Multidimensional Uncertainty Quantification of Microwave/RF Networks Using Linear Regression and Optimal Design of Experiments

I was attracted to this paper because I'm interested in polynomial chaos (PC) as well as Design of Experiments (DOE)

• MC is not optimum approach
• PC is good but literature is dominated by "highly accurate but intrusive stochastic Galerkin (SG) approach"  And if the network is nonlinear then the PC "inner product operations need to be approximated using a quadrature method where each quadrature term is represented busing a large number of additional voltage/current sources".  These approaches can not use SPICE simulators.
• non-intrusive PC approaches include stochastic collocation (SC), pseudo-spectral collocation approach, PC linear regression approach.  These approaches can use SPICE simulators.
• They are advocating an DOE approach to get the PC coefficients.  Apparently, "blindly choosing the DoE can lead to inaccurate evaluation of the PC coefficient" and says that no guidance is given in literature.  I think that they aim to remedy this.
• They introduce some D-optimal approach to find a good DoE for PC applications.   It sounds like they get a whole set of separate DoE's and quantify their quality (D-optimal), discard the worst ones, generate new ones and continue until the best D-optimal one is found. They say that they give lots of details on how to do this.  It sounds exactly like what JMP does, so I'm not terribly impressed.
• If I was trying to write my own D-Optimal DoE creation script I would certainly read and use this paper.
• II. Review of PC Approaches for Microwave/RF Circuit Simulation
• "the goal of uncertainty quantification is to compute all statistical moments of the unknown network response in an accurate yet efficient manner."
• represent the output network response with PC expansion (a projection using orthogonal multivariate polynomials).  This results in the PC coefficients.
• Approaches to obtain PC coefficients
• nonintrusive linear regression with DoE (their approach)
• Once the PC coefficients are known, the mean and variance of the a given network output can be calculated.
• They give lots of good examples of applications

A. K. Prasad, M. Ahadi and S. Roy, "Multidimensional Uncertainty Quantification of Microwave/RF Networks Using Linear Regression and Optimal Design of Experiments," in IEEE Transactions on Microwave Theory and Techniques, vol. 64, no. 8, pp. 2433-2446, Aug. 2016.

NOTE: I just noticed that I reviewed this paper on 8/16/16 as well.  I had this paper in my browser tab for a long time and thought that I needed to review it.

REDUCED NOISE SENSITIVITY IN INVERSE SCATTERING THROUGH FILTERING

These guys have an improvement of the TDR impedance peeling algorithm to deal with the inherit measurement or numerical round-off noise

• They call the peeling algorithm "inverse scattering, modified deconvolution"
• "multiple reflections" column 4, line 19
• Reference "Inverse Scattering for Discrete Transmission-line Models", 51 AM Rev., 1987,
• This might be an interesting read

Reduced noise sensitivity in inverse scattering through filtering
US 5321365 A
1994 - Textronix

https://www.google.com/patents/US5321365

OVERCOMING MULTIPLE REFLECTIONS IN PACKAGES AND CONNECTORS AT HIGH SPEED BROADBAND SIGNAL ROUTING

• concerned with "multiple reflections" -- abstract
• "cause multi-frequency reflections to destructively interfere" -- abstract
• add a stub to a trace where "the stub is configured to reduce broadband reflections in the trace by causing multi-frequency reflections that destructively interfere with at least some broadband reflections"  Paragraph [0005] see also [0044]
• the stub length, impedance and position are key [0006]
• "Due to the above stated impedance discontinuities, a high speed broadband signal propagating through the transmission line 120 is severely degraded. The signal degradation is caused by frequency dependent distortion that is introduced by the multiple reflections between the trace 120 and discontinuities along or at both ends of the trace such as introduced by 130 and 110. Specifically, the discontinuities cause multiple reflections of the broadband signal within the trace, which results in signal distortion and degrades the output signal quality. To mitigate this effect, stubs 107A and 107B are formed along the trace transmission line, such as along a longitudinal length of the transmission line 120 -- 0025
• "For each stub 107A and 107B, the stub length, stub impedance, stub width as well as the location of the stub along the longitudinal length of the transmission line 120 is selected such that the stubs are configured to reduce broadband reflections in the trace by causing multi-frequency reflections that destructively interfere with at least some broadband reflections, to be added to signals transmitted along the trace 120 --026
• "It is noted that numerous techniques can be used to determine stub parameters to overcome broadband reflections of a broadband signal that propagates through a trace of a semiconductor package. FIG. 5 is a flowchart depicting one representative technique."
• They don't really describe an optimization technique, the summary of the described approach is to twiddle the inputs and select outputs which look good.  

Summary, insert a stub in a transmission line which improves low frequency loss at the cost of increasing high frequency loss.  This is due to multiple reflections.  The reflections cancel at low frequency and combine catastrophically at high frequencies.

Overcoming multiple reflections in packages and connectors at high speed broadband signal routing
US 20140016686 A1
https://www.google.com/patents/US20140016686

CMOS ADC-based receivers for high-speed electrical and optical links

• They contrast ADC receivers to mixed-signal receivers.  They state that ADC receiver can more easily support PAM4 and robustness to PVT variations.  
• The downside of ADC receivers is that they consume more power.  
• Their research is about "techniques to improve ADC energy efficiency and novel approaches to reduce DSP complexity and power consumption."
• TI (time-interleaving) converters
• partial embedded analog equalization -- sounds interesting
• SNDR - signal-to-noise-plus-distortion ratio -- also sounds interesting
• SAR converters - successive approximation registers, a piece of the ADC
• TI (time-interleaving)
• When you do time-interleaving, conversion errors can occur due to mismataches between the parallel sub-ADC.  They note that these effects are typically calibrated out.
• Also need to calibrate/correct the sampling offset error.
• Also bandwidth mismatches between the parallel sub-ADC.  One benefit of an ADC is that each branch can be equalized independently which somewhat addresses any bandwidth mismatch.
• skew errors
• Equalization
• "After quantization, ADC-based receivers typically remove ISI in the digital domain with combination of a FFE and a DFE before symbol detection."
• "A major difference between ADC-based and mixed-signal receivers is the quantization noise introduced by the ADC, which can ultimately limit the BER."
• partial analog equalization means that they do some analog equalization (such as a FFE) before the ADC where they do more equalization.
• The quality of the equalization impacts the required ADC resolution.
• "A higher ADC resolution is required as channel loss increases due to reduced signal power and the impact of quantization noise amplification by the stronger FFE tap settings, with the DSP-only receiver requiring an 8-bit ADC resolution to compensate for the highest loss channel.
• " High-speed linearity is of particular importance with more spectrally efficient modulation, such as PAM4, further motivating ADCbased front-ends that allow for precise digital FFE and DFE computation"
• Hybrid ADC-Based Receiver

S. Palermo et al., "CMOS ADC-based receivers for high-speed electrical and optical links," in IEEE Communications Magazine, vol. 54, no. 10, pp. 168-175, October 2016.

Exerpts from --> Sapiens: A Brief History of Humankind

This is a provocative book which I am attracted to because my interests in understanding and building cultures.

• 87 - One of history's few iron laws is that luxuries tend to become necessities and to spawn new obligations.
• 164 - Equality and liberty, cognitive dissonance, the best way to understand a culture is to understand its cognitive dissonance.

The first section of chapter 9:

After the agricultural revolution, human societies grew larger and more complex, while the imagined constructs sustaining the social order also became more elaborate.  Myths and fictions accustomed people, nearly from the moment of birth, to think in certain ways, to behave in accordance with certain standards, to want certain things, and to observe certain rules. They thereby created artificial instincts that enable millions of strangers to cooperate effectively.  This network of artificial instincts is called 'culture'.

