DesignCon 2013 Comparison of Two Statistical Methods for High Speed Serial Link Simulation Masashi Shimanouchi, Altera Corporation mshimano@alatera.com Mike Peng Li, Altera Corporation mpli@altera.com Hsinho Wu, Altera Corporation hwu@altera.com
Abstract High speed serial link simulation at multi-gbps and beyond at behavior-level instead of transistor-level including bounded and unbounded jitter and noise has become essential part of the design process. There are two distinctive methods for behavior-level simulation, which are waveform-based method and statistical method. While waveform-based method can simulate not only linear behavior but also nonlinear behavior and therefore is more accurate, statistical method is the only way to directly simulate ISI due to channel and jitter and noise characteristic of tranceiver for very low BER down to 10-12 to 10-15. Furthermore, there are two statistical simulation methods whose difference becomes very distinctive when simulating equalization by Tx PreEmphasis and Rx CTLE. We will discuss how each statistical simulation method works and their pros and cons in order to avoid wrong use or expectation in each simulation method and to interpret the simulation results appropriately so that we can make best use of high speed serial link simulation tools. Authors Biography Masashi Shimanouchi is a senior member of technical staff at Altera Corporation. His work on high-speed serial links of FPGA products includes link system and component architecture, modeling, characterization, and link jitter and BER simulation tools development with expertise in signal processing, signal integrity and jitter area. Dr. Mike Peng Li is a fellow at Altera Corporation. He is a corporate expert and adviser on jitter, noise, signal integrity, high-speed link, SERDES and on-die instrumentation (ODI), electrical and optical signaling, silicon photonics, and optical FPGA. Dr. Li was the Chief Technology Officer (CTO) for Wavecrest Corporation from 2000-2007, where he led the technology roadmap, leadership, vision, and product developments. Dr. Hsinho Wu is a principal engineer at Altera Corporation. He presently works on high-speed communication systems of FPGA products. His development and research interests include signal integrity, clock and data recovery, equalizations, system modeling and simulation techniques. Dr. Wu was with National Semiconductor from 1999-2011 where he developed high speed interface products.
1. Introduction Simulating high speed serial link for its performance parameters such as eye diagram and BER down to 10-12 to 10-15 before its physical design and characterization by measurement is essential part of the design process these days for multi Gbps data rate and above. Though SPICE simulation at transistor-level is most accurate to simulate tranceiver device behavior, this method is not practical for the entire link performance simulation because the simulation may not converge or it would take too much time (many hours to days) if it converges. Useful link performance simulation must include the critical parameters such as additional bounded and unbounded jitter and noise, effect of complicated bits pattern such as PRBS2 31-1, cascaded segments of frequency dependent passive signal path elements such as PCB traces and vias, connectors in addition to transceiver devices. In order to overcome the simulation time issue, behavior-level simulation such as StatEye, peak distortion analysis and IBIS-AMI instead transistor-level simulation has begun gaining popularity [1][2]. In section 2, we will review the basics of behavior-level simulation. There are two distinctive methods for behavior-level simulation. One is waveform-based method, and another is statistical method. In section 3, we will discuss these behavior-level simulation methods including jitter and noise. Each method has pros and cons, and our challenge is to make a better use of each method s strength and come up with a better overall solution. In section 4, we will compare fully statistical method with a hybrid approach which combines statistical method and waveform-based method. In section 5, we will summarize our discussion and conclude. 2. Basics of Behavior-level Simulation Simulating ISI due to the basic link components is of our primary interest in this section. 2.1 Link System Model and Waveform-based Method A typical high speed serial link consists of transceiver (TX), channel, and receiver (RX). Its behavior-level model with major components is illustrated in Figure 1. TPs are typical test points at which signals are to be simulated and evaluated. Ref Clock PLL Vod & PreEmp Buffer Non-Linearity TX TP Channel RX Pkg S TX Pkg L TP RX CTLE N-Lin TP DFE N-Lin TP Pattern Figure 1 Major Components of a Typical High Speed Serial Link Model CDR General operation of the link is the following. TX receives bit pattern from the core logic circuit, and sends them out to the channel at a certain output voltage level and with pre-
emphasis to alleviate the ISI caused by the lossy channel. RX receives the signal at the channel output, and applies equalization to the received signal to further improve signal quality. External clock provides reference timing with TX. RX CDR recovers timing information from the received signal. TX model: TX output waveform shape nonlinearly depends on the amount of the preemphasis in general Typical behavior-level model is mostly linear with some nonlinearity correction Though actual output impedance varies during its state transition, it is usually modeled as time-invariant linear system RX model: RX equalizer usually consists of analog equalizer and DFE Although analog equalizer is often referred as CTLE (continuous-time linear equalizer), some of actual devices are not linear, and mainly affected by gain compression Although DFE is not linear system, it may be simulated within a framework