et

4 al [6]. In the KR spec, measurements of the total crosstalk are upper bounded by the crosstalk mask (see D Ambrosia et al [3, 4]). The insertion-loss-to-crosstalk (ICR) ratio is lower bounded by the ICR mask (see IEEE KR Spec [1], D Ambrosia [3]). 2.2 PDF calculation for each crosstalk In the time domain, crosstalk is captured as a step response and is used to derive the pulse response. The pulse response represents the interference of a bit through the transmit driver and coupled to the victim. A pattern of bits therefore gives rise to a sequence of pulse responses with delay of 1 UI in succession. The combined effect represents the total crosstalk effect on the victim, due to one aggressor, on 1 bit. The goal here is to calculate the probability of such crosstalk effect from billions of bit-pattern combinations. The end result is captured in a PDF function. Casper et al [9] and Moore [11, 12] have both calculated crosstalk PDF using this method. Treat each bit in the aggressor as a random variable (RV), and since the Tx bits are scrambled, any two RVs are independent. Consider only these 2 RVs, the summed effect has a density function which is the convolution of each density function according to Papoulis [14]. Let x be one crosstalk cursor in the pulse response, and y be another crosstalk cursor, then the sum of the 2 crosstalk cursors on the victim is the convolution of individual PDF: ( ) f ( z) = f z y f ( y) dy x Since we are dealing with discrete type, the PDF should be expressed as probability mass function (PMF) as: ( ) = ( ) ( ) P n P n k P k z x y k = For convenience of notation, the rest of the paper refers PDF and PMF collectively as PDF since the notation and understanding are now known to the reader. Now consider a pattern of independent bits in the aggressor. The combined crosstalk effect of this aggressor on the victim can be characterized by the convoluted PDF as: y f ( x) = f ( x) f ( x) L f ( x) 1 2 N where N is the number of bits in a pattern in the aggressor. This forms the PDF of an aggressor. The method of calculating PDF from individual cursors in general can be found in Ahmad [7, 8], Casper [9], and Stojanovic [10].

5 2.3 PDF calculation for aggregate crosstalk In backplane interconnect applications, multiple aggressors aggregate and impact the victim in a statistical manner. The challenge is how to model such aggregation in a tractable, yet accurate manner. Questions arise around whether the addition of aggressors will be constructive, or whether they will cancel each other and product only minimal effect. Due to its statistical nature, the best way to tackle this complex matter is through the use of PDF calculation. The PDF of the aggregate crosstalk can be again calculated through the use of PDF convolution. In Section 3, we will see results for a total of 4 aggressors, 2 NEXT and 2 FEXT. In this case, the aggregate is the summation of four crosstalks, with aggregate PDF as: f ( x) = f ( x) f ( x) f ( x) f ( x) aggregate NEXT1 NEXT 2 FEXT1 FEXT 2 More PDF convolution means that crosstalk noise can aggregate to a very large amplitude and distort the eye opening (peak distortion) extensively. But such an occurrence has a very low probability, due to the coherent destructive effect from each aggressor happening at the same time. The aggregate PDF faggregate( x ) directly addresses these 2 factors: (1) noise amplitude A, and (2) probability of having combined noise larger than this amplitude, i.e., Pr( A> A o ) (tail integration), on the current bit of the victim. With this equation, we can calculate BER due to the impact of each type or combined aggressor, providing significant insight into the backplane system evaluation. 3. Lumped crosstalk PDFs vs Gaussian PDF 3.1 Crosstalk measurements and calculated PDFs Figure 1 shows the pulse response of 2 NEXT and 2 FEXT that have been derived from the captured step responses, respectively. Figure 2 shows the corresponding calculated PDF, with Gaussian PDF overlaid as a comparison. To calculate the PDF, a bit pattern of 35 is used. This pattern is chosen because it represents slightly worse than PRBS31, which is standard for 10G Ethernet traffic and also yields close to 1e-12 BER rate.