• 179 - Us and Them
• 172 - Potential universal orders 1) Monetary 2) Political 3) religion
• 180 - Money is the most universal and most efficient system of mutual trust ever devised.
• 185 - Christians and Muslims who could not agree on religious beliefs could nevertheless agree on a monetary belief, because whereas religion asks us to believe in something, money asks us to believe that other people believe in something.
• 259 - In 1620 Francis Bacon published a scientific manifesto titled The New Instrument.  In it he argued that 'knowledge is power'.  The real test of 'knowledge' is not whether it is true, but whether it empowers us.  Scientists usually assume that no theory is 100% correct.  Consequently, truth is a poor test for knowledge.  The real test is utility.  A theory that enables us to do new things constitutes knowledge.
• 264 - Until the scientific revolution most human cultures did not believe in progress.  They thought the golden age was in the past, and that the world was stagnant, if not deteriorating. 
• 273 - To channel limited resources we must answer questions such as 'What is more Important?' and 'What is good?' And these are not scientific questions.  Science can explain what exists in the world, how things work, and what must be in the future.  By definition, it has no pretensions to knowing what should be in the future.  Only religions and ideologies see to answer such questions.
• 274 - scientific research can flourish only in alliance with some religion or ideology.
• 282 - The Chinese and Persians did not lack technical inventions such as steam engines (which could be freely copied or bought).  They lacked the values, myths, judicial apparatus and sociopolitical structures that took centuries to form and mature in the West and which could not be copied and internalized rapidly.  France and the United States quickly followed in Britains's footsteps because the French and Americans already shared the most important British myths and social structures.  The Chinese and Persians could not catch up as quickly because they thought and organized their societies differently.  . . .  [the difference was ] modern science and capitalism.
• 283 - The key factor was that the plant seeking botanist and the colony seeking naval officer shared a similar mindset.  Both scientist and conqueror began by admitting ignorance - they both said, 'I don't know what's out there.' They both felt compelled to go out and make new discoveries.  And they both hoped the new knowledge acquired would make them masters of the world.
• 303 - People continue to conduct a heroic struggle against racism without noticing that the battlefront has shifted, and that the place of racism in imperial ideology has now been replaced by 'culturism'.  . . . Among today's elites, assertions about the contrasting merits of diverse human groups are almost always couched in terms of historical differences between cultures rather than biological differences between races.
• 309 - If the global pie stayed the same size, there was no margin for credit.  Credit is the difference between today's pie and tomorrow's pie.  If the pie stays the same, why extend credit?

Sapiens: A Brief History of Humankind (By: Yuval Noah Harari) [published: September, 2014]

Saturating Amplifiers

I asked Dr. Ross Walker to help me find a paper which discusses the issues of saturating amplifiers and this is what he gave me.  I think that the background section will be most helpful.

This is a good paper.  This is exactly what I needed. I should read this again someday and really try to understand circuit design better.

• "CMOS scaling has led to greatly reduced supply voltages  and transistors with poor intrinsic gain (g_m*r_o ) [1] and calls for fundamentally different approaches to analog design
• "Conventional design techniques that use operational transconductance amplifiers (OTAs) at low supply voltages (<< 1.2 V) suffer from reduced signal swings and poor distortion
• "Negative feedback is often employed in both architectures to improve linearity, but at the expense of bandwidth.

• OTA: operational transconductance amplifiers

B. Vigraham, J. Kuppambatti and P. R. Kinget, "Switched-Mode Operational Amplifiers and Their Application to Continuous-Time Filters in Nanoscale CMOS," in IEEE Journal of Solid-State Circuits, vol. 49, no. 12, pp. 2758-2772, Dec. 2014.

Strategies for Coping with Nonlinear and Time Variant Behavior for High Speed Serial Buffer Modeling -- DesignCon 2008

• It seems that their objective is to really show the boundary (or tolerance) where nonlinear behavior significantly impact the statistical BER eye estimations.
• They have a nice history of computer interfaces.  I should definitely use this paper as a reference.  They say that as speeds increased, reflections became more of a concern and it forced the Tx designers to make the drive impedance constant which makes the system linear!  Which of course leads to great analytic approaches.
• seems to be primarily concerned about Tx nonlinearity . . .

R. I. Mellitz, M. Tsuk, T. Donisi, and S. G. Pytel, "Strategies for Coping with Nonlinear and Time Variant Behavior for High Speed Serial Buffer Modeling," paper resented at DesignCon, Santa Clara, CA, Feb 4-8, 2008.

I found another BER estimation of nonlinear link paper!

I would be very curious as to how well our method works for single ended signaling.  I would expect that it would work very well.

• Mentions that MER is mainly focused on modeling the nonlinearity of the TX but their method aims to capture TX and RX characteristics.
• Requires an intrusive analytical closed form equation of the nonlinear rx buffer stage.
• "compact waveform sets"? --> consists of all possible waveforms for 1 UI.  This is somehow derived from the LTI channel pulse response.  Section IIIA doesn't help much more.  "main wave" and "ISI waves"?  
• This compact waveform set seems like a folded pulse response waveform . . .
• A non-compact way of representing this would be to estimate the length of the channel response and then get a PRBS of this length (since it guarantees to cover all patterns) and then fold this waveform up . . . 
• They use a two segment PWL representation of the nonlinear IV curves.
• They then have some analytic expressions for their receiver and this somehow transforms the input 'compact waveform sets' into the output 'compact waveform set' which somehow leads to the statistical eye.
• They validate their method by comparing to a long simulation which they then dual-dirac extrapolate.  The irony is that they earlier dismissed this idea . . .
• I do like how they do a comparison of computational time of their reference vs their method.

Overall, this is an intrusive method that require intimate understanding of the circuit design.  While it is interesting it isn't practical.  It does serve the purpose of showing that the nonlinear BER problem is real.

H. Kim et al., "A fast and accurate statistical eye-diagram estimation method for high-speed channel including non-linear receiver buffer circuit," 2015 Asia-Pacific Symposium on Electromagnetic Compatibility (APEMC), Taipei, 2015, pp. 94-97.

BER analysis of high speed links with nonlinearity

I found another paper on my topic!

This was a very interesting paper obviously written by the circuit designers from a PDF perspective.  They model the non-linearity as a odd powered polynomial and use some interesting math to go from the linear PDF (or BER) to transform it to include the non-linear saturating response.  I felt that the validation of their method could have been stronger (they say they have a perfect match!) and if I wanted to replicate their results I will have to do a lot of work.

Steps:

• Determine the polynomial coefficients of the non-linearity
• determine the PDF at the input of the non-linearity f_x
• transform f_x to f_y (the output PDF) using a transform.

Overall, I'm quite impressed with the paper and their 'intrusive' solution to the problem.  I call this an 'intrusive' solution because a priori information about the non-linearity must be known to contrast this to our 'outside' solution.

[9/11/16 updates]  Notes from 2nd read through

• Statistical reminders:
• The PDF of the sum of two or more independent random variables is the convolution of their individual PDFs.
• Probability of an error event can be computed using the theorem on total probability
• I would like to learn more about the statistical property that lead to equation (17).  I have sent an email to the authors and hopefully they can help me out.

G. Malhotra and J. Kamali, "BER analysis of high speed links with nonlinearity," 2015 49th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, 2015, pp. 635-640.

No Noise is Good Noise

This article is a little out of my area of expertise but that is why I wanted to read it.

• LNA = low noise amplifier
• Two design approaches, noise matching and noise canceling.  This paper discusses this and shows that they are more similar than people previously thought.
• They use signal flow graphs but overall I'm not interested in this topic after all. 

L. Belostotski, "No Noise Is Good Noise: Noise Matching, Noise Canceling, and Maybe a Bit of Both for Wide-Band LNAs," in IEEE Microwave Magazine, vol. 17, no. 8, pp. 28-40, Aug. 2016.

Polynomial Chaos, Design of Experiments and Circuit simulations

This paper is about Polynomial Chaos (PC), DoE and Circuits.  All things I am interested in.

• PC methods can be divided into intrusive and non-intrusive.
• "On the other hand, nonintrusive PC approaches such as the stochastic collocation (SC) approach, pseudo-spectral collocation approach, and linear regression approach, among others, have recently been explored for circuit and EM problems [23]–[36]. These nonintrusive approaches evaluate the PC coefficients by solving the original stochastic network model at a sparse set of nodes located in the random space [43]. The solution at each node translates into a deterministic simulation. The deterministic simulation of the network at each node can be performed using SPICE-like legacy solvers directly without requiring any intrusive coding. In addition, the relevant deterministic simulations can be parallelized unlike the conventional SG approach where the network equations are always coupled. Among nonintrusive approaches, the linear regression approach has been found to be highly popular
• RJA: this sounds very much like the DOE approach I have done for years.  It is interesting to see the connection between my approach and what they describe. I should follow some of the references to understand better the connection of PC and DOE.
• RJA: They use D-optimal DoE and this somehow works well for PC coefficient determination.  It sounds like they spend a lot of time explaining their algorithm for obtaining the D-optimal DoE.
• RJA: Perhaps I can use their references to learn more about PC.

A. K. Prasad, M. Ahadi and S. Roy, "Multidimensional Uncertainty Quantification of Microwave/RF Networks Using Linear Regression and Optimal Design of Experiments," in IEEE Transactions on Microwave Theory and Techniques, vol. 64, no. 8, pp. 2433-2446, Aug. 2016.

Joel Harley's "Predictive Guided Wave Models Through Sparse Modal Representations"

I took the course "Advanced Signal Processing" from Dr. Harley and really enjoyed it.  I think that his paper may have some nuggets of wisdom that I could use.