of linear system since it is located at the end of the link Input impedance is usually modeled as time-invariant linear system, which is generally a good assumption Channel model: Most complete modeling method is to cascade TX output impedance, channel consisting of TX and RX packages and all the signal path segments, and RX input impedance at S-parameter level considering both insertion/return loss and odd/even mode Reference clock model: This becomes critical when its phase noise is considered in simulation Bit pattern: This is not relevant in fully statistical simulation while it is essential for waveform-based simulation The simulation flow of waveform-based method is shown in Figure 2. The behavior model can be either linear or nonlinear, with which the link system response is simulated as waveform for a given input bit pattern. Actual calculation of the response is done either fully in time domain or in frequency domain combined with time domain. In order to simulate nonlinear behavior, time domain simulation is needed while linear part of the system can be simulated in frequency domain too. From the simulated waveform, eye diagram is created, from which link performance parameters such as eye width and eye height, BER, signal-to-noise ratio etc. are obtained.
behavior-level non-linear model simulated waveform eye diagram created from waveform performance parameters (EW/EH, BER etc.) Figure 2 Waveform-based Simulation Flow An example of the simulated waveform, pdf eye diagram and BER eye diagram is shown in Figure.3. (a) Simulated Waveform (b) PDF Eye Diagram (c) BER Eye Diagram Figure 3 Example Outputs of Waveform-based Method
2.2 Linear Link System Model and Statistical Method Fully statistical method requires linear link model. An example is illustrated in Figure 4. Linear TX model would provide more optimistic high frequency boost than nonlinear model if it is not compensated by reducing the pre-emphasis amount in the linear model. When the nonlinearity of RX is strong, which is more often the case than TX, link performance could be highly overestimated, especially eye height because gain compression could significantly reduce eye height. No bit pattern is used though the number of pre-cursors and the number of post cursors need to be specified instead. While it s possible to include reference clock, TX PLL and RX CDR in fully statistical simulation to some extent, they are discussed in section 3 in order to focus on the essential point of linear model on ISI here. Though DFE is nonlinear circuit, it can be modeled within the framework of linear modeling discussed below because it is located at the end of the link system. Therefore DFE is not discussed in this paper. TX TP PreEmp Vod & TX Pkg Channel RX Pkg Buffer S CTLE DFE L Figure 4 Major Components of a Typical Linear Link Model The principle of single bit response (SBR)-based statistical simulation method is the following. When a single bit pulse is sent to a linear system, which is basically a low pass filter in our application, the output pulse from the system spreads over many bit-durations as illustrated in Figure 5. The signal with the largest amplitude is called main cursor, the signals before the main cursor are called pre-cursors, and the signals after the main cursor are called post-cursors. Depending on the subsequent signal processing, one sample for each bit, or many samples (tens to hundreds) for each bit are used. TP RX TP TP single bit input single bit output (response) Linear System pre-cursors post-cursors main-cursor Figure 5 Single Bit Response of a Linear System When the input signal to a linear system is random bit stream, the eye diagram of the output signal can be constructed by overlaying all the possible combination of the pre and post
cursors as illustrated in Figure 6. This operation is mathematically the convolution of the pdf of each cursor, and therefore this method is called statistical method. main cursor largest EH/2 from b +1 from b -1 from b -2 from b -3 smallest EH/2 (a) Pre, Main and Post Cursors (b) Statistical Accumulation of Cursors Figure 6 SBR Cursors and Their Statistical Accumulation The simulation flow of statistical method is shown in Figure 7. Linear link model is used to generate SBR, with which link system response is simulated. By statistically combining various cursors (portions of SBR), eye diagram is created, from which link performance parameters such as eye width and eye height, BER, signal-to-noise ratio, etc. are obtained. behavior-level linear model simulated SBR eye diagram created from SBR performance parameters (EW/EH, BER etc.) Figure 7 SBR-based Simulation Flow
An example of the simulated pdf eye diagram and BER eye diagram is shown in Figure.8. (a) PDF Eye Diagram (b) BER Eye Diagram Figure 8 Example Outputs of Statistical Method 3. Behavior-level Simulation Including Jitter and Noise In order to simulate more realistic link performance or worst case scenarios, various types of jitter and noise need to be considered. How these jitter and noise are simulated depends on each simulation method. 3.1 Waveform-based Method Typical jitter and noise to be considered in link simulation is summarized in Table 1. Though waveform-based method can simulate all the types of jitter and noise, very low BER such as 10-12 due to random jitter and noise cannot be directly simulated in practical time. For example, simulating a few million bits provides merely ~10-6 of BER, which is much larger than desired. In this case, we have to estimate the link performance at very low BER by extrapolating the raw simulation results [3]. This procedure is shown in Figure 9 (b). Since linear regression can be used in Q-scale, simulated raw BER is converted to Q- scale, and Q values are linearly extrapolated. Then, the extrapolated Q-values are converted back to BER values. An example result is shown in Figure 10.