6 cha NEXT1 cha NEXT2 cha FEXT1 cha FEXT2 Fig. 1: Pulse response of 2 NEXT & 2 FEXT aggressors, for the victim Channel A. Step response was captured in lab and transformed to pulse response for calculation. Red: Gaussian overlaid same sigma sigma = 2.38 sigma = 2.23 cha NEXT1 cha NEXT WC = WC = sigma = 1.37 sigma = 0.88 cha FEXT1 cha FEXT WC = WC = Fig. 2: PDF of 2 NEXT & 2 FEXT aggressors, for the victim Channel A. Gaussian PDF is plotted using the same sigma as calculated from each xtalk PDF. The worst-case xtalk amplitude in each aggressor is also shown. The worst case corresponds to PRBS35.

7 3.2 Figure of merit for comparison with Gaussian PDF It is important to understand the difference between crosstalk noise PDF and the wideband noise Gaussian PDF. It is equally important to see the progressive approximation of the aggregate crosstalk PDF to the Gaussian shape. Unlike Gaussian noise, crosstalk noise is bounded and is a function of the coupling and data pattern of the aggressor. However, as we will see in Section 3.3, the resemblance to the Gaussian shape is more and more obvious when more aggregation is performed. There are several methods to evaluate the differences between each crosstalk PDF and the Gaussian function. For example, Papoulis [14] describes an error term between a PDF and Gaussian with respect to moments and Hermite polynomials. Another method plots the data using a normal probability plot, where a straight line gives perfect Gaussian, and any deviation is represented by the skewness factor. We propose a third way to describe the difference, with respect to a communications problem as the way BER is calculated. Using this, the tail difference can be seen clearly. In a communications channel under Gaussian noise with rms value = σ, BER = 1e-12 occurs when amplitude of the signal at eye center is at least 7.03σ from the decision threshold. That is, Pr( A> 7.03 σ ) = 1e 12 We can interpret A in one of the two ways: (1) A is the noise amplitude in a Gaussian noise PDF, or (2) A is the signal amplitude at eye center at least with a distance dmin = 7.03σ from the decision threshold. We can transfer this formulation to the crosstalk noise PDF. We first calculate the rms of the crosstalk and then integrate the tail of the PDF such that the probability in the tail is 1e-12. Mathematically, Pr( A> Kσ ) = 1e 12 The rms value of the crosstalk noise PDF is calculated as: σ B x 2 f x B crosstalk = ( ) where x is the x-axis variable of the PDF fcrosstalk ( x ), and B is the bounded value of the crosstalk noise, which can be revealed in the PDF after cursors convolution as described in Section 2.2. With this interpretation, we define the figure of merit to indicate how different the crosstalk PDF from Gaussian as:

8 FOM = σ = K σ K The σ that is calculated from the crosstalk noise PDF is used in Gaussian noise PDF. Both are identical and after cancellation, the FOM is a simple ratio. Intuitively K < 7.03 because crosstalk noise is bounded while Gaussian noise is unbounded. Therefore, FOM 1. As crosstalk noise gets aggregated and its PDF is approaching Gaussian, FOM 1. When crosstalk noise is perfectly Gaussian, FOM = 1. If the FOM = 1.25, and the Gaussian approximation is used, crosstalk will be overestimated by 25%. For ideal FOM = 1, the Gaussian approximation is just as accurate as the PDF calculations. Integration of Gaussian PDF tail can be carried out using the Q function as QAσ ( / ) where 2 1 x 2 QA ( / σ ) = e dx A σ 2π or using the complementary error function which is readily available as a function or macro in computing packages as: 1 A QAσ ( / ) = erfc 2 2σ or simply by means of integration. The integration of crosstalk PDF is carried out by means of direct tail integration. Figure 3 shows the results for NEXT1, NEXT2, FEXT1, and FEXT2. FOM is between 1.55 and 1.65 for NEXT, and about 2.0 for FEXT. The difference in the tail from Gaussian is clearly seen in these plots and in the FOM. This says that it is not accurate to assume individual crosstalk PDF as Gaussian. The analysis is 55% to 100% pessimistic.

