## Linear Feedback Shift Registers for the Uninitiated, Part II: libgf2 and Primitive Polynomials

Last time, we looked at the basics of LFSRs and finite fields formed by the quotient ring \( GF(2)[x]/p(x) \).

LFSRs can be described by a list of binary coefficients, sometimes referred as the polynomial, since they correspond directly to the characteristic polynomial of the quotient ring.

Today we’re going to look at how to perform certain practical calculations in these finite fields. I maintain a Python library on bitbucket called...

## Linear Feedback Shift Registers for the Uninitiated, Part I: Ex-Pralite Monks and Finite Fields

Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.

— Évariste Galois, May 29, 1832

I was going to call this short series of articles “LFSRs for Dummies”, but thought better of it. What is a linear feedback shift register? If you want the short answer, the Wikipedia article is a decent introduction. But these articles are aimed at those of you who want a little bit deeper mathematical understanding,...

## Ten Little Algorithms, Part 6: Green’s Theorem and Swept-Area Detection

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 2: The Single-Pole Low-Pass Filter
- Part 3: Welford's Method (And Friends)
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method

This article is mainly an excuse to scribble down some cryptic-looking mathematics — Don’t panic! Close your eyes and scroll down if you feel nauseous — and...

## Donald Knuth Is the Root of All Premature Optimization

This article is about something profound that a brilliant young professor at Stanford wrote nearly 45 years ago, and now we’re all stuck with it.

TL;DRThe idea, basically, is that even though optimization of computer software to execute faster is a noble goal, with tangible benefits, this costs time and effort up front, and therefore the decision to do so should not be made on whims and intuition, but instead should be made after some kind of analysis to show that it has net...

## Zebras Hate You For No Reason: Why Amdahl's Law is Misleading in a World of Cats (And Maybe in Ours Too)

I’ve been wasting far too much of my free time lately on this stupid addicting game called the Kittens Game. It starts so innocently. You are a kitten in a catnip forest. Gather catnip.

And you click on Gather catnip and off you go. Soon you’re hunting unicorns and building Huts and studying Mathematics and Theology and so on. AND IT’S JUST A TEXT GAME! HTML and Javascript, that’s it, no pictures. It’s an example of an

## The Other Kind of Bypass Capacitor

There’s a type of bypass capacitor I’d like to talk about today.

It’s not the usual power supply bypass capacitor, aka decoupling capacitor, which is used to provide local charge storage to an integrated circuit, so that the high-frequency supply currents to the IC can bypass (hence the name) all the series resistance and inductance from the power supply. This reduces the noise on a DC voltage supply. I’ve...

## How to Succeed in Motor Control: Olaus Magnus, Donald Rumsfeld, and YouTube

Almost four years ago, I had this insight — we were doing it wrong! Most of the application notes on motor control were about the core algorithms: various six-step or field-oriented control methods, with Park and Clarke transforms, sensorless estimators, and whatnot. It was kind of like a driving school would be, if they taught you how the accelerator and brake pedal worked, and how the four-stroke Otto cycle works in internal combustion engines, and handed you a written...

## Round Round Get Around: Why Fixed-Point Right-Shifts Are Just Fine

Today’s topic is rounding in embedded systems, or more specifically, why you don’t need to worry about it in many cases.

One of the issues faced in computer arithmetic is that exact arithmetic requires an ever-increasing bit length to avoid overflow. Adding or subtracting two 16-bit integers produces a 17-bit result; multiplying two 16-bit integers produces a 32-bit result. In fixed-point arithmetic we typically multiply and shift right; for example, if we wanted to multiply some...

## Scorchers, Part 1: Tools and Burn Rate

This is a short article about one aspect of purchasing, for engineers.

I had an engineering manager once — I’ll leave his real name out of it, but let’s call him Barney — who had a catchy response to the question “Can I buy XYZ?”, where XYZ was some piece of test equipment, like an oscilloscope or multimeter. Barney said, “Get what you need, need what you get.” We used purchase orders, which when I started in 1996 were these quaint forms on...

## Padé Delay is Okay Today

This article is going to be somewhat different in that I’m not really writing it for the typical embedded systems engineer. Rather it’s kind of a specialized topic, so don’t be surprised if you get bored and move on to something else. That’s fine by me.

Anyway, let’s just jump ahead to the punchline. Here’s a numerical simulation of a step response to a \( p=126, q=130 \) Padé approximation of a time delay:

Impressed? Maybe you should be. This...

## The Least Interesting Circuit in the World

It does nothing, most of the time.