• "The goal of this paper is to demonstrate how sparse wavenumber analysis can be expanded to new structures and materials. We discuss the process of defining a wave propagation model and integrating the model with sparse wavenumber analysis for three different systems: 1) standing waves in a 1-D fixed string;
• "Sparse Representation of Waves: We analytically express the impulse response of waves traveling between any two points in a medium by their Green’s function. In many cases, a Green’s function can be written as a linear combination of location-dependent eigenvectors, also known as eigenmodes
• Y = A(V), Y is the known data, V is the data's sparse representation (what is trying to be found) and A is typically an under-determined operator.
• Sparse Recovery Algorithms:
• Greedy methods
• L1- based optimization methods
• Bayesian methods
• He uses 'orthogonal matching pursuit' a greedy algorithm.
• He first applied 'orthogonal matching pursuit' to a simple fixed string system.  This would be a good opportunity to try and recreate his work on this simple system to learn more about the sparse recovery algorithm.

J. B. Harley, "Predictive Guided Wave Models Through Sparse Modal Representations," in Proceedings of the IEEE, vol. 104, no. 8, pp. 1604-1619, Aug. 2016.

Reduced -cost surrogate modeling and design optimization of dual-band antennas using response features

• Summary

• more efficient optimization methods, two classes
• surrogate-assisted techniques (local and global)
• gradient-based methods with adjoint sensitivities.
• two major classes of surrogates include 
• data driven: constructed by approximating sampled EM simulation data. Neural networks, radial-bases functions, kriging and support vector regression.
• They are fast to evaluate but require large amounts of tranining data to ensure usable accuracy.
• The number of required data samples grows quickly with number of parameters and ranges
• physics based
• exhibit better generalization (due to the embedded knowledge in the low-fidelity model)
• Do not require as many training samples
• For EM antennas, the surrogate is expensive to evaluate.
• Their method
• response feature sets are build around the coordinates of antenna resonances and supplemented with additions (infill) points
• 
  
• Here are my comments to the author
• Is there a type on page 6, line 46?   s_l,j(x) --> s_L,j(x)
• Figure 6.  Is there a reason you did not include some of the surrogate data points on top of the high fidelity response? I think that showing the data points would be a powerful way of demonstrating the technique.
• The surrogate captured the magnitude of S11 but what about the phase?  Was this not important since the structures are electrically short?  Please discuss.
• Overall, I liked the paper.  It communicated clearly and think that it contributes interesting concepts to the community.  That said, if I wanted to replicate the authors results, I don't think that the body of the paper contains enough information for me to do so.  I would like to see an explicit example of equations 1 to 3 and more explanation of kriging interpolation for those not familiar with it.

A Quasi-Analytical Tool for the Characterization of Transmission Lines at High Frequencies

I subscribe to some table-of-contents alerts from IEEE.  There are several journals and magazines which I could potentially have interest but most of the time the articles are not what I am looking for.  Yesterday, I saw this notice for the Antennas and Propagation Magazine.  Here are my notes from reading it:

• Above 50 GHz higher order modes are excited and the quasi-static assumptions which drive many of the transmission line tools of today are not that accurate.
• They have a free tool which does this.
• assume conductors are infinitesimal in thickness
• Three characteristic propagation modes are of interest
• bounded modes: nonradiative modes such as the main propagating mode in a microstrip
• surface-wave modes: propagating modes exciting a surface wave instrinsic to the stratification
• leaky-wave modes: propagating modes radiating in a dense infinite medium, as in the case of circuits placed at the bottom of a dielectric lens.
• gamma = beta(f) + alpha(f)*j
• The real part of the complex wavenumber beta is the phase constant of the main propagating mode.
• the imaginary part of the wavenumber alpha is the superposition of attenuation due to conductor, dielectric and radiative losses.
• If I ever need to worry about these higher order modes, I should review this paper again.

S. L. van Berkel, A. Garufo, N. Llombart and A. Neto, "A Quasi-Analytical Tool for the Characterization of Transmission Lines at High Frequencies [EM Programmer's Notebook]," in IEEE Antennas and Propagation Magazine, vol. 58, no. 3, pp. 82-90, June 2016.

A Stochastic Collocation Algorithm for Uncertainty Analysis

I am starting to think that Polynomial Chaos Expansion is not the right tool for my Ripple Analysis uncertainty analysis.  The previous post gave me the idea to look into stochastic collocation as a possible alternative to Monte Carlo type approach.

This paper is a NASA paper from 2003 and I hope that it is a good overview of the topic.

• Abstract
• This report describes a stochastic collocation method to adequately handle a physically intrinsic uncertainty in the variables of a numerical simulation. For instance, while the standard Galerkin approach to Polynomial Chaos requires multi-dimensional summations over the stochastic basis functions, the stochastic collocation method enables to collapse those summations to a one-dimensional summation only. This report furnishes the essential algorithmic details of the new stochastic collocation method and provides as a numerical example the solution of the Riemann problem with the stochastic collocation method used for the discretization of the stochastic parameters.
• In background section, they mention a 'perturbation technique'
• "that basically involves expanding the variables of the problem in terms of Taylor series around their mean value. Though effective, this technique is limited to Gaussian or weakly non-Gaussian processes due to the difficulty in incorporating terms of order higher than two."
• In background section, they mention Monte Carlo
• "It is considered as an “exact” method for accounting for uncertainty in the sense that it does not require any approximations nor assumptions."
• "the resulting approximated variance of a σ^2 -variance random process is σ^2_MC = σ^2/n where n is the number of independent samples"
• "Despite some techniques which improve the convergence rate (Latin Hypercube, importance sampling, etc.), this last point prevents its use for most of the practical problems currently studied by" the authors
• They have a nice discussion of polynomial chaos expansion before stating that they don't know how to apply it to their application.
• One sentence definition of stochastic collocation method
• In the stochastic collocation method (SCM) one lets the Probability Density Function (PDF) of the random variable ξ serve as the basis for the transformation between the physical random variable ξ and its artificial stochastic space
• Bijective (from wikipedia)
• In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set.
• They go through some math comparing SCM to PCE but I think overall it is straightforward.  They analyze a simple example of f = u^3, where u is distributed as Gaussian and they calculate the distribution of f.  This would be a good exercise to replicate.

A Stochastic Collocation Algorithm for Uncertainty Analysis
Lionel Mathelin and M. Yousuff Hussaini Florida State University, Tallahassee, Florida
https://perso.limsi.fr/mathelin/research/SC_CR_Mathelin.pdf

Uncertainty Quantification in Computational Science

This post is about some slides.  Although I don't have a lot of text to rely on I can tell that the slides are a nice introduction to polynomial chaos expansion.  Here are some of the key highlights:

• Difference between intrusive and non-intrusive methods:
• Nonintrusive methods only require (multiple) solutions of the original (deterministic) model
• Intrusive methods require the formulation and solution of a stochastic version of the original model
• Polynomial Chaos is an intrusive method
• Steps
• consider a truncated spectral expansion of the solution in the random space
• plug the truncated expansion in the original PDE
• multiply by phi_k(zeta)
• Integrate over the probability space (Galerkin Projection)
• Pull the deterministic components out of the integrand
• use the orthogonality properties to arrive at a system of uncoupled and deterministic equations.
• Conclusions
• The use of polynomial expansions transform the original stochastic problem into a more complex deterministic problem
• Polynomials are only one of the possible basis. wavelet are another popular choice.
• This forces us to interpret the resulting mathematical structure with potential enormous gains w.r.t. non-intrusive approaches
• It also forces you to rewrite codes!
• They go on to describe stochastic collocation methods (nonintrusive methods)
• Typical sampling methods (Monte Carlo)
• Sample the random input parameter vector according to its probability distributions
• Perform a sequence of independent simulations
• Compute statistics of the quantity of interest
• Now we will introduce an alternative way of computing the output statistics without random sampling!
• The idea is noticing that the mean and variance (Which are the first and second moments) are integrals in the probability space.  The idea is to directly calculate these integrals (moments) with advanced numerical techniques with as few of samples as possible yet still arrive at very accurate results!
• Given that the true underlying distribution is of some order of polynomial, then this result can give the exact result.  With more points sampled, the error can be estimated.
• Here is the overall summary slide:
• Sampling methods compute statistics and pdf by interrogating the parameter space
• SC (stochastic collocation) and PC (polynomial chaos) are radically different, because the moments are computed explicitly
• SC: from quadrature rules
• PC: by construction from the assumed solution representation
• How do we compute the pdf in SC and PC? 
• We interpret both SC and PC as approximate (surrogate) representations of the solution that depend continuously on the input random variables

Here is the link to the Uncertainty quantification lab at standford.  http://web.stanford.edu/group/uq/uq_home.html
It hasn't seen any attention for a few years but I might be able to find more information there.