Ref Clock TX RX RJ SJs (spurs) PJ/SSC DJ PWJ RJ DCD DN RN DJ PWJ RJ DCD DN RN Waveform w/extrapolation Table 1 Jitter and Noise Injected by Waveform-based Method w/extrapolation w/extrapolation w/extrapolation w/extrapolation Channel BUN (xtalk) ISI with Non-lin DJ/DN Bathtub Curve Simulated by Waveform-based Method Extrapolation in Q-scale to Lower BER Extrapolated BER via Q-scale Extrapolation Evaluate Link Performance (a) Simulated BER by Waveform-based Method (b) BER Extrapolation Procedure Figure 9 Waveform-based BER Simulation and Need of Extrapolation
(a) Extrapolated in Q-scale (b) Extrapolated BER via Q-scale Figure 10 BER Extrapolation via Q-scale 3.2 Statistical Method How statistical method simulates the jitter and noise is summarized in Table 2 along with waveform-based method. Statistical method is not suited to simulate time-dependent nature such as periodic jitter and spread spectrum. Other deterministic jitter and noise such as DJ and DN also depends on time, and therefore statistical method is not suited well for them. They may be simulated by convolving their probability density functions with the SBR. Though random jitter and random noise also change their states from time to time, statistical method can simulate them more accurately than DJ and DN because certain assumption on their spectra (power spectral densities to be more precise) works relatively well. As discussed later in this subsection, Normal distribution is assumed for random jitter and random noise when they are convolved with other signals, and therefore the BER is directly simulated to very small values. When nonlinearity of TX and/or RX is relatively strong, and statistical method is still preferred, gross nonlinearity correction can be made though high accuracy may not be expected depending on actual circuit behavior. An example of RX CTLE gain compression correction effect is shown in Figure 11.
Ref Clock TX RX Waveform Statistical RJ w/extrapolation PSD/pdf SJs (spurs) PSD/pdf PJ/SSC DJ pdf PWJ RJ w/extrapolation PSD/pdf DCD pdf DN RN w/extrapolation PSD/pdf DJ pdf PWJ RJ w/extrapolation PSD/pdf DCD pdf DN pdf RN w/extrapolation PSD/pdf Channel BUN (xtalk) pdf ISI with Non-lin DJ/DN Table 2 Jitter and Noise Injected by Waveform-based Method vs. Statistical Method pdf w/work around (not so accurate) (a) No RX CTLE Gain Compression (b) With Gain Compression Correction Figure 11 PDF Eye Diagrams by Statistical Method with/without Gross Gain Compression Correction
One way to simulate random jitter in statistical method is illustrated in Figure 12. If TX s inherent RJ or/and reference clock RJ is provided as rms values, white phase noise power spectral density (PSD) is assumed with user-specified frequency range, and its interaction with TX PLL and RX CDR is simulated in frequency domain. If they are specified as phase noise spectra, they can be directly used in the frequency domain simulation. Once the corresponding power spectrum is obtained, its rms value is calculated by integrating the power spectrum. When the resulting random jitter is convolved with the other signals, Normal distribution is assumed. Note that RX RJ is added after CDR since the dominant RX RJ source is usually the CDR s VCO. Convolve at TX out and CH out Convolve at RX out pdf N(0,RJ tx ) pdf N(0,RJ rx ) RJ tx RJ rx RJ txi (RJ ref ) PSD txi PSD ref PLL LPF PSD tx CDR HPF PSD cdr Figure 12 Statistical Random Jitter Modeling RJ cdr RJ rxi One way to simulate random noise in statistical method is illustrated in Figure 13. TX RN and RX RN are provided as rms values. In order to simulate the effect of lossy channel and RX CTLE on TX RN and RX RN, the power spectral densities of these noise are calculated first assuming while noise. Then their interaction is calculated in frequency domain. When the resulting random noise is convolved with the other signals, Normal distribution is assumed. Convolve at TX out pdf N(0,RJ txi ) Convolve at CH out pdf N(0,RN cho ) Convolve at RX out pdf N(0,RN rxo ) RN cho RN rxo RN txi H rntxi RN rxext RN rxin H rnrxin H rnrxo H ch RN rxi H rxpkg & rx Figure 13 Statistical Random Noise Modeling H ctle
