## Linear Feedback Shift Registers for the Uninitiated, Part IX: Decimation, Trace Parity, and Cyclotomic Cosets

Last time we looked at matrix methods and how they can be used to analyze two important aspects of LFSRs:

- time shifts
- state recovery from LFSR output

In both cases we were able to use a finite field or bitwise approach to arrive at the same result as a matrix-based approach. The matrix approach is more expensive in terms of execution time and memory storage, but in some cases is conceptually simpler.

This article will be covering some concepts that are useful for studying the...

## Linear Feedback Shift Registers for the Uninitiated, Part VIII: Matrix Methods and State Recovery

Last time we looked at a dsPIC implementation of LFSR updates. Now we’re going to go back to basics and look at some matrix methods, which is the third approach to represent LFSRs that I mentioned in Part I. And we’re going to explore the problem of converting from LFSR output to LFSR state.

Matrices: Beloved Historical DregsElwyn Berlekamp’s 1966 paper Non-Binary BCH Encoding covers some work on

## Obsolete? Yes. Still in use? Yes. How do you use it? Ummm...

In today's world of constantly changing technology, quick parts availability, and seemingly endless options, some things can't change. It isn't a big deal to wait a day or less for a computer upgrade to arrive. It seems program size increases proportionally to hard drive size. The old is discarded and replaced with the new. Hard drives can hold terrabytes and even SD cards can hold gigabytes of information.

Now, suppose a system can't be changed. It is still...

## Linear Feedback Shift Registers for the Uninitiated, Part VII: LFSR Implementations, Idiomatic C, and Compiler Explorer

The last four articles were on algorithms used to compute with finite fields and shift registers:

- multiplicative inverse
- discrete logarithm
- determining characteristic polynomial from the LFSR output

Today we’re going to come back down to earth and show how to implement LFSR updates on a microcontroller. We’ll also talk a little bit about something called “idiomatic C” and a neat online tool for experimenting with the C compiler.

## Lazy Properties in Python Using Descriptors

This is a bit of a side tangent from my normal at-least-vaguely-embedded-related articles, but I wanted to share a moment of enlightenment I had recently about descriptors in Python. The easiest way to explain a descriptor is a way to outsource attribute lookup and modification.

Python has a bunch of “magic” methods that are hooks into various object-oriented mechanisms that let you do all sorts of ridiculously clever things. Whether or not they’re a good idea is another...

## Android for Embedded Devices - 5 Reasons why Android is used in Embedded Devices

The embedded purists are going to hate me for this. How can you even think of using Android on an embedded system ? It’s after all a mobile phone operating system/software.

Sigh !! Yes I did not like Android to begin with, as well - for use on an Embedded System. But sometimes I think the market and needs decide what has to be used and what should not be. This is one such thing. Over the past few years, I have learned to love Android as an embedded operating system....

## Linear Feedback Shift Registers for the Uninitiated, Part VI: Sing Along with the Berlekamp-Massey Algorithm

The last two articles were on discrete logarithms in finite fields — in practical terms, how to take the state \( S \) of an LFSR and its characteristic polynomial \( p(x) \) and figure out how many shift steps are required to go from the state 000...001 to \( S \). If we consider \( S \) as a polynomial bit vector such that \( S = x^k \bmod p(x) \), then this is equivalent to the task of figuring out \( k \) from \( S \) and \( p(x) \).

This time we’re tackling something...

## Linear Feedback Shift Registers for the Uninitiated, Part V: Difficult Discrete Logarithms and Pollard's Kangaroo Method

Last time we talked about discrete logarithms which are easy when the group in question has an order which is a smooth number, namely the product of small prime factors. Just as a reminder, the goal here is to find \( k \) if you are given some finite multiplicative group (or a finite field, since it has a multiplicative group) with elements \( y \) and \( g \), and you know you can express \( y = g^k \) for some unknown integer \( k \). The value \( k \) is the discrete logarithm of \( y \)...

## Introduction to Deep Insight Analysis for RTOS Based Applications

Over the past several years, embedded systems have become extremely complex. As systems become more complex, they become harder and more time consuming to debug. It isn’t uncommon for development teams to spend more than 40% development cycle time just debugging their systems. This is where deep insight analysis has the potential to dramatically decrease costs and time to market.

Defining Deep Insight Analysis

Deep insight analysis is a set of tools and techniques that can be...

## Linear Feedback Shift Registers for the Uninitiated, Part IV: Easy Discrete Logarithms and the Silver-Pohlig-Hellman Algorithm

Last time we talked about the multiplicative inverse in finite fields, which is rather boring and mundane, and has an easy solution with Blankinship’s algorithm.

Discrete logarithms, on the other hand, are much more interesting, and this article covers only the tip of the iceberg.

What is a Discrete Logarithm, Anyway?Regular logarithms are something that you’re probably familiar with: let’s say you have some number \( y = b^x \) and you know \( y \) and \( b \) but...

## The DSP Online Conference - Right Around the Corner!

It is Sunday night as I write this blog post with a few days to go before the virtual doors of the very first DSP Online Conference open..

It all started with a post in the DSPRelated forum about three months ago. We had just had a blast running the 2020 Embedded Online Conference and we thought it could be fun to organize a smaller event dedicated to the DSP community. So my goal with the post in the forum was to see if...

## 10 Circuit Components You Should Know

Chefs have their miscellaneous ingredients, like condensed milk, cream of tartar, and xanthan gum. As engineers, we too have quite our pick of circuits, and a good circuit designer should know what's out there. Not just the bread and butter ingredients like resistors, capacitors, op-amps, and comparators, but the miscellaneous "gadget" components as well.