Uncertainty Quantification in Computational Science, Gianluca Iaccarino, Department of Mechanical Engineering, Stanford University, An Introductory Lecture Series, July 1 - 8, 2011

http://web.stanford.edu/group/uq/events/pdfs/lecture_3.pdf

Wiener's Polynomial Chaos for the Analysis and Control of Nonlinear Dynamical Systems with Probabilistic Uncertainties

• This is an IEEE Control Systems Magazine introduction to the topic of Polynomial Chaos.
• They make a good point that "uncertainties in the observed data, model parameters, and implemented inputs, if not taken into account, may result in failure to realize the benefits of using optimal control"  I take this to mean, that there needs to be a good way of propagating from a system's input to its output the uncertainty of the input parameters.
• Polynomial Chaos Expansion (PCE) is based on a 1938 paper by Norbert Wiener an American mathematician and philosopher.
• I want to apply PCE to Ripple Analysis.  The type of analysis which I believe that I am interested in is "Variance analysis: The simplest way of quantifying uncertainty propagation is to compute the variance of the system response around its mean value"
• "This variance analysis can provide important information for robust design and optimization and can be used to characterize the robustness of the prediction, characterize the reachability and controllability of the system, and compute confidence levels of associated predictions."
• This other application to PCE sounds like what I am after as well "Risk analysis: Apart from variance analysis, determining probabilities that certain system characteristics exceed critical values or safety thresholds has significant importance in risk and reliability assessment."
• Here is their one sentence summary of what this paper is about
• For probabilistic uncertainty quantification in dynamical uncertain systems, this column focuses on polynomial chaos and its generalizations with intrusive projection methods, called Galerkin projections, to identify associated surrogate models, which are computationally efficient approximations of the original system. 
• Here is some good news, after mentioning that they will use formal mathematical language to define PCE they say:
• PCEs are so closely related to numerical methods for function approximation that they are regularly applied by practitioners who only have a mathematical background typical of a first-year engineering graduate student.
• But after they say that, they jump right into some sigma-algebra notation.  I think I am following but may not understand all of the implications . . .
• Step 1) transform input parameter to some normalized form
• "For a given probability distribution of theta, the first step of the polynomial chaos analysis is to transform the parameters to a set of independent random variables that are normalized. "
• The overall concept of PCE is the realization that 
• "The probability space is decoupled from the deterministic spatial and temporal spaces" which allows for cool analysis . . .
• I believe they have a typo in Example 1 where they apply the Galerkin projection to equation (18).  It looks to me that they swapped s_2 with s_3.
• Example 2 was easier to follow but they didn't go into detail of how the PCE basis functions were derived but simply announced their values.
• Overall, this is the best PCE paper so far but I still don't know how to apply it to my problem.  I think that after reading a few other papers that I will come back to this one for a re-read of page 5.

K. K. K. Kim, D. E. Shen, Z. K. Nagy and R. D. Braatz, "Wiener's Polynomial Chaos for the Analysis and Control of Nonlinear Dynamical Systems with Probabilistic Uncertainties [Historical Perspectives]," in IEEE Control Systems, vol. 33, no. 5, pp. 58-67, Oct. 2013.

http://web.mit.edu/braatzgroup/Wiener/Wiener_PCE2.pdf

Stochastic Formulation of SPICE-Type Electronic Circuit Simulation with Polynomial Chaos

At the SPI2016 there were a few talks about polynomial chaos to study variational analysis.  Jan Preibisch of Hamburg Germany gave a talk that I really liked.  I reached out to him for resources on polynomial chaos and this is a paper that he suggested I study on the subject.

• "method of efficient tolerance analysis of electric circuits based on nonsampling stochastic simulation of transients."
• I wonder what is meant by 'nonsampling' is this to contrast it with a DOE type approach?
• "faster than the Monte Carlo method" (this answers my question 'nonsampling' question above)
• PC is a spectral approach and related to the Fourier series somehow.
• "Hermite polynomial chaos is associated with the Gaussian distribution"
• "In the same way as the Fourier series makes use of orthogonal complex exponentials to describe periodic functions, Hermite polynomial chaos makes use of orthogonal Hermite polynomials to describe a stochastic process X (θ)"
• "The randomness resides in the polynomial bases. It is this characteristic that allows one to transform a stochastic problem, into a deterministic counterpart and so supports the problem-solving procedure."
• Got lost on page 7, equation (12).  I'm not sure how they arrive at this. Will come back to this paper once I've understood PCE better.

Kai Strunz and Qianli Su. 2008. Stochastic formulation of SPICE-type electronic circuit simulation with polynomial chaos. ACM Trans. Model. Comput. Simul. 18, 4, Article 15 (September 2008), 23 pages.

Efficient Computation of Localized Fields for Through Silicon Via Modeling Up to 500 GHz

I recently attended the IEEE Workshop on Signal and Power Integrity (spi2016.org) and one of the presenters (D. Dahl) mentioned that they have an analytic via solver and this paper details this work.

• Focused on through silicon vias (TSV) which are used to connect IC dies
• Compares their work to and FEM analysis
• this paper focuses on the localized near field electromagnetic properties of TSV transitions in closed cavities.
• Richard: One major difference between TSV and PCB vias is the PCB via traverses several cavities created by ground and power planes.
• FDFD (Finite Difference Frequency Domain) simplification to Maxwell's equations are used to determine the physics bases behavior.
• Use the PEC (Perfect electric conductor) assumption for the metal with the justification of a FEM study showing that above 100 MHz that this is a good approximation.
• Using this assumptions they are able to find expressions for the Magnetic field in the phi direction and the port Admittance.  The expressions are in the form of sparse matrices.
• Reference [7] is really loved by the author.  This seems to be a foundation to much of the work done here.  A. G. Williamson, "Radial-line/coaxial-line junctions: Analysis and equivalent circuits", Int. J. Electron., vol. 58, no. 1, pp. 91-104, 1985
• There are four ports in the via model, A is a circle around the outer edge of the the top reference anti-pad metal, B is a circle around the bottom edge of the the top reference anti-pad metal, C is the radial surface of the signal via barrel and D is the radial surface of the ground via barrel.
• Figure 8 shows S-parameter comparison between the proposed method and several other analytic and numerical results.  Overall, the comparison shows that the proposed method matches the other approaches.  Note the upper bound of 500 GHz!
• They have comparisons of their via across variations in plane perforations and signal/ground via placements
• They conclude that their FDFD method is an efficient way of determining the near-field properties of TSV.
• Note: In the authors SPI 2016 paper they use these results to calculate the impact of huge arrays of TSV.

D. Dahl, X. Duan, I. Ndip, K. D. Lang and C. Schuster, "Efficient Computation of Localized Fields for Through Silicon Via Modeling Up to 500 GHz," in IEEE Transactions on Components, Packaging and Manufacturing Technology, vol. 5, no. 12, pp. 1793-1801, Dec. 2015.

Multi-level pseudo-random signal design and “model-on-demand” estimation applied to nonlinear identification of a RTP wafer reactor

• "Multi-level pseudo-random sequences (m-level PRS), in contrast, allow the user to highlight nonlinear system behavior while manipulating the harmonic content of the signal to enable unbiased estimation of the linear dynamics in the presence of nonlinearities [ref].
• A problem with excitation patterns is quantified in the variance/bias tradeoff
• They state that m-level PRS have an autocorrelation function similar to white noise.  I've requested their reference on this point from the library.  Perturbation Signals For System Identification.  I'm curious how they prove the auto correlation.
• The number of levels m should be at least one greater than the nonlinearity order in the candidate model.
• They talk about suppressing harmonics but I'm not sure what they mean by this.

Overall, this looks like a nice application of the m-level PRS.  I wonder how we can quantify the nonlinearity order of our target model.

M. W. Braun, D. E. Rivera, A. Stenman, W. Foslien and C. Hrenya, "Multi-level pseudo-random signal design and “model-on-demand” estimation applied to nonlinear identification of a RTP wafer reactor," American Control Conference, 1999. Proceedings of the 1999, San Diego, CA, 1999, pp. 1573-1577 vol.3.

Multi-Level Pseudo Random Sequence Generation for Coherent Optical Transmission Systems

I was attracted to this paper because it involved multi-level pseudo sequences for communication links.