4. Waveform-based Method, Statistical Method and Hybrid Approach We discussed waveform-based method and statistical method, and their pros and cons in section 2 and section 3. We discuss one way to improve statistical method in this section utilizing an advantage of waveform-based method. 4.1 Hybrid Approach Fully statistical method requires linear system assumption, and therefore high accuracy cannot be expected when actual device exhibits strong nonlinear behavior such as RX CTLE gain compression. One way to significantly alleviate this limitation in statistical method is that when ISI is simulated, waveform-based simulation result is combined with statistical result such as SBR-based ISI simulation result. Using this hybrid approach, the inaccuracy due to nonlinearity is significantly improved, and inability to simulate the interaction between PWJ and DCD can be overcome to some extent. Thus, the capability and coverage of statistical method can be extended as summarized in Table 3. Waveform Statistical Hybrid Ref Clock RJ w/extrapolation PSD/pdf PSD/pdf SJs (spurs) PSD/pdf PSD/pdf PJ/SSC DJ pdf pdf PWJ partial PWJ TX RJ w/extrapolation PSD/pdf PSD/pdf DCD pdf DCD DN RN w/extrapolation PSD/pdf PSD/pdf DJ pdf pdf PWJ partial PWJ RX RJ w/extrapolation PSD/pdf PSD/pdf DCD pdf DCD DN pdf pdf RN w/extrapolation PSD/pdf PSD/pdf Channel BUN (xtalk) pdf pdf ISI with Non-lin DJ/DN pdf w/work around (not so accurate) Table 3 Jitter and Noise Injection by Three Methods wvfm & pdf
4.2 SBR-based vs. Step Response-based ISI Simulation Let s assume that total of N bits (80 bits for example) block is to be simulated. In SBRbased fully statistical simulation, 2 N-1 combination of bits for each main bit (0 and 1) is simulated utilizing convolution of their pdfs. In the hybrid approach discussed in the previous subsection, a block consisting of a few bits including the main bit is simulated at waveform level, and the rest of the part is statistically simulated as illustrated in Figure 14. The waveforms of all the bit patterns for the central block are simulated, and the corresponding pdf eye is calculated. Then, this central block pdf eye is convolved with each cursor-pdf from the other part of the N bits to obtain the final result. Note that nonlinear model is used for the waveform level simulation for the central block. Since large portion of the overall signal magnitude are generated from a few cursors around main bit, this hybrid approach significantly alleviates the limitation for nonlinear behavior simulation in statistical method. total number of bits to simulate main bit Statistically simulated (e.g. by SBR-based) all possible bit patterns simulated by step-resp Statistically simulated (e.g. by SBR-based) Figure 14 Combining Statistical Simulation and Waveform-based Simulation There are basically two different methods to perform the waveform level simulation for the central block. One is SBR-based, and another is step response-based. The idea of SBRbased method is illustrated in Figure 15 for an example pattern (0110101) which has four post-cursor causing bits and two pre-cursor causing bits. Though each bit response is not rounded in Figure 15, they are to clarify the concept, and actual bit response is low pass filtered, and therefore has smooth shape. For each bit location, SBR is required if bit=1. To simulate overall response, all the SBRs are added together. cause post cursors cause pre cursors main bit add these waveforms Figure 15 SBR-based Waveform Simulation
The idea of step response-based method is illustrated in Figure 16 for the example pattern as for SBR-based method. For each rising edge, positive step response is required. For each falling edge, negative step response is required. To simulated overall response, all the step responses are added together. cause post cursors cause pre cursors main bit add these waveforms Figure 16 Step Response-based Waveform Simulation If DCD and/or PWJ are directly simulated at SBR, there would be overlap or gap between adjacent SBRs, and therefore SBR-based method cannot simulate DCD and PWJ even at waveform level. On the other hand, step response-based method can easily simulate DCD and PWJ. Though PWJ simulation is limited to a few bits of maximum number of contiguous identical bits, step response method can simulate the interaction between DCD and PWJ, in which DCD and PWJ would partially cancel each other, or they would add up in worse direction resulting in worst case eye opening. 