9 cha NEXT1 cha NEXT2 Gaussian Xtalk 1e-12 FOM=1.65 1e-12 FOM=1.55 cha FEXT1 cha FEXT2 1e-12 FOM=2.00 1e-12 FOM=1.97 Fig. 3: Probability (tail integration of PDF) for each xtalk, and its comparison with Gaussian PDF. FOM is defined in Section 3.2. This plot examines how much tail difference is, between xtalk and Gaussian PDF, more clearly. 3.3 Lumped crosstalk PDF Using the derivation in Section 2.3, we can now aggregate each of the four crosstalks in sequence. Figure 4 shows the aggregation of NEXT1 and NEXT2, forming a larger rms value. The worst-case noise amplitude is apparently the sum of individual crosstalk worst-case noise amplitude. Figure 5 further aggregates with FEXT1, and Figure 6 performs the last aggregation with FEXT2, forming the total aggregation of data from the four crosstalks. The calculated FOM = 1.23, and we can see from Figure 7 that the tail approaches Gaussian. Table 1 summarizes the growth of σ and max noise amplitude (worst-case noise amplitude). Although the worst-case noise amplitude is a rare event, it is real. It is the tail of this aggregate PDF that gives rise to transmission bit errors at 1e-12 to 1e-15 error rate.

10 Lump NEXT1 & 2 Red: Gaussian overlaid sigma = WC = Fig. 4: Aggregation of NEXT1 and NEXT2 through PDF convolution. Larger sigma and longer tail are the results. Lump NEXT1,2 & FEXT1 Red: Gaussian overlaid sigma = WC = Fig. 5: Aggregation of NEXT1, NEXT2, and FEXT1 through PDF convolution. Larger sigma and longer tail are the results.

11 Lump NEXT1,2 & FEXT1,2 Red: Gaussian overlaid sigma = WC = (see next figure for zoom in) Fig. 6: Aggregation of NEXT1, NEXT2, FEXT1, and FEXT2 through PDF convolution. Larger sigma and longer tail are the results. Red: Gaussian overlaid cha NEXT1, NEXT2, FEXT1, FEXT2 Very close to Gaussian sigma = WC = Fig. 7: Aggregation of 4 xtalk PDFs, forming a PDF that is very close to Gaussian distribution.

12 cha Aggregate xtalk PDF Gaussian pdf sigma = sigma x 7.03 = for Gaussian Compare to conv xtalk PDF Pr (A > ) = 1e-12 Very close to Gaussian FOM=1.23 Fig. 8: Probability (tail integration of PDF) for the aggregate xtalk, and its comparison with Gaussian PDF. For perfect resemblance of Gaussian, FOM = 1. More aggregation of xtalk approach to this more and more. Table 1: Show the growth in σ and max amplitude after each convolution. Crosstalk σ, Max amplitude, NEXT NEXT FEXT FEXT Lump NEXT1, NEXT Lump NEXT1, 2, FEXT Lump all Table 1 illustrates two interesting outcomes:

13 (1) Variance of convoluted PDF is equal to the sum of individual variance, that is, σ = σ + σ + L + σ aggregate N (2) Max amplitude of convoluted PDF is equal to the sum of individual max amplitude A = A + A + L + A max( aggregate) max(1) max(2) max( N ) The bounded value of the crosstalk noise amplitude increases quickly with convolutions. As the FOM and Section 3.4 both show, this is closer and closer to the Gaussian distribution. 3.4 Theoretical distribution and central limit theorem The central limit theorem is well described in many texts such as Papoulis [14], so rather than repeat this, this section will show an interesting way to quantify the FOM that was calculated earlier in Section 3.3. In Section 3.3, individual FOM for NEXT ranges from 1.55 to 1.65, for FEXT about 2.0. After convolving four crosstalk PDFs, the resulting FOM is Let s use uniform PDF as individual PDF to do convolution in this section to compare the FOM obtained earlier. From this, we have an idea what these FOM values represent. A uniform PDF can be thought as extreme case of bounded noise. Figure 9 shows the convolution of 1, 2, 4, and 16 uniform PDFs, and Gaussian is overlaid on the results. The difference between the uniform PDF and Gaussian is remarkable, and the resemblance is remarkable after convolving the uniform PDF just a few times. Though Figure 9 shows the remarkable resemblance at the center of the PDF, we are more interested in the tail difference. Since the PDF does not show enough detail at the tails, we will use the BER-like curve to illustrate the differences at the tails. Figure 10 shows the corresponding FOMs and adds the result for 32 uniform PDFs. As N increases, FOM approaches 1.0. Table 2 summarizes the FOM values. By comparing the FOM = 1.23 due to aggregate crosstalk and the FOM = 1.22 due to the convolution of 16 uniform PDFs, we can conclude that the aggregate crosstalk is indeed very similar to Gaussian even in the tail. And of course the PDF due to the aggregate crosstalk is clipped to the max value = as shown in Table 1.