It cannot compute pi. It won’t oscillate. It doesn’t light up.

Often it makes other circuits stop working.

It is… the least interesting circuit in the world.

What is it?

About 25 years ago, I took a digital computer architecture course, and we were each given use of an ugly briefcase containing a bunch of solderless breadboards and a power supply and switches and LEDs — and a bunch of

## Complexity in Consumer Electronics Considered Harmful

I recently returned from a visit to my grandmother, who lives in an assisted living community, and got to observe both her and my frustration first-hand with a new TV. This was a Vizio flatscreen TV that was fairly easy to set up, and the picture quality was good. But here's what the remote control looks like:

You will note:

- the small lettering (the number buttons are just under 1/4 inch in diameter)
- a typeface chosen for marketing purposes (matching Vizio's "futuristic" corporate...

## Linear Feedback Shift Registers for the Uninitiated, Part XVIII: Primitive Polynomial Generation

Last time we figured out how to reverse-engineer parameters of an unknown CRC computation by providing sample inputs and analyzing the corresponding outputs. One of the things we discovered was that the polynomial \( x^{16} + x^{12} + x^5 + 1 \) used in the 16-bit X.25 CRC is not primitive — which just means that all the nonzero elements in the corresponding quotient ring can’t be generated by powers of \( x \), and therefore the corresponding 16-bit LFSR with taps in bits 0, 5,...

## Fluxions for Fun and Profit: Euler, Trapezoidal, Verlet, or Runge-Kutta?

Today we're going to take another diversion from embedded systems, and into the world of differential equations, modeling, and computer simulation.

DON'T PANIC!First of all, just pretend I didn't bring up anything complicated. We're exposed to the effects of differential equations every day, whether we realize it or not. Your car speedometer and odometer are related by a differential equation, and whether you like math or not, you probably have some comprehension of what's going on: you...

## Someday We’ll Find It, The Kelvin Connection

You’d think it wouldn’t be too hard to measure electrical resistance accurately. And it’s really not, at least according to wikiHow.com: you just follow these easy steps:

- Choose the item whose resistance you wish to measure.
- Plug the probes into the correct test sockets.
- Turn on the multimeter.
- Select the best testing range.
- Touch the multimeter probes to the item you wish to measure.
- Set the multimeter to a high voltage range after finishing the...

## Supply Chain Games: What Have We Learned From the Great Semiconductor Shortage of 2021? (Part 4)

Today we’re going to look at what’s been going on this past year in the chip shortage, particularly in the automotive markets. I’m going to share some recent events and statements that may shed some light on what’s been happening.

In Part Three we went through a deep dive on some aspects of Moore’s Law, the semiconductor foundries, and semiconductor economics, and we looked at the game Supply Chain Idle. We touched on a couple of important points about the...

## Lost Secrets of the H-Bridge, Part III: Practical Issues of Inductor and Capacitor Ripple Current

We've been analyzing the ripple current in an H-bridge, both in an inductive load and the DC link capacitor. Here's a really quick recap; if you want to get into more details, go back and read part I and part II until you've got equations coming out of your ears. I promise there will be a lot less grungy math in this post. So let's get most of it out of the way:

Switches QAH and QAL are being turned on and off with pulse-width modulation (PWM), to produce an average voltage DaVdc on...

## Wye Delta Tee Pi: Observations on Three-Terminal Networks

Today I’m going to talk a little bit about three-terminal linear passive networks. These generally come in two flavors, wye and delta.

Why Wye?The town of Why, Arizona has a strange name that comes from the shape of the original road junction of Arizona State Highways 85 and 86, which was shaped like the letter Y. This is no longer the case, because the state highway department reconfigured the intersection

## Important Programming Concepts (Even on Embedded Systems) Part II: Immutability

Other articles in this series:

- Part I: Idempotence
- Part III: Volatility
- Part IV: Singletons
- Part V: State Machines
- Part VI: Abstraction

This article will discuss immutability, and some of its variations in the topic of functional programming.

There are a whole series of benefits to using program variables that… well, that aren’t actually variable, but instead are immutable. The impact of...

## Supply Chain Games: What Have We Learned From the Great Semiconductor Shortage of 2021? (Part 1)

So by now I’m sure you’ve heard about the semiconductor shortage of 2021. For a few complicated reasons, demand is greater than supply, and not everybody who wants to buy integrated circuits can do so. Today we’re going to try to answer some hard questions:

- Why are we in the middle of a semiconductor shortage?
- Why is it taking so long to get my [insert part number here]?
- Did this shortage suddenly sneak up on everybody? If not, what were the signs, and why...

