More Reading

• When Shorter isn't Better, DesignCon 2010, Mike Steinberger + Cray.
• This was a long paper which built up to a very small conclusion.  The conclusion was sometimes you have resonant behavior which is hard to find and suggest a metric (unequalizable energy ) to maybe identify it.  Resonance is something which is easily attributed but hard to pin down.  I would have liked to see more insight into the problem and solution.  There was not mention of the attributed lengths and the wavelength of the fundamental frequency, no mention of loops in smith charts and no mention of Q factor.  If they used PDA they would have identified the worst case conditions before they built a whole system.
• Design of High Speed I/O Interfaces for High Performance Microprocessors, by Ankur Agrawal, Harvard Dissertation 2010.  Chapters 1 & 2.
• I found this paper when searching for information on clock recovery dithering noise.  It turns out to very interesting and very applicable to my current work.  I have only read chapters 1 and 2 which are the introductory sections but I am already impressed.  Overall it is well written and describes the problems of high speed I/O with clarity and precision.  His major contribution is a concept of collaborative timing recovery where for a multi-lane system you combine the CDR circuitry for area, power and accuracy improvement.  He also heavily references the CDR paper by Sonntag and Stonick which I have studied extensively.  His insights helped me understand it better.  Overall, I would recommend this as a good tutorial.
• An Introduction to Clock Recovery: Critical Aspects in Test and Measurement in High-speed Serial Designs.  Tektronix publication 2007.
• This was a fine primer on CDR concepts although sometimes they glossed over topics I wanted more on (derivation of jitter transfer function) and spent too much time on other topics (tabulating all the specs of various standards).  They had some pretty pictures and my favorite was one which showed the various stress jitter types used in testing a system.  Surprisingly they didn't even mention a specific Tek product to use.  Overall it is a good primer on the topic.

Crosstalk and Jitter

I have been learning a new signal integrity tool lately called Quantum Channel Designer (QCD) which has nothing to do with quantum behavior  . . .   but I digress.  I have been working on understanding how it works and why it works.  Here are some readings which I have done:  

• Comparison of BER Estimation Methods which Account for Crosstalk - DesignCon 2009 - Michael Steinberger, Barry Katz, Walter Katz, Todd Westerhoff.
• This is a nice paper but I would have liked a comparison to PDA.
• Exploration of Deterministic Jitter Distributions - DesignCon 2008 - Michael Steinberger and Christopher White.
• This paper had an interesting but flawed approach to discuss the jitter distribution of ISI and crosstalk.  But they did arrive at the conclusion that crosstalk roughly follows a bounded Gaussian distribution . . . since this agrees with my experience and research the flawed approach didn't bother me that much. :)
• Demonstration of SerDes Modeling using the Algorithmic Model Interface (AMI) Standard.  Designcon 2008, Michael Steinberger, Todd Westerhoff, Christopher White
• This was a fine introduction to AMI model usage but nothing ground breaking.  One good comment was on LTI analysis: 
• The very fact that the analysis takes so many different possible messages into account masks the effect of any one message.
• This is the problem/feature of statistical LTI analysis.
• Channel Compliance Testing Utilizing Novel Statistical Eye Methodology, DesignCon 2004, Anthony Sanders, Mike Resso, John D’Ambrosia.
• Wow!  This is a great paper and I can't believe that I haven't read it before.  This paper describes the algorithm behind StatEye (and QCD) and a statistical approach to calculate the Bit Error Ratio.  I would very much like to implement this so I can better understand it and to compare it to some of my own ideas on how to quickly calculate the BER.  I would highly recommend this paper.

Reading as of late

I have been very busy lately and haven't had time to summarize all of my reading.  The list below are some of the papers that I have read in the past week:

• Modeling and Analysis of High-Speed I/O Links, Balamurugan, G.;   Casper, B.;   Jaussi, J.E.;   Mansuri, M.;   O'Mahony, F.;   Kennedy, J.;   Microprocessor Technol. Labs., Intel Corp., Hillsboro, OR, IEEE Transactions on Advanced Packaging, May 2009, Volume: 32 Issue:2, On page(s): 237 - 247
• This was a great paper discussing how sigsim3 does it's simulations.  One of the interesting conclusions is that Tx jitter amplification can occur which is still a point of debate in some circles.
• A Digital Clock and Data Recovery Architecture for Multi-Gigabit/s Binary Links, Sonntag, J.L.;   Stonick, J.;   IEEE Journal of Solid-State Circuits, Issue Date: Aug. 2006, Volume: 41 Issue:8, On page(s): 1867 - 1875 
• This is an excellent tutorial on CDR design and analysis.  I was introduced to the concepts of input-referred jitter, input-reflected jitter and limit cycle behavior.
• Signal Integrity Design for High-Speed Digital Circuits: Progress and Directions, Jun Fan;   Xiaoning Ye;   Jingook Kim;   Archambeault, B.;   Orlandi, A.; IEEE Transactions on Electromagnetic Compatibility, Issue Date: May 2010, Volume: 52 Issue:2, On page(s): 392 - 400 
• This is a nice summary of the state of the art signal integrity techniques.  They have extensive references and would be a good place to start some literature research.  One point they did omit was Paul Huray's work on surface roughness, besides this I thought the coverage was good.
• Predicting BER with IBIS-AMI: experiences correlating SerDes simulations and measurement, Todd Westerhoff, Adge Hawes, Dr. Michael Steinberger, Kent Dramstad, Dr. Walter Katz, Barry Katz, DesignCon 2010
• This was a nice paper which showed simulation correlation between the commercially available tool QCD and IBM's HSSCDR tool.  They have a good flow and approach for any simulation correlation activity.
• Using IBIS-AMI Models to Study Clock Recovery Loop Performance, Barry Katz, Dr. Michael Steinberger.  Unpublished SiSoft paper.
• Two ways of estimating the bit error ratio of a high speed serial link are (quoting the paper)
• jitter based analysis:  the underlying hypothesis is that an error will occur whenever the clock to data timing at the decision latch is violated, and so the goal of the analysis is to estimate the probability density function of the clock to data timing.
• conditional probability calculation: the probability of error given a Constanta clock to data timing is estimated for each possible clock to data timing, resulting in a conditional probability curve.  The clock to data timing is then assumed to be independent of the data, and the probability of error is estimated by evaluating the integral of the conditional probability times the PDF of the clock to data timing.
• This second approach is the one that QCD uses and its use is the subject of the paper.  I would like to understand better how exactly this works but the paper does a good job discussing its application.

So that is my summary of what I read last week . . .

When intuition and math probably look wrong

Here are two riddles with different answers:

I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?

Suppose that Mr. Smith has two children, at least one of whom is a son. What is the probability both children are boys?

The question is, why are they different?  The article on information and probability in Science News was very interesting.

Memristors: Memory Resistors

'Memristive' switches enable 'stateful' logic operations via material implication

Just read a cool article in Nature about memresistors, or memory resistors, whose resistance can be set and recorded later.  This has the same functioning of a memory device but has the form of a resistor.  Traditional memory devices use voltage or charge as the physical state variable but these devices use resistance as their physical state variable.  They discuss how they have built these devices and by how utilizing a new logic scheme they can create circuits which can either act as logic or memory.

I have just started a new book at home, Introduction to Real Analysis by Michael J. Schramm, and they have been talking about logic but one thing which I had not seen before was this "implication" operation.  p-->q   <=> not(p) or q.  This is an operation that EE's do not see in their classes.  Well, these memresistor guys use this material implication operation along with the false operation as a computationally complete logic basis.  This is so cool that I've copied the supplemental information into the graphic below:

The author's work on memesistor usage is very important but I found the material implication idea more immediately applicable.  Let me finish with the following quote from the paper:

Instead of forcing these familiar Boolean logic shcemes onto new nanoscale devices in doomed competition with ultrahigh-performance silicon integrated circuits, memresistice IMP provides a stateful logic that is an unconventional computation framework determined more by the nano-device properties than any pre-conceived logic architecture.

Fractional Distances and Moments, Part 1

I've just pulled up out of an obsessive compulsive research project, so hold on as this will be a long post.  I work doing signal integrity for computer system interfaces and one of the critical things we do is evaluate the performance of a bus by quantifying the minimum eye diagram width and height.  The quick and dirty way of obtaining this minimum eye diagram is through Peak Distortion Analysis (PDA) description here and the original PDA paper here.  One of the problems is that for a multi-line system the chances of actually getting this worst case eye diagram is very small and to constrain design based on this alone would be crippling to progress and cost.  So instead the signal integrity community is moving to evaluating the Bit Error Rate (BER) to quantify performance but this is much more complex and  computationally expensive.

Through my research at Intel (DesignCon paper and presentation), I've shown that PDA is just a periodic one-norm and thus is subject to triangle inequality property.  A norm is a  measure of distance and the two-norm or Euclidean norm is the distance metric that most people are familiar with.  A one-norm is the distance one would  travel between two points if you are constrained to some coordinate system.  For instance, a taxi cab driver in Manhattan will charge you the distance he drove you through the city blocks rather than the distance it would take a crow to fly between two points.  The distance the taxi-cab drove is an example of the one-norm.

My obsessive compulsive research was based on the observation that I could replace the 1-norm in the PDA equations with a higher order fractional norm and then then relate this to the BER response of the system.  This would enable a quick and dirty BER calculation and would be really cool.  There is a clear correspondence between one-norm and worst BER value but I had difficulty in mapping the 2-norm value to a specific BER value.  Without a second point I can not relate the two and have concluded that this is just an interesting dead end.   In an effort to justify the countless hours I spent I am going to describe some of my peripheral findings. (That's right, I'm just getting started.)