• discuss that while PRQS and higher level sequences can be generated with Galois Fields it is not practical to do so on chip.
• One option is to use multiple PRBS sequences of the same order with different delays.  With the problem that 'maximal symbol sequence coverage is not guaranteed 'which leads to 'inconsistent or inaccurate BER estimates' [see reference]
• Their contribution is that they have a straightforward way of determining the needed delays to have maximal coverage.
• "a de Bruijn sequence is required for maximal symbol sequence coverage to characterize a link that uses multiple-bit symbols.
• "Using the proposed algorithm for an N-bit PRBS shown in Fig. 2, the percentage of m-bit symbols generated out of 2m−1 possible symbols can be determined. This algorithm can be extended to a more general case where an N-bit PRBS is used for generating m-bit symbol sequences for non-zero dispersion, which causes ISI due to memory in the channel
• They have a slick algorithm which will let you know what percentage of sequences are included with a given set of delays.  The objective is to find a maximal length sequence

These guys have a nice algorithm for finding maximal length m-level sequences from PRBS sequences.

M. Singh and S. Gupta, "Multi-Level Pseudo Random Sequence Generation for Coherent Optical Transmission Systems," in IEEE Communications Letters, vol. 18, no. 10, pp. 1723-1726, Oct. 2014.

Volterra Series: Wikipedia

Notes:

• "The Volterra series is a model for non-linear behavior similar to the Taylor series
• "It differs from the Taylor series in its ability to capture 'memory' effects.
• "Used in electrical engineering to model intermodulation distortion in . . .  frequency mixers.
• "Its main advantage lies in its generality: it can represent a wide range of systems.  Thus it is sometimes considered a non-parametric model.

We typically model non-linear behavior as a 'memory-less non-linearity'. I wonder if we do this to avoid the direct use of Volterra series.

h_n are the Volterra coefficients

• "If N is finite, the series is said to be truncated.  If a, b, and N are finite, the series is called doubly finite.
• "Estimating the Volterra coefficients individually is complicated since the basis functional of the Volterra series are correlated. 

They have some in depth math on how to use the correlation approach to estimate the Volterra coefficients.  I would really like to have a concrete example of how it works.  I've requested two books from the library on Volterra series and hopefully they will be useful.

Nonlinear system identification with pseudorandom multilevel excitation sequences

• I need to understand better what a Volterra series is
• Wikipedia article looks interesting.  Mentioned that EE use the Volterra series to model intermodulation distortion, which is essentially what I'm up against.
• what is a 'finite degree of nonlinearity'?  How does one quantify the degrees of a nonlinearity?
• If the non-linearity is of polynomial degreee N, then the muli-level sequence needs N+1 distinct levels for 'complete' identification.
• They say that have a least squares identification algorithm which avoids forming the inverse of the data matrix!
• They use the Kronecker Product
• From wikipedia: If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is the mp × nq block matrix"
• I skimmed the rest of the paper, lots of heavy math.  I'm not sure how much of this applied to me.  They fixate on the maximal length properties of the multilevel sequences but not on their autocorrelation.
• Overall, this was an interesting perspective on the problem.  I need to determine if sending a N-level signal into a SerDes will send it off into the weeds.  Perhaps I would need to fix everything (including CDR) before this.

R. D. Nowak and B. D. Van Veen, "Nonlinear system identification with pseudorandom multilevel excitation sequences," Acoustics, Speech, and Signal Processing, 1993. ICASSP-93., 1993 IEEE International Conference on, Minneapolis, MN, USA, 1993, pp. 456-459 vol.4.

Identification of Nonlinear Systems by Heinz Ubehauen

This book chapter (well the first 8 pages) was a very interesting read and good overview.  One of the first things to catch my eye was the following statement:

For the selection of the input signal or input sequence two cases have to be considered: Only signals of normal operation of a system are allowed to be used.  Application of specific test signals, as for example, a pseudo-random binary sequence (PRBS) for linear models and pseudo-random multi-level sequences (PRMLS) for nonlinear models is allowed.

I find the standard use of PRMLS very interesting and need to dig deeper into the literature to see if I can find more information on this.

He has a fabulous table where he gives twenty examples of non-linear behavior.  Currently I am only considering the saturation non-linearity but I need to keep the others in mind as well.

I also liked his 'easy and rough' nonlinearity tests, which I've done before but his perspective gives clarity to the procedure.  One simply plots the input voltage vs the output voltage but you delay the input voltage by the delay of the system.  If the resulting scatter plot is a perfect line, then the system is linear and non-linear otherwise.  The point which I was reminded of is how important the delay is in this plot.  One usually should search around for the best delay before making any analysis decisions.

He has a nice tree of analysis approaches of which he lists "Volterra Series Model", "Generalized Frequency Response Model", "Step response Model" and "Phase Plane Model" under the nonparametric approaches.  These are all avenues of further literature research.

The link at the bottom leads to the first 8 pages of the chapter.  I requested through the University inter-library loan the full chapter which they did send to me but the scan is of such poor quality that most of it was illegible (especially the equations and figures).  If given the opportunity I would like to read the whole chapter as it seems to be an excellent summary of the topic.

[Ubehauen] H. Ubehauen, "Identification of Nonlinear Systems," in Control Systems, Robotics and Automation, Vol VI, UNESCO-EOLSS publishers.

http://www.eolss.net/sample-chapters/c18/e6-43-10-00.pdf  [First 8 pages]

System Identification, Theory for the User, 2nd Ed. Lennart Ljung

This was a very interesting book.  I obviously just jumped around and read the sections which were immediately relevant to me but would like to go back and give it a good cover to cover treatment.

Section 5.2: I learned about the Wiener and Hammerstein parametric non-linear models, which represent the system as a non-linear function followed by a linear function (Wiener) and vice verse for Hammerstein.  I've use this type of approach before in modeling non-linear behavior of IBIS-AMI model.  But this isn't why I'm reading the book.

The realization that I am not interested in parametric system identification (like I would be if I were estimating the UI-spaced FIR taps of a system) but non-parametric system identification.  Chapter 6 addressed this approach and there I found sections on impulse-response analysis and step-response analysis. He makes an interesting comment in discussing step responses after analyzing its error:

If we really aim at determining the impulse-response coefficients using [step response equation], we would suffer from large errors in most practical applications.  However, if the goal is to determine some basic control-related characteristics, such as delay time, static gain, and dominating time constants, step responses can very well furnish that information to a sufficient degree of accuracy.

This is the exact perspective we are taking but are using the waveform to estimate the system BER.

Overall, this book pointed me in the correct direction of investigating non-linear non-parametric system identification.

[Ljung1999] L. Ljung, System Identification, Theory for the User, 2nd Ed., NJ: Prentice Hall, 1999.

What should I call it?

What should I call it?

• reflective distortion
• reflective interference -- I think that this term is the most direct and communicable term.
• resonant interference
• standing wave distortion
• Most accurate description but not really a familiar enough term for signal integrity engineers
• standing wave interference
• reflective noise
• noise has a connotation of randomness but the reflections are not random!
• inter-symbol interference
• Most people interpret ISI to be from loss
• reflective inter-symbol interference
• descriptive options
• resonant ISI
• would also work
• resonance
• resonant distortion
• multiple reflection noise
• multiple reflection interference

Wikipedia: Grey box model

https://en.wikipedia.org/wiki/Grey_box_model

A Grey box model:

combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of the result, to models that need only parameter values from data or existing literature.  Thus, almost all models are grey box models as opposed to black box where no model form is assumed or white box models that are purely theoretical.

This sounds a lot like my inversion theory class.

They have a good point about model validation.  In our approach we only have a single data set from which we identify the system.  Therefore, our extracted model perfectly predicts this data set.  Perhaps we need a few more data sets by which we can quantify the error inherit in the model estimation.

They reference a work which basically does L1 regularization on their examples of Grey-box modeling.

Approaches to Identification of Nonlinear Systems

Apparently this guy has (literally) written the book on system identification and this paper is his synopsis of non-linear identification techniques.

Question.  Is our approach a parametric or non-parametric method?  Definitely parametric since we form a model of the impulse response.  Therefore, section 3 "Parametric Methods: A Palette of Grey Shades" is applicable to my application.  There is a lot of confusion on the topic, partly because of the

negative definition ("non"-linear): it has been commented that this area is as huge as "non-elephant zoology" (Quote attributed to mathematician/physicist Stan Ulam)."

Here is how he categorizes the approaches

• White models (based on known physics)
• Off-white models: Parameterized Physical Models: (based on partially known physics, i.e. some parameters may not be known)
• Smokey-Grey Model: Semi-physical Modeling: "finding nonlinear transformations of the measured data, so that the transformed data stands a better chance to describe the system in a linear relationship" -- this sounds closest to our application.  He references his own book so I'll pick it up from the library tomorrow to explore some examples.
• Steel-Grey Models - different models are needed depending on the operation conditions
• Composite Local Models: linearization around a working point.  Form a family of local models and then compose a global model from these.  Sometimes called "local linear models"
• Linear Parameter Varying (LPV) Models: ?
• Slate-Grey Models: variant of LPV models?
• Hybrid Models: A piecewise linear model which switches between different states as the state or operating point changes.  Obviously related to the previous 3 types.
• Block-oriented Models: 
• Black Models: Basis Function Expansion:  

Overall, the time spent reading section 3 was well worth it as it hopefully will help me connect our method to literature of the same kind.