## Linear Feedback Shift Registers for the Uninitiated, Part XVI: Reed-Solomon Error Correction

Last time, we talked about error correction and detection, covering some basics like Hamming distance, CRCs, and Hamming codes. If you are new to this topic, I would strongly suggest going back to read that article before this one.

This time we are going to cover Reed-Solomon codes. (I had meant to cover this topic in Part XV, but the article was getting to be too long, so I’ve split it roughly in half.) These are one of the workhorses of error-correction, and they are used in...

## Padé Delay is Okay Today

This article is going to be somewhat different in that I’m not really writing it for the typical embedded systems engineer. Rather it’s kind of a specialized topic, so don’t be surprised if you get bored and move on to something else. That’s fine by me.

Anyway, let’s just jump ahead to the punchline. Here’s a numerical simulation of a step response to a \( p=126, q=130 \) Padé approximation of a time delay:

Impressed? Maybe you should be. This...

## Lessons Learned from Embedded Code Reviews (Including Some Surprises)

My software team recently finished a round of code reviews for some of our motor controller code. I learned a lot from the experience, most notably why you would want to have code reviews in the first place.

My background is originally from the medical device industry. In the United States, software in medical devices gets a lot of scrutiny from the Food and Drug Administration, and for good reason; it’s a place for complexity to hide latent bugs. (Can you say “

## First-Order Systems: The Happy Family

Все счастли́вые се́мьи похо́жи друг на дру́га, ка́ждая несчастли́вая семья́ несчастли́ва по-сво́ему.— Лев Николаевич Толстой, Анна Каренина

Happy families are all alike; every unhappy family is unhappy in its own way.— Lev Nicholaevich Tolstoy, Anna Karenina

I was going to write an article about second-order systems, but then realized that it would be...

## The CRC Wild Goose Chase: PPP Does What?!?!?!

I got a bad feeling yesterday when I had to include reference information about a 16-bit CRC in a serial protocol document I was writing. And I knew it wasn’t going to end well.

The last time I looked into CRC algorithms was about five years ago. And the time before that… sometime back in 2004 or 2005? It seems like it comes up periodically, like the seventeen-year locust or sunspots or El Niño,...

## Signal Processing Contest in Python (PREVIEW): The Worst Encoder in the World

When I posted an article on estimating velocity from a position encoder, I got a number of responses. A few of them were of the form "Well, it's an interesting article, but at slow speeds why can't you just take the time between the encoder edges, and then...." My point was that there are lots of people out there which take this approach, and don't take into account that the time between encoder edges varies due to manufacturing errors in the encoder. For some reason this is a hard concept...

## Second-Order Systems, Part I: Boing!!

I’ve already written about the unexciting (but useful) 1st-order system, and about slew-rate limiting. So now it’s time to cover second-order systems.

The most common second-order systems are RLC circuits and spring-mass-damper systems.

Spring-mass-damper systems are fairly common; you’ve seen these before, whether you realize it or not. One household example of these is the spring doorstop (BOING!!):

(For what it’s worth: the spring...

## Donald Knuth Is the Root of All Premature Optimization

This article is about something profound that a brilliant young professor at Stanford wrote nearly 45 years ago, and now we’re all stuck with it.

TL;DRThe idea, basically, is that even though optimization of computer software to execute faster is a noble goal, with tangible benefits, this costs time and effort up front, and therefore the decision to do so should not be made on whims and intuition, but instead should be made after some kind of analysis to show that it has net...

## Bad Hash Functions and Other Stories: Trapped in a Cage of Irresponsibility and Garden Rakes

I was recently using the publish() function in MATLAB to develop some documentation, and I ran into a problem caused by a bad hash function.

In a resource-limited embedded system, you aren't likely to run into hash functions. They have three major applications: cryptography, data integrity, and data structures. In all these cases, hash functions are used to take some type of data, and deterministically boil it down to a fixed-size "fingerprint" or "hash" of the original data, such that...

## 10 More (Obscure) Circuit Components You Should Know

The interest in my previous article on obscure but useful electronics parts, "10 Circuit Components You Should Know" was encouraging enough that I thought I would write a followup. So here are another 10:

1. "Ideal Diode" controllers

Load-sharing circuits use diodes tied together at their cathode terminal to take the most positive voltage among the sources and connect it to a load. Works great: you have a DC/DC power supply, a battery, and a solar cell, and it will use whichever output is...