5. Summary and Conclusion We discussed two distinctive simulation methods, which are waveform-based method and statistical method, and their pros and cons in section 2 and section 3. Since waveform-based method requires least assumptions and/or simplifications such as linear system assumption and use of convolution at pdf-level, it would provide best accuracy in most of real world applications. One downside of this method is that link performance at very low BER needs to be estimated by extrapolating the simulated raw data because the simulated time duration is not long enough to comprehend very low occurrence due to random jitter and random noise. This is not a problem in statistical method because it directly simulates the probability of such occurrence assuming normal distribution of random jitter and random noise. On the other hand, statistical method requires linear system assumption, and therefore high accuracy may not be expected when actual device exhibits strong nonlinear behavior such as RX
CTLE gain compression. One way to significantly alleviate this limitation in statistical method instead of using gross nonlinearity correction as a work around is that when ISI is simulated, waveform-level simulation is combined with statistical method such as SBRbased simulation. Using this hybrid method, the capability and coverage of statistical method can be extended. Table 4 summarizes the features of each simulation method, which would help avoid wrong use or expectation in each simulation method and to interpret the simulation results appropriately so that we can make best use of high speed serial link simulation tools. Pattern Length Jitter and Noise Device Nonlinearity BER SPICE Full-wvfm Full-stat Transistorlevel Behavior-level [1] [2] [3] [4] Partialwvfm + stat ~1e3 (or BER~1e-3 due to DDJ) 1e3~1e6 (or BER~1e-6 due to DDJ) 1e6~ (or BER<1e-6 due to DDJ) Bounded Unbounded spectra/dd (5) pdf (5) spectra/dd pdf Strong nlin and high accuracy need (4) Moderate nlin & accuracy need (3) Linear or not much accuracy need Directly sim to low BER (2) (2) Extrapolate to low BER (1) Simulation Time Very long (hours ~ days) Medium (minutes ~ hours) (2) (2) Short (~ a few minutes) (1) Very low BER such as 10-12 and 10-15 is mainly caused by unbounded jitter and noise after long run duration. Since fully waveform-based simulation cannot run long enough to provide statistically meaningful result, the simulated result needs to be extrapolated to estimate the link performance at the target BER. (2) RJ spectrum can be considered, and the effect of TX PLL and RX CDR on RJ can be simulated in statistical method. Likewise, RN spectrum can be considered, and the effect of channel loss and CTLE frequency response can be simulated in statistical method. (3) When nonlinearity is moderate and very high accuracy is not expected, gross modeling for nonlinearity effect may be applied to alleviate nonlinearity constraint. (4) Large portion of signal characteristic including nonlinearity is generated from a few bits around the main bit, and therefore nonlinearity effect can be simulated with certain accuracy by combining waveform-level simulation and SBR-based statistical method. (5) Simulating RJ PSD and its interaction with TX PLL and RX CDR in frequency domain, and converting the result to a new RJ value of a normal distribution, RJ behavior in a link system can be simulated well in statistical method. Likewise, RN PSD and its interaction with lossy channel and CTLE is simulated well. Table 4 Summary of Four Simulation Methods
6. References [1] A.Sanders, M.Resso, J.D Ambrosia, Channel Compliance Testing Utilizing Novel Statistical Eye Methodology, DesignCon 2004 [2] B.K.Casper, M.Haycock, R.Mooney, An Accurate and Efficient Analysis Method for Multi- Gb/s Chip-to-chip Signaling Schemes, IEEE Symposium On VLSI Circuit Design, 2002 [3] M.Shimanouchi, M.P.Li, D.Chow, New Modeling Methods for Bounded Gaussian Jitter (BGJ)/Noise(BGN) and Their Applications in Jitter/Noise Estimation/Testing, IEEE International Test Conference, 2009