14 Red: Gaussian overlaid 1 uniform conv 2 uniforms conv 4 uniforms conv 16 uniforms Fig. 9: Illustrating concept of central limit theorem convolution of 1, 2, 4, and 16 uniform distributions and studying their resemblances to Gaussian PDF. Fig. 10: Probability (tail integration of PDF) for each case of convolution, of a number of uniform PDFs convolutions.

15 Table 2: Compare FOM from actual measurements with uniform PDF convolutions. The following compares the Gaussian nature of the crosstalk measurements: Crosstalk FOM NEXT NEXT FEXT1 2.0 FEXT Lump all uniforms uniforms uniforms Worst-case vs. optimistic scenarios A crosstalk coupling by itself may not indicate how much of an impact it will have on a a timedomain signal at a certain time. The severity of its impact depends on the bit pattern in the aggressor. In this section we will study what the worst-case noise amplitude would be by deriving the worst-case bit pattern. The worst-case pattern may happen rarely but the probability is finite and it is real. The advantage of deriving the worst-case pattern is that we do not need to exercise the simulation indefinitely long in order to wait for this to happen. We will also study crosstalk impact by using a random pattern. Depending on the number of simulations carried out, the results can be optimistic. The two scenarios, namely, using worstcase pattern and a short random pattern, provide a sharp contrast to each other. It should also be noted that depending on the signature of the crosstalk, the worst-case pattern will be unique for every aggressor and victim coupling. One worst-case pattern cannot be used for all channels. This drives to a conclusion that a more realistic backplane system evaluation is needed to consider the probability of events. The aggregate PDF directly provides such information, and this is the subject of discussion in Section NEXT measurement Figure 11 shows a channel with NEXT to be amplified by a factor of 4 to try to match the total crosstalk noise mask. The quality of the matching is not the focus here. In reality, a number of crosstalk aggregate as shown in Section 3, but here, for the sake of illustration, one NEXT is scaled up by factor of 4 so that the dramatic effect on the victim can be shown.

16 chd NEXT1 Fig. 11: One NEXT is amplified by 4 to try to show the effect on the victim channel. The intent is to show a relative but observable difference between the use of a WC pattern and a random pattern. 4.2 Derivation of worst-case crosstalk using worst-case pattern Worst-case crosstalk here means the largest crosstalk noise is at the eye center;,.in other words, the smallest eye trace is caused due to a particular bit pattern. Using the pulse response, the precursor, main cursor, and postcursor are known and that determines the future, current, and past bits of the aggressor, which together give the maximum destructive interference to the current bit in the victim. Using the NEXT as shown in Figure 11, the derived 35-bit pattern (future 4 bits, current 1 bit, and past 30 bits) is determined. Going from past bits to future bits: Worst-case bit pattern = [ ] Clearly, any filters including transmit de-emphasis and receive filters are part of the crosstalk end-to-end channel so a change in the parameters will effect a corresponding change to the worst-case bit pattern. 4.3 Comparison of worst-case scenario with short simulation The worst-case crosstalk pattern as derived in Section 4.2 is embedded in the PRBS7 pattern and the crosstalk thus generated is used as an aggressor added to the receiver input. Figure 12 shows the worst-case crosstalk with an amplitude of -47. The ideal 1 signal, due to ISI of the channel, is further pulled down by the crosstalk noise to UI. Whereas on the right, random crosstalk which merely has UI produces a minor noise effect on the received signal.