L. Ljung, "Approaches to identification of nonlinear systems," Control Conference (CCC), 2010 29th Chinese, Beijing, 2010, pp. 1-5.

Available online at: http://users.isy.liu.se/en/rt/ljung/ccc/ljung_paper.pdf

Inversion of Circulant

If your matrix is circulant then the eigenvalue decomposition is practically free since the eigenvectors are known a priori.

D. Bozkurt, "On the Determinants and Inverses of Circulant Matrices with a General Number Sequence," Feb. 2012, http://arxiv.org/abs/1202.1068

I.J. Good, “On the inversion of circulant matricies,” Biometrika, Vol 37, No. 1/2 June 1950, pp. 185-186

DC Extrapolation with causality constraint

This paper has already pointed me to two references, one from 1996 which used "causality base interpolation and extrapolation" and 2006 (Triverio).

They describe an iterative process which uses  causality as a constraint in the DC extrapolation.  It merits a more in depth look as they seem to describe their algorithm really well.

H. Shi, "A Refinement Procedure for S-Parameter DC Extrapolation based on Sampling Theorem and Causality," Electrical Performance of Electronic Packaging, 2007 IEEE, Atlanta, GA, 2007, pp. 43-46.

An efficient and simple algorithm to restore passivity for measured S-parameters data

These guys have an interesting approach to correcting the passivity violation of S-parameters directly.  The naive approach is to simply scale the S matrix at those frequencies which have a max eigenvalue greater than 1 until the max eigenvalue is less than or equal to 1.  This causes discontinuities and really isn't that helpful.

What these authors do is to select a region around the discontinuity and scale the whole region down.  This results in a smoother max eigenvalue response.

My problem with these approaches is that if your measurements are this bad you should just accept that above some some frequency you have bad data and truncate it.  The reality is, more information about the structure of the network is required to make these kinds of corrections.

A. Chakrabarti, S. Agili, A. Morales and M. Resso, "An efficient and simple algorithm to restore passivity for measured S-parameters data," Consumer Electronics (ISCE), 2015 IEEE International Symposium on, Madrid, 2015, pp. 1-2.

A new method for causality enforcement of DRAM package models using discrete hilbert transforms

Avoids the vector fitting approach by extrapolating the frequency response.  Correctly notes that the extrapolation reduces the boundary artifacts around the original max frequency.  They don't give many clues as to how they do the extrapolation . . . but I have a good guess of what they are up to.

I wonder how this extrapolation method compares to the filtered dispersion relations approach.

H. Aboutaleb, L. L. Baranny, A. Elshabini and F. Barlow, "A new method for causality enforcement of DRAM package models using discrete hilbert transforms," Microelectronics and Electron Devices (WMED), 2013 IEEE Workshop on, Boise, ID, 2013, pp. 21-24.

An accurate, robust and intuitive technique to detect causality violations in broadband frequency measurements

From the abstract: The use of a low pass filter makes the dispersion relationships much easier to implement and understand than previous approaches.  If this is easier to understand and implement then they certainly have caught my attention as their 2008 was very complex.

Sources of error in applying the Dispersion Relations

• truncation error. 
• s-parameters do not necessarily decay to zero (example is S11) as frequency increases.

He calls his 2008 approach 'Dispersion relations with subtractions'.  He says that this was an improvement but is complicated and "does not provide an accurate estimation of the detected violation."

His 2013 paper introduced 'filtered dispersion relations'.  "Through the use of a low-pass filter instead of subtraction points, this method features a rigorous minimization and estimation of truncation artifacts."

The choice of filter type and order is a trade off between truncation error and a distortion factor (introduced by the filter).  He tests his algorithm by creating non-causal s-parameters.

P. Triverio, "An accurate, robust and intuitive technique to detect causality violations in broadband frequency measurements," Electromagnetic Compatibility (EMC), 2014 IEEE International Symposium on, Raleigh, NC, 2014, pp. 815-820.

Robust Causality Characterization via Generalized Dispersion Relations

On the simplistic causality check: "Apply the dispersion relations and take the difference between the result and the original data . . . .The approximation errors due to the finite set of available samples may be so large to compromise the resolution of the causality test."

Two conditions must hold for insuring a sound numerical causality test. First, the numerical error in the evaluation of the dispersion relations must be small. Second, a good estimate or a bound for this error must be available, in order to quantify the numerical resolution of the test. Causality violations will be detectable only when larger than this numerical resolution.

This is a classic paper and will need to be studied carefully.

P. Triverio and S. Grivet-Talocia, "Robust Causality Characterization via Generalized Dispersion Relations," in IEEE Transactions on Advanced Packaging, vol. 31, no. 3, pp. 579-593, Aug. 2008.

Causality of tabulated S-Parameters

Lalgudi has a pair of papers (conference and transaction) dealing with a way to test the causality of tabulated S-parameters.  Here I focus on the conference paper since it is shorter but intend to come back to the transaction paper later.

The author see their Causality checking solution to account for interpolation error which they say Triverio in his 2008 "robust causality characterization via generalized dispersion relations" does not account for.  It might be interesting to trace the back and forth between the two authors as Triverio is one of the few who reference Lalgudi's transaction paper.

I wonder if the dueling papers can be see as two approaches to time domain simulation from S-parameters.  1) the convolution approach by using the iFFT and 2) the macro-modeling approach using rational transfer functions.  Here is the authors summary:

In this paper, the work in [3] is extended to use a global (i.e., nonpiecewise) rational function approximation for tabulated frequency responses. As the tabulated frequency response can be noncausal, the rational function is allowed to have both positive and negative real parts. There are some advantages to this extension: (a) Because of better approximation properties of rational functions over piecewise polynomials, the resolution of the checker (defined as smallest noncausal violations that can be detected for the given tabulated frequencies) is improved. (b) The closed-form expressions for the numerical GHT in [3] can produce unbounded values if tabulated frequencies are highly nonuniformly spaced, like in a logarithmically-spaced frequencies. However, this problem does not occur with the expression using global rational functions. (c) Though global rational functions lead to an approximation problem (as opposed to an interpolation problem), they lead to a more accurate GHT result.

I wish someone would write a paper showing the impact of non-causal responses on the time domain simulation results.  I have a nagging feeling that while it can be severe in artificial or contrived cases in general it isn't that bad.

S. Lalgudi, S. Asgari and M. Tsuk, "Improved procedure to test causality of tabulated S-parameters," Electrical Performance of Electronic Packaging and Systems (EPEPS), 2012 IEEE 21st Conference on, Tempe, AZ, 2012, pp. 191-194.

S. Lalgudi, "On Checking Causality of Tabulated S-Parameters," in IEEE Transactions on Components, Packaging and Manufacturing Technology, vol. 3, no. 7, pp. 1204-1217, July 2013.

Least squares convolution: A method to improve the fidelity of convolution in transient circuit simulation

This paper by Ansoft discusses some of the problems with the convolution approach.  They acknowledge that it seems to be very straightforward but note subtleties.  They state that the computation of the impulse response through a direct iFFT of any finite bandwidth is non-causal except in special cases.  They note that one common way of overcoming this is to extrapolate to high frequencies to mitigate this issue.

They have a good discussion of the properties and assumptions of the FFT and how it applies to the simulation of time domain waveforms.  Interesting approach.  I wonder if this is the engine they use in the Ansys (formerly Ansoft) time domain solver.

M. Tsuk and S. Lalgudi, "Least squares convolution: A method to improve the fidelity of convolution in transient circuit simulation," Electrical Performance of Electronic Packaging and Systems, 2009. EPEPS '09. IEEE 18th Conference on, Portland, OR, 2009, pp. 73-76.

NTS: I need to follow up and read their reference #1 which they heavily rely on.

Mismatch Uncertainty

They are interested in measuring the group delay ripple of their networks.

Interesting connections.  They state that the Hilbert transform of the magnitude of the frequency response is used in the minimum phase assumption.  I hadn't connected the two before but it makes perfect sense, if you know the magnitude then you can predict the phase via the Hilbert transform.  Minimum phase assumption only work for those structures which have all their poles and zeros in the left-hand side of the complex frequency plane, including the jw axis.

They are primarily interested in microwave circuits so they have to qualify where this Hilbert transform can be used as they state that some circuits are non-minimum phase.  Luckily for me, they state that all passive circuits with a single input and output can be assumed to be minimum phase.