Here are ten circuit components you may not have heard of, but which are occasionally quite useful.

1. Multifunction gate (

## Ten Little Algorithms, Part 1: Russian Peasant Multiplication

This blog needs some short posts to balance out the long ones, so I thought I’d cover some of the algorithms I’ve used over the years. Like the Euclidean algorithm and Extended Euclidean algorithm and Newton’s method — except those you should know already, and if not, you should be locked in a room until you do. Someday one of them may save your life. Well, you never know.

Other articles in this series:

- Part 1:

## An overview of Linux Boot Process for Embedded Systems

This Text provides an insight in to the Embedded Linux Boot Process. Reader should have a basic Knowledge of Boot Process in general and should be familiar with Embedded Linux Boot Process.

.................PART-A................(1) Software components Involved in Embedded Linux Boot Process (a) Bootloader (b) kernel Image (c) root file system - either an initrd image or a NFS location(2) Steps during Booting process of a conventional...## Lost Secrets of the H-Bridge, Part I: Ripple Current in Inductive Loads

So you think you know about H-bridges? They're something I mentioned in my last post about signal processing with Python.

Here we have a typical H-bridge with an inductive load. (Mmmmm ahhh! It's good to draw by hand every once in a while!) There are four power switches: QAH and QAL connecting node A to the DC link, and QBH and QBL connecting node B to the DC link. The load is connected between nodes A and B, and here is represented by an inductive load in series with something else. We...

## PID Without a PhD

I both consult and teach in the area of digital control. Through both of these efforts, I have found that while there certainly are control problems that require all the expertise I can bring to bear, there are a great number of control problems that can be solved with the most basic knowledge of simple controllers, without resort to any formal control theory at all.

This article will tell you how to implement a simple controller in software and how to tune it without getting into heavy...

## Introduction to Microcontrollers - 7-segment displays & Multiplexing

Doing the 7 Segment ShuffleThe 7 segment display is ubiquitous in the modern world. Just about every digital clock, calculator and movie bomb has one. The treadmills at my gym have 6 or 7, each one displaying 3 or 4 digits. What makes the 7-seg interesting is that it presents an opportunity to make a trade off between GPIO (output pins) for time. Every 7-seg display requires 8 outputs (the 7 segments and usually either a decimal point or a...

## Best Firmware Architecture Attributes

Architecture of a firmware (FW) in a way defines the life-cycle of your product. Often companies start with a simple-version of a product as a response to the time-to-market caveat of the business, make some cash out of the product with a simple feature set. It takes only less than 2-3 years to reach a point where the company needs to develop multiple products derived from the same code base and multiple teams need to develop...

## Free Goodies from Embedded World - What to Do Next?

I told you I would go on a hunt for free stuff at Embedded World in order to build a bundle for someone to win.

## 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...

## 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...

## Introduction to Microcontrollers - More On Interrupts

A Little More Detail About The Interrupt MechanismIt's time to look a little closer at what happens in an interrupt request and response. Again this is in general terms, and different microcontroller designs may do things somewhat differently, but the basics remain the same. Most but not all interrupt requests are latched, which means the interrupt event sets a flag that stays set even if the interrupt event then goes away. It is this latched flag...

## VHDL tutorial - A practical example - part 1 - Hardware

In previous posts I described some simple VHDL examples. This time let's try something a little more complex. This is part one of a multiple part article. This is intended to be a detailed description of one of several initial designs that I developed for a client. This design never made it into a product, but a similar design was used and is currently being produced. As a considerable amount of work was put into this effort, I decided to share this design...

## Byte and Switch (Part 2)

In part 1 we talked about the use of a MOSFET for a power switch. Here's a different circuit that also uses a MOSFET, this time as a switch for signals:

We have a thermistor Rth that is located somewhere in an assembly, that connects to a circuit board. This acts as a variable resistor that changes with temperature. If we use it in a voltage divider, the midpoint of the voltage divider has a voltage that depends on temperature. Resistors R3 and R4 form our reference resistance; when...

## Ten Little Algorithms, Part 4: Topological Sort

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 5: Quadratic Extremum Interpolation and Chandrupatla's Method
- Part 6: Green’s Theorem and Swept-Area Detection

Today we’re going to take a break from my usual focus on signal processing or numerical algorithms, and focus on...

## 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...

## From bare-metal to RTOS: 5 Reasons to use an RTOS

Developers can come up with amazing and convoluted reasons to not use an RTOS. I have heard excuses ranging from they are too expensive (despite open source solutions) all the way to they aren’t efficient and use too much memory. In some circumstances some excuses are justified but there are many reasons why a developer should look to an RTOS to help with their real-time scheduling needs.

From bare-metal to RTOS Quick Links## Another 10 Circuit Components You Should Know

It's that time again to review all the oddball goodies available in electronic components. These are things you should have in your bag of tricks when you need to design a circuit board. If you read my previous posts and were looking forward to more, this article's for you!

1. Bus switches

I can't believe I haven't mentioned bus switches before. What is a bus switch?

There are lots of different options for switches:

- mechanical switch / relay: All purpose, two...

## 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

## Lazy Properties in Python Using Descriptors

This is a bit of a side tangent from my normal at-least-vaguely-embedded-related articles, but I wanted to share a moment of enlightenment I had recently about descriptors in Python. The easiest way to explain a descriptor is a way to outsource attribute lookup and modification.

Python has a bunch of “magic” methods that are hooks into various object-oriented mechanisms that let you do all sorts of ridiculously clever things. Whether or not they’re a good idea is another...