17 With WC xtalk generated by WC pattern pessimistic but can be real Before equalization, supposed a 1 With random xtalk short-simulation scenario -47 WC xtalk effect Only -8 random xtalk UI Fig. 12: Comparison of effect of worst-case xtalk and random xtalk. WC xtalk is generated from WC pattern, derived from the knowledge of pulse response. Random xtalk is generated from a random pattern. The difference is significant between these 2 extremes. Figure 13 compares the eye diagram obtained after DFE equalization. On the right when there is no crosstalk noise, the inner eye opening is 216 mvppd with an eye area of 11.8 V.ps. With worst-case crosstalk applied, the inner eye is degraded to 130 mvppd (4.4 db penalty), and eye area to 6.1 V.ps (5.7 db penalty). DFE equalization does not deal with crosstalk noise reduction and therefore about the same amount of noise exists after equalization. This impacts performance by 5 to 6 db and that is the maximum SNR margin we can lose.

18 After DFE equalization reference chd NEXT1 4.4 db penalty in Eye Vert 5.7 db penalty in Eye Area No xtalk With WC xtalk Eye metrics: 85/216/11.8 Eye metrics: 73/130/6.1 Diff in Eye Vert = = 86 mvppd 86 / 2 = 43 close to 47 WC xtalk added Fig. 13: Comparison of no xtalk (left) with added worst-case xtalk (right). Both results are obtained after DFE equalization. WC xtalk amplitude = 47. After equalization, it reduces eye size by 43. Jitter is more and eye area is seriously affected by 5.7 db. Figure 14(a) compares 3 cases in the time domain: no crosstalk, worst-case crosstalk, and random crosstalk. The difference in impact is clearly seen. The signal has been equalized by a DFE. The corresponding eye diagrams are shown in Figure 14(b). Random crosstalk, which is generated using the same crosstalk pulse response but with a random bit pattern, produces <1 db of degradation to the eye, as opposed to 5 to 6 db via the WC pattern. From this, we can see that: Simulations need not run indefinitely long. We simply need to derive the WC pattern to find the WC crosstalk effect. Probability of the occurrence of such an event can be calculated easily. Running short simulations may not convey the right result. As in the case of using a random pattern, optimistic results can be misleading, and can be off by as much as 4 to 5 db when compared to the worst case. These results could be very frustrating because significantly more errors will be seen at the prototype stage of the product. Instead of focusing on the two extremes of the spectrum, in Section 5 we consider events with probability. We answer system evaluation questions by attaching probability and one of the best tools for this is the aggregate crosstalk PDF that we have generated in Section 3.

19 After DFE equalization No xtalk With WC xtalk With random xtalk WC can make situation pessimistic, but is real Random can make situation optimistic Only 6 effect = 43 WC xtalk always remained Only = 6 random xtalk Fig. 14(a): Comparison of time-domain signal, for no xtalk, worst-case xtalk, and random xtalk addition. It is a comparison of pessimistic (using WC pattern) vs. optimistic (short simulation) results. However, the pessimistic scenario is real. After DFE equalization No xtalk With WC xtalk With random xtalk V eye = 108 x 2 = 216 mvppd V eye = 65 x 2 = 130 V eye = 102 x 2 = 204 reference 4.4 db penalty in vert 5.7 db penalty in area 0.6 db penalty in vert 0.9 db penalty in area Eye metrics: 85/216/ /130/6.1 79/204/10.6 Fig. 14(b): Comparison of eye, for no xtalk, worst-case xtalk, and random xtalk addition. WC xtalk impacts the most with 5.7 db penalty in eye area, vs. a merely 0.9 db with random xtalk. The small penalty often leads to wrong conclusions in backplane evaluation.