They make the interesting observation that if in the production of microwave filter, first measure it with an expensive VNA and see if the minimum phase relationship holds.  If it does then the less expensive scalar network analyzer (SNA) can be used on the production floor.

In discussing the error sources of their approach they diagram the multiple reflections of the system of three s-parameters!  They call this 'mismatch uncertainty'. This is really done in the context of the VNA calibration but interesting none the less.

P. Perry and T. J. Brazil, "Hilbert-transform-derived relative group delay," in IEEE Transactions on Microwave Theory and Techniques, vol. 45, no. 8, pp. 1214-1225, Aug 1997.

Forcing causality on S-parameter data using the Hilbert transform

This is an early paper discussing the issues of using S-Parameters in Time Domain simulations via the FFT.  They note the periodic assumption, and Gibbs phenomenon (typically reduced by windowing) and causality issues.

It appears that they were dealing with end point discontinuity with the FFT.  For some reason they were applying the FFT to only the positive frequencies and of course there was a mismatch!  The example they give was a S11 response and they interpolated so that the highest frequency and lowest frequency magnitude were the same.  I don't think that this would work that well for S21 where there is no expectation that the highest and lowest frequency magnitudes should be the same.

They do mention that they use the chip-z transform to go from the frequency domain to the time domain.  I should familiarize myself with this approach.

P. A. Perry and T. J. Brazil, "Forcing causality on S-parameter data using the Hilbert transform," in IEEE Microwave and Guided Wave Letters, vol. 8, no. 11, pp. 378-380, Nov 1998.

Fast Passivity Enforcement for S-Parameter Models by Perturbation of Residue Matrix Eigenvalues

This is an algorithm to enforce passivity in macromodels.  His literature review is excellent as is usual for Gustavsen.  In the review he mentions Hamiltonian matrix eigenvalues, I really need to figure out what these things are.  He does mention the previously reviewed paper by Sarawat2010 but goes well beyond that.  He mentions that all elements of the algorithm has been discussed elsewhere but he brings it together and provides many of the implementation details.

He talks about constraints, least squares and other approaches he used so when I get around to implementing my own technology, re-reading this paper will be useful.

B. Gustavsen, "Fast Passivity Enforcement for S-Parameter Models by Perturbation of Residue Matrix Eigenvalues," in IEEE Transactions on Advanced Packaging, vol. 33, no. 1, pp. 257-265, Feb. 2010.

Restoration of passivity in S-parameter data of microwave measurements

They discuss the two ways of using S-parameter data in simulations!  Up to this point I have only seen the macromodel approach acknowledged but myself is really only familiar with the convolution approach.

Transient analysis involving tabulated data can be mainly accomplished via convolution based approaches [8], [9] or passive macromodels [3]–[4] [5] [6] [7]. If the convolution based approach is used, the frequency-domain S-parameter data is converted into time-domain using the IFFT algorithm. Subsequently, transient responses are evaluated by convoluting the IFFT response with the input responses. However this can run into convergence problems if the tabulated data is non-passive.

On the other hand, transient analysis using macromodels requires that the passivity of the macromodel is ensured, to guarantee asymptotic stability. Passive macromodeling algorithms available in the literature for this purpose are based on approximating the tabulated data through rational-functions and subsequently checking/compensating for any passivity violation [3]–[4] [5] [6] [7]. The time/effort spent in compensating the rational function model depends on the amount of the passivity violation in the rational-function fitted model. This in turn is influenced by the amount of the passivity violation present in the original tabulated data.

Their technique is to analyze the non-passive frequencies through eigenvalue analysis and perturb the S matrix such that the max eigenvalue is reduced.  Overall, they do some nice analysis that is straightforward to understand and looks like they got good results.

D. Saraswat, R. Achar and M. Nakhla, "Restoration of passivity in S-parameter data of microwave measurements," Microwave Symposium Digest, 2005 IEEE MTT-S International, 2005, pp. 4 pp.-.

Enforcing Passivity for S-Parameter Raw Data

While searching for references on the passivity correction of S-parameter I found this interesting presentation from 2010.  The slide below shows two common approaches to passivity correction but also illustrates that if such a passivity violation occurs more than just the violating frequencies need to be corrected.

This goes further to motivate why the passivity correction of the S-parameters themselves is not sufficient but it needs to be addressed for the whole system as can be seen below.

It will be interesting to see how my technology stacks up against these approaches.

Source: https://ibis.org/summits/nov10b/tseng.pdf

An improved phase de-embedding technique for high speed connectors

This short paper addresses the issue of the sensitivity of the de-embedded DUT S-Parameter due to the test fixture S-Parameter uncertainty.  They show that the S11 and S22 terms are twice as sensitive to the length uncertainty of the fixture S-Parameter as the S21 and S12 terms.

This is a good supporting paper for my literature review as it motivates the issue that while the de-embedding process is deterministic, the inputs are not and therefore this de-embedding is not a stable process.

D. Campbell, A. Morales and S. Agili, "An improved phase de-embedding technique for high speed connectors," Consumer Electronics (ICCE), 2010 Digest of Technical Papers International Conference on, Las Vegas, NV, 2010, pp. 211-212.

A Comparative Study of Passivity Enforcement Schemes for Linear Lumped Macromodels

This paper deals with the passivity of macro-models but gives hints that they also consider passivity correction at discrete frequencies which I take to apply directly to S-Parameters themselves.  Further reading shows that they sample the macro-model to determine the discrete frequencies.

S. Grivet-Talocia and A. Ubolli, "A Comparative Study of Passivity Enforcement Schemes for Linear Lumped Macromodels," in IEEE Transactions on Advanced Packaging, vol. 31, no. 4, pp. 673-683, Nov. 2008.

Passivity Check of S-Parameter Descriptor Systems via S-Parameter Generalized Hamiltonian Methods

This paper caught my eye because I believe that I understand well how to check for passivity violations in S-parameters, i.e. simply check that the max eigenvalue from S*S' is less than 1 at each frequency but this paper is taking a very different perspective to the problem.

They introduced me to 'Descriptor Systems' (DSs) which I take as some generalization of the S-parameters and appear to come to use from control theory.  I am trying to figure out how this perspective is useful.

One interesting side note is that they briefly discuss fitting macro-models to S-parameter data and state that

Recently, the Loewner matrix interpolation technique [16], [17] has been advocated to fit measured/simulated data of electronic circuits/systems to produce the corresponding DS. Such framework is superior to the traditional vector fitting approach in the sense that no manual pole initialization is needed and that the optimal model order can be automatically extracted.

I find this interesting that they think that this Loewner matrix approach is much superior to the vector fitting and requires more investigation.

After reading the introduction section (they have 35 references of background info!) I still can't understand the motivation or what they are doing . . . I will put Descriptor Systems on my list of things to further investigate.

Z. Zhang and N. Wong, "Passivity Check of S-Parameter Descriptor Systems via S-Parameter Generalized Hamiltonian Methods," in IEEE Transactions on Advanced Packaging, vol. 33, no. 4, pp. 1034-1042, Nov. 2010.

Floating Tap DFE paper from LSI

Notes:

• "To remove reflection-induced ISI due to impedance discontinuities in the media, the DFE must cover tap positions higher than 40 UI which is beyond tap range of the DFEs reported in previous work."  (Are these other work just fixed taps?)
• "Key contributions of this work are: 1) a 14.025 Gb/s half-rate 10-tap DFE which employs direct feedback for the first tap, and four floating taps with positions independently adapted to any four out of the 32 tap positions ranging from 7 to 38.
• Figure 2 is a nice block diagram of the their receiver.

F. Zhong et al., "A 1.0625  14.025 Gb/s Multi-Media Transceiver With Full-Rate Source-Series-Terminated Transmit Driver and Floating-Tap Decision-Feedback Equalizer in 40 nm CMOS," in IEEE Journal of Solid-State Circuits, vol. 46, no. 12, pp. 3126-3139, Dec. 2011.

Telian Papers on Trouble Shooting Reflective Channels

Simulating Large Systems with Thousands of Serial Links, section 3.3 "Discontinuity-Induced Resonances"

An excellent example of analyzing a system with reflective noise is provided by Donald Telian in 2012.  After identifying a worst case performing channel in a system simulation with thousands of links, he first analyzes the TDR waveform and relates the largest impedance discontinuities to the PCB vias on either side of a connector model and rightly identifies the reflective noise as a discontinuity induced resonance.  The particular connector has numerous rows each with a different length and thus different electrical delay.  He relates electrical delay to resonant frequency and picks out the connector rows which most impact the data rate they are targeting.  Overall, this is a good example of how to analyze reflective noise, relate it to physical structures so that redesign is done systematically.