There are many example of fractional approximations to otherwise discrete relations and equations.  I first became aware of such relations in high school while reading about the chaos theory, fractals and fractional dimensions.  Other well defined examples are fractional calculus, where you can take a 1/2 or 1/3 derivative of a function and the gamma function which is the continuous version of the factorial.  A visual approach to these fractional and higher order ideas is shown with superellipses.  These shapes are the result of the following equation:

|y|^p+|x|^p = 1

Here we see that we get the unit circle for p=2, which is a nice visualization of how euclidean or two-norms measure distance.  A unit circle shows all the points equidistant from the origin.

For p=1 we get the one norm diamond, for p=infinity we get a square and with p=4 we get the squircle (it is seriously called that) which is half way between a circle and square.  With each distinct p value, a different space is mathematically defined.  Each of these spaces has their own concept of distance (generalized p-norm) as well as its own concept of 'center' which is called a moment.

The p-norm is notated as ||x|_p and is defined for any p greater or equal to one.  You can quickly see that for p = 2 that the equation simplifies to the Euclidean distance equation.  There is a relation that higher order norms (higher p value) are smaller in magnitude than lower order norms for the same data sequence xi.  This is important to remember,  useful in many situations and summarized by the following equation:

The generalized mean equation (also called the power mean) is shown next.  If the target vector x is composed by subtracting a measure of central tendency (mean, median, mode, etc) it becomes a generalized moment equation  (more on this in Part 2).  If you set p to 1, the generalized mean becomes the familiar averaging function, if p is allowed to approach 0, the equation becomes the geometric mean. (I am still working on carrying out this simplification) and if p=-1, then the equation becomes the harmonic mean (for the EE's think of resistors in parallel).  The first four moments, which correspond to p = 1, 2, 3 and 4, are the arithmetic mean, standard deviation, asymmetry (skewness) and sharpness (kurtosis).

With this setup we are ready to go into Part 2 where I'll discuss fractional norms and fractional moments.

Deviation

Ever wonder why the standard deviation contains the squared and square root terms? Why is it like that? What makes it so standard? Well it turns out that there are some alternative deviation metrics such as the mean absolute deviation. This deviation metric takes the average of the absolute deviation of a vector (or random variable). This target vector could be a difference vector of the vector of interest and some measure of central tendency. It turns out the mean minimizes the standard deviation whereas the median minimizes the mean absolute deviation.

The paper "Revisiting a 90-year-old debate: the advantages of the mean deviation" by Stephen Gorard of the University of York is excellent argument for the proper use of the mean absolute deviation. He discusses the historical reasons for the dominance of the standard deviation, noting that

The standard deviation, by squaring the values concerned, gives us a distorted view of the amount of dispersion in our figures. The act of squaring makes each unit of distance from the mean exponentially (rather than additively) greater, and the act of square-rooting the sum of squares does not completely eliminate this bias.

He uses this point to show that the standard deviation is more sensitive to noise in the data than the mean absolute deviation and claims that the mean absolute deviation is better suited to all distribution other than the normal distribution. Finally he concludes that the mean absolute deviation has a simpler intuitive interpretation and thus more approachable for students.

So what properties must a metric have to be called a "deviation"? What other alternatives are available for our analysis?

Robust Causality Characterization via Generalized Dispersion Relations

P. Triverio, S. Grivet-Talocia, "Robust Causality Characterization via Generalized Dispersion Relations," IEEE Transactions on Advanced Packaging, VOL. 32, NO. 3, pp. 579-593, August 2008

This extremely well written and organized paper treats the causality of S-parameter data as throughly as any up to this time. He proposes a generalized Hilbert transform, i.e. extending the traditional Hilbert transform by including Lagrange polynomial to minimize the truncation and discretization error of the discrete frequency S-parameter data. He not only discusses this new approach but also bounds the error so that you know exactly the bounds of your assumptions. Given the needed time, I would like to implement his algorithms.

Wavefront Error

Just read a cool article on a subject that I didn't know exist and I bet it will come in handy someday.

IEEE Spectrum Magazine, March 2010, pages 46-53

Wavefront Error of a telescope: Aberrations in the curvature of a lens or mirror can cause telescopes in space and on the ground to suffer from blurry vision.

The guys at JPL have a new way of fixing blurry images.

Hartman screen test: 1904

Shack-Hartmann wavefront sensor: 1960

Gerchberg-Saxton algorithm: 2000s

It sounds like an adaptive telescope optical transfer function identifier. They create non-rigid telescope lenses which they purposely distort to correct for the optical distortion of atmosphere. Can create better images on earth with this approach than Hubble!

They claim that LASIK eye procedures use a form of wavefront distortion sensors to figure out how your eye is deformed!

Analysis by Its History

I've always wanted to take an Analysis class so when I saw this book referenced in the "Interpolate then Sample" paper below I had to get it. I'm only on page 135 of 350 but it is great so far. The authors go through the development of mathematical thought, describe how the Arabs solved quadratic equations, how the binomial theorem arose and how trigonometric functions came to be. I really like how the authors give lots of examples from published papers at the time. They weave letters from Euler, Fermat, Newton and the Bernoulli's into the mathematical development. This certainly makes it more entertaining and give a small amount of context to help remember the theorems. I've spent quite a bit of time solving the problems at the end of each section and given enough time I aim to complete them all.