20 5. Realistic scenarios a matter of probability tc12 NEXT1, NEXT2, FEXT1, FEXT2 sigma = sigma x 7.03 = for Gaussian Compare to Xtalk PDF Pr (A > ) = 1e-12 not far from Gaussian Pr (A > ) = 1e-12 Pr (A = ) = 7.7e WC = Fig. 15: PDF of aggregate crosstalk, focusing on the probability having WC amplitude, and the noise amplitude at which probability = 1e-12. Figure 15 demonstrates that the WC noise amplitude of the aggregate crosstalk can happen, but does so very rarely with a probability of only 7.7e-43. Since we are mostly interested in 1e-12 BER, WC is too pessimistic. In the case of single channel we recommended using the WC pattern. In the case of aggregate crosstalk, we recommend using the PDF and the crosstalk estimate related to 1e-12 occurrence. This will be a relevant amount of crosstalk. From a communications point of view, this means that as more crosstalk aggregates, in order to maintain the same BER = 1e-12, more transmit signal power is needed to cover the SNR margin loss. Or in terms of PDF interpretation, d min needs to be larger (farther away from the decision threshold) in order to maintain the same BER. There are two ways to keep d min larger: (1) higher transmission power, or (2) lower-noise receiver design. Since crosstalk will also increase with the larger transmit signal in a symmetrical backplane, only a lower noise receiver design helps.

21 6. PRBS7 validity for lab measurements In lab measurements, running billions of different bit patterns can take a long time, and the analysis of the measurements can take even longer. Because of this, PRBS7 pattern is often used as a substitute. Here we investigate the validity of using PRBS7, and how time-domain methods using the same tools as needed for BER tests and debugging (a high-speed scope and a BERT) can fully characterize the system without the need of a VNA (Vector Network Analyzer), which is generally used for frequency domain or S parameters of a system. A VNA is not going to be used for any other purpose in an all data-communication lab. We prefer the time-domain method as it eliminated the need for unnecessary equipment, and thus lowers costs. It also provides deeper insights into the system characteristics and is more accurate. This is due to the fact that no frequency-magnitude and frequency-phase data must be transformed into the time domain where small truncation errors can result in unrealistic behavior. Table 3 shows that the σ obtained from just using PRBS7. It is not that different from using the step response and PDF method, despite the fact that the max noise amplitude can be quite different because the PDF contains the WC bit pattern result. Table 3 also shows that when the four crosstalks aggregate, the σ obtained are also comparable with each other. This says that once σ is obtained, by the use of central limit theorem, the statistic of the crosstalk noise distribution is largely governed by the Gaussian PDF. And that PRBS7 therefore is a valid pattern to find this statistic. By measuring σ using PRBS7 method, one does not have to capture the step response and compute PDF. This way the measurement can be very short yet very accurate. The example shows only 3% inaccuracy. Table 3: Compare σ obtained from using the crosstalk PDF and using the time-domain PRBS7 simulations. σ, Using PDF, Using PRBS7, NEXT NEXT FEXT FEXT Aggregate

22 7. Conclusions There are 6 main conclusions from this statistical analysis using realistic crosstalk measurements: 1. It is rigorous to use the aggregate crosstalk PDF to predict performance of a complex backplane interconnect system. 2. The aggregate crosstalk PDF is close to a Gaussian PDF; the more aggregation, the more Gaussian it will become. 3. When using just the Gaussian PDF with the same σ to predict BER performance, a 23 % overestimation (pessimistic) of error rate occurred. Our example, though realistic, used just four channels for predicting 1e-12 BER. In a real backplane, there will be upwards of 10 channels adding to the crosstalk. Though it is hard to say that all backplanes will exhibit the same characteristics, it can be said that it is very likely that the results will be similar due to central limit theorem. 4. Using only a PRBS7 pattern and standard lab equipment (high-speed BERT/scope) and not VNA, we can measure σ for individual crosstalk and aggregate the results. In the chosen example, the prediction underestimated crosstalk within the range of 3 % when compared to the PDF method. We used the same method on several other channels and found very similar results. 5. The advantage of (4) above is that we can save cost (equipment/engineering) and time (data collection/analysis). With calibration, results can be obtained and analyzed in a time window that is several orders of magnitude faster than traditional methods. 6. Overall, if we use PRBS7 in time-domain lab measurements, assume the presence of multiple sources of crosstalk, and use the Gaussian assumption, the prediction of performance will be within ~20% on the pessimistic side. However, faster lab decisions can be made with appropriate calibration and without the need for PDF calculation in each run, while keeping rigorous PDF calculations as presented in this paper for post-processing as needed. Acknowledgement The authors wish to thank Gurpreet Bhullar of Gennum Corporation for his input and timedomain lab measurements on crosstalk aggregation using a generic backplane at 10GHz data rate.

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