[Telian2012a] D. Telian, S. Camerlo, M. Steinberger, B. Katz, W. Katz, "Simulating Large Systems with Thousands of Serial Links," presented at the DesignCon 2012 Conference, Santa Clara, CA.
(link to paper)

Moving Higher Data Rate Serial Links into Production - Issues & Solutions, section 2.1 "Removing Discontinuities" and section 2.3.2 "Layout Adjustments: Irregular and Discontinuous Distributed Impedances"

Section 2.1 Removing Discontinuities

Notes that SerDes equalization is primarily aimed at compensating for loss in longer channels and not for reflective noise.  In his situation, the short reflective channels are performing the worst so the channels need to be redesigned by removing the impedance discontinuities to achieve desired performance targets.

He shows how if the impedance discontinuities are removed the performance improves drastically (he does not state how they are removed, but it has to be either the removal of the problematic via block or the theoretical redesign of the via block).  Here is a great quote: "This performance improvement is impressive when you consider it required only a slight adjustment to 1% of the channel’s interconnect, demonstrating that the challenge is knowing which elements to adapt and in what way."

Section 2.3.2 Layout Adjustments: Irregular and Discontinuous Distributed Impedances

A TDR of a measured PCB trace is shown and it is noted that the impedance profile is not consistent.  Four possible sources of this inconsistency is noted, 1) serpentine routes (and thus common to differential mode conversion noise) 2) reference plane voids, 3) fiberglass weave, and 4) reflections.

Another measured TDR is shown with features (i.e. the discontinuities) of the physical structure identified.  He notes that skill must be used to relate the TDR to structure and surmise the sources of the wiggles (my word) of the TDR.

He lastly notes that some of the wiggles in the TDR are due to multiple reflections and relates some of these wiggles to various round trip resonance times.  The wiggles identified are out in time from the main structures and multiple reflections are the only explanation for them.  It would be interesting to see understand how many of the wiggles in the main section of the TDR are due to multiple reflections as well (my analysis).

[Telian2014]  D. Telian, S. Camerlo, K. Matta, M. Steinberger, B. Katz, W. Katz, "Moving Higher Data Rate Serial Links into Production - Issues & Solutions," presented at the DesignCon 2014 Conference, Santa Clara, CA.
(link to paper)

Signal Flow Graph

How Many Loops in the following Signal Flow Graph?

Example of cascading s-parameters with slight numerical irregularities that the combined S-parameter have issues with passivity and causality

This paper gives a short example of cascading 52 s-parameters with slight numerical irregularities that the combined S-parameter have issues with passivity and causality.  They also perform some causality enforcement which looks suspect.

This would be a good paper to motivate the need for my technology.

Cocchini, M.; Becker, W.D.; Katopis, G.; Pytel, S.G., "Time-domain simulation of system interconnect using convolution and Newton-Raphson iteration methods," in Electronic Components and Technology Conference (ECTC), 2010 Proceedings 60th , vol., no., pp.646-651, 1-4 June 2010

De-embedding method and segmentation approach using full wave solver for high speed channels in PCB/package co-design

The question this paper addresses is, can a electrical models of sub sections be solved separately and then cascaded together to represent the whole system?  How does it compare to a simulation of the whole system.

For the segmentation to work, each EM simulation port of the sub section needs to be excited by an TEM wave.  But sometimes the structure of the sub-section doesn't allow for this.  Therefore, extend the port so that it can support a TEM wave and then de-embed the extension.  Actually, that isn't what they are doing.  I'm not entirely sure of what they are doing but they are de-embedding structures.

Essentially from my perspective, this is an application of de-embedding where small errors could easily creep into the final result.

LianKheng Teoh; ChunTong Chiang; Scogna, A.C.; Khrone, K.; HsuenYen Lee, "De-embedding method and segmentation approach using full wave solver for high speed channels in PCB/package co-design," in Electrical Design of Advanced Packaging & Systems Symposium (EDAPS), 2014 IEEE , vol., no., pp.33-36, 14-16 Dec. 2014

A de-embedding method for extracting S-parameters of vertical interconnect in advanced packaging

Most IEEE papers on De-embedding are focused on silicon measurements.  This was one of the few papers I found that focused on interconnect de-embedding.

Notes: 

Uses T-Matrix De-embedding.  Claims that method doesn't have frequency limitations.  Obtained fixture S-parameter from EM simulation.  

If I can get my technology working, this paper shows that it would likely be useful to ensure that the DUT S-parameter is a compromise between accuracy, passivity and causality.

Yin-Cheng Chang; Hsu, S.S.H.; Da-Chiang Chang; Jeng-Hung Lee; Shuw-Guann Lin; Juang, Y.-Z., "A de-embedding method for extracting S-parameters of vertical interconnect in advanced packaging," in Electrical Performance of Electronic Packaging and Systems (EPEPS), 2011 IEEE 20th Conference on , vol., no., pp.219-222, 23-26 Oct. 2011

De-embedding in High Speed Design - DesignCon 2012

Objective: Extracting the DUT response from the fixture + DUT + fixture system response.

The classic de-embedding problem is, given the measured S-Parameters of the composite system (DUT + fixture) recover the S-Parameters of the DUT alone.  He spends a lot of time (rightly) on how to obtain good S-parameter estimation of the fixture.

De-embedding may be thought of as the process that extends the Calibration Reference Plane to the DUT Reference Plane.

He uses a matrix notation to do the de-embedding calculation.  While somewhat cumbersome for 1 or 2 ports it proves useful for N ports.

The rest of the paper is devoted to ways of obtaining the fixture S-parameter so it can be de-embedded.  He uses TRL, second tier calibration and time gating.

Overall, this was a nice review of the technology.

[DeGroot2012] D. DeGroot, K. Doshi, D. Dunham, P. Pupalaikis, “De-embedding in High Speed Design,” presented at the DesignCon 2012 Conference, Santa Clara, CA.

A VERSATILE SPECTRUM SHAPING SCHEME FOR COMMUNICATING BEYOND NOTCHES IN MULTI-DROP INTERFACES - Ali Hormati -- DesignCon 2016

So what do you do when you have really large resonances in your system?  One option, as proposed in this paper is to code it such that you send no energy at the notch frequency.  One major benefit is that no fancy equalization or receiver circuit modification are needed.  The idea is to use the channel reflections to our advantage.  If the notch frequency is known, then the data is sent twice so that the noise from the first repetition adds constructively with the second set.  This results in two closed eyes, followed by two open eyes.

The use of coding to resolve the reflection noise issue is certainly a novel approach.

A. Hormati, "A Versatile Spectrum Shaping Scheme for Communicating Beyond Notches in Mulit-Drop Interfaces," presented at the DesignCon 2016 Conference, Santa Clara, CA.

Don't let your Transmission Lines be resonant lengths

This early paper focuses on "reflection noise and its reduction" which is the same subject for my research.  The whole idea is that in 1996 the transmitters and receivers were terminated on the PCB and thus a large stub which created resonances was a big issue.  Their approach was to figure out which lengths to avoid on the PCB so that the reflective noise didn't arrive at the receiver near the sampling time.  Now days this is obvious but likely a novel and unique insight back in the 90's.

Blennemann, H.; Yao-Chao Yang; Nikel, R., "Off-chip 400 Mbps signal transmission: noise reduction using non-resonant lengths and other techniques," in Multi-Chip Module Conference, 1996. MCMC-96, Proceedings., 1996 IEEE , vol., no., pp.138-142, 6-7 Feb 1996

Genetic Algorithm for Transmission Line Design

I have been interested in Genetic Algorithms (GA) for quite a while now, so this was very interesting to see its application to transmission line design.  They motivate the paper by stating that traditional impedance matching techniques, popular in microwave circuit design, are targeted to a single frequency and are not broadband as needed for baseband wire signaling.  They demonstrate how they represent a transmission line as a chromosome of impedance and lengths and discuss how they perform selection, crossover and mutation of the population.  While the impedances of each section are independent, the lengths are not, so they used what they called a BX-alpha crossover scheme which was well described.  Lastly, the fitness of each member was determined by comparing the resulting waveform to an ideal waveform of a single bit pulse response and integrating the error between the two. They report the successful application of the GA to the design of a DDR DIMM and a backplane and even xtalk reduction.

I would love to be able to play with this and see for myself how well it works.  My questions involve, how well does it work under manufacturing variation?  How well do manufactures respond to such complex PCB structures?  Can it work well for multi-drop DDR systems too, i.e. can you simultaneously optimize all the receivers?

Overall, this was an interesting and well written paper.

Yasunaga, M.; Shimada, H.; Seki, K.; Yoshihara, I., "Segmental transmission line: Its practical application the optimized PCB trace design using a genetic algorithm," in Evolvable Systems (ICES), 2014 IEEE International Conference on , vol., no., pp.23-30, 9-12 Dec. 2014