Linear Feedback Shift Registers for the Uninitiated, Part XI: Pseudorandom Number Generation

Jason Sachs December 20, 2017

Last time we looked at the use of LFSRs in counters and position encoders.

This time we’re going to look at pseudorandom number generation, and why you may — or may not — want to use LFSRs for this purpose.

But first — an aside:

Science Fair 1983

When I was in fourth grade, my father bought a Timex/Sinclair 1000. This was one of several personal computers introduced in 1982, along with the Commodore 64. The...

Linear Feedback Shift Registers for the Uninitiated, Part X: Counters and Encoders

Jason Sachs December 9, 2017

Last time we looked at LFSR output decimation and the computation of trace parity.

Today we are starting to look in detail at some applications of LFSRs, namely counters and encoders.


I mentioned counters briefly in the article on easy discrete logarithms. The idea here is that the propagation delay in an LFSR is smaller than in a counter, since the logic to compute the next LFSR state is simpler than in an ordinary counter. All you need to construct an LFSR is

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

Jason Sachs December 3, 2017

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

Jason Sachs November 21, 2017

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 Dregs

Elwyn Berlekamp’s 1966 paper Non-Binary BCH Encoding covers some work on

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

Ed Nutter November 14, 20175 comments

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

Jason Sachs November 13, 2017

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

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

Jason Sachs November 7, 2017

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

Maharajan Veerabahu November 6, 20173 comments

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

Jason Sachs October 18, 2017

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

Jason Sachs October 1, 2017

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 \( y \)...

VHDL tutorial - A practical example - part 2 - VHDL coding

Gene Breniman May 27, 2011

In part 1 of this series we focused on the hardware design, including some of the VHDL definitions of the I/O characteristics of the CPLD part.  In part 2, we will describe the VHDL logic of the CPLD for this design.

With any design, the first step to gather the requirements for the job at hand.  From part 1 of this article, I have copied two sections that address some of the requirements for the CPLD design.

The data acquisition engine has the...

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

Jason Sachs February 27, 20171 comment

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

Analog-to-Digital Confusion: Pitfalls of Driving an ADC

Jason Sachs November 19, 20117 comments

Imagine the following scenario:You're a successful engineer (sounds nice, doesn't it!) working on a project with three or four circuit boards. More than even you can handle, so you give one of them over to your coworker Wayne to design. Wayne graduated two years ago from college. He's smart, he's a quick learner, and he's really fast at designing schematics and laying out circuit boards. It's just that sometimes he takes some shortcuts... but in this case the circuit board is just something...

10 Circuit Components You Should Know

Jason Sachs November 27, 20111 comment

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 (

Byte and Switch (Part 1)

Jason Sachs April 26, 201112 comments

Imagine for a minute you have an electromagnet, and a microcontroller, and you want to use the microcontroller to turn the electromagnet on and off. Sounds pretty typical, right?We ask this question on our interviews of entry-level electrical engineers: what do you put between the microcontroller and the electromagnet?We used to think this kind of question was too easy, but there are a surprising number of subtleties here (and maybe a surprising number of job candidates that were missing...

Introduction to Microcontrollers - Interrupts

Mike Silva September 18, 20136 comments

It's Too Soon To Talk About Interrupts!

That, at least, could be one reaction to this chapter.  But over the years I've become convinced that new microcontroller programmers should understand interrupts before being introduced to any complex peripherals such as timers, UARTs, ADCs, and all the other powerful function blocks found on a modern microcontroller.  Since these peripherals are commonly used with interrupts, any introduction to them that does not...

VHDL tutorial - Creating a hierarchical design

Gene Breniman May 22, 20086 comments

In earlier blog entries I introduced some of the basic VHDL concepts. First, developing a function ('VHDL tutorial') and later verifying and refining it ('VHDL tutorial - part 2 - Testbench' and 'VHDL tutorial - combining clocked and sequential logic'). In this entry I will describe how to...

Help, My Serial Data Has Been Framed: How To Handle Packets When All You Have Are Streams

Jason Sachs December 11, 201110 comments

Today we're going to talk about data framing and something called COBS, which will make your life easier the next time you use serial communications on an embedded system -- but first, here's a quiz:

Quick Diversion, Part I: Which of the following is the toughest area of electrical engineering? analog circuit design digital circuit design power electronics communications radiofrequency (RF) circuit design electromagnetic...

Arduino robotics #1 - motor control

Lonnie Honeycutt October 13, 20133 comments
Arduino Robotics

Beginner robotics is a series of article chronicling my first autonomous robot build, Clusterbot.  This build is meant to be affordable, relatively easy and instructive.  The total cost of the build is around $50.  

1. Arduino robotics - motor control2. Arduino robotics - chassis, locomotion and power3. Arduino robotics - wiring, coding and a test run4. 

Data Hiding in C

Stephen Friederichs April 20, 201317 comments

Strictly speaking, C is not an object-oriented language. Although it provides some features that fit into the object-oriented paradigm it has never had the full object-oriented focus that its successor C++ offers. C++ introduced some very useful concepts and abilities that I miss when I’m developing in ANSI C. One such concept is protected member variables and functions.

When you declare a class in C++ you can also declare member variables and functions as part of that class. Often, these...

Two jobs

Stephane Boucher December 5, 201223 comments

For those of you following closely embeddedrelated and the other related sites, you might have noticed that I have been less active for the last couple of months, and I will use this blog post to explain why. The main reason is that I got myself involved into a project that ended up using a better part of my cpu than I originally thought it would.

I currently have two jobs: one as an electrical/dsp engineer recycled as a web publisher and the other as a parent of three kids. My job...

October winner announced

Stephane Boucher November 15, 20121 comment

If you are a regular visitor of EmbeddedRelated, you are most likely aware that I have been running monthly draws lately for users of the site who are helping me to clean up the archives by rating threads in the forums section.  

For the month of August, the member "Cryptoman" won a iPad, and for the month of September, 10 members won $50 each.  

For October, the winner of the new iPod Touch is the member with the username "hssathya".

The winner of the next draw will win...

Behold, the New Comments System!

Stephane Boucher September 18, 201229 comments

I have just finished implementing a new system for commenting the blogs.  It uses Ajax extensively, so the page won't reload if you post a comment.  And it is a 'threaded' system, which means that if you post a reply to a comment, it will be attached to it.  

What do you think?  I personally love it.  Please go ahead and test it with a quick comment.

Although it is better to be logged in to post a comment, non-registered users can also comment, but they will have...

Best Embedded Systems pdf Documents Out There

Stephane Boucher September 11, 20128 comments

There are thousands of pdf documents related to Embedded Systems available online.  In fact, when I do a search in Google for:

embedded systems filetype:pdf

I get 4,340,000 results! A huge mix of articles, promotional documents, theses, etc.

Out of these 4 millions+ documents, I suspect that there must be at least a few hundreds jewels that deserve to be given more visibility.

Today, I am asking for your help (again!) to build a directory of some of the most useful pdf files out...

New Discussion Group for Users of TI ARM based MCUs

Stephane Boucher June 14, 2010

If you are a user of an ARM based TI Microcontroller, please feel free to join the new "TI ARM processors MCUs" discussion group by sending a blank email to: This discussion group will be moderated, so you don't have to worry about receiving more spam than you probably already get. It usually takes a few weeks for a group to gain momentum, so don't worry if the activity level is low for a little while, but make sure to join so you don't miss the good...

New TI MCU Resource Center

Stephane Boucher April 1, 2010

I am happy to announce the publication of the new "TI MCU Resource Center" on, where TI will regularly add videos and articles to keep you informed on their latest and greatest MCU related products.

To access the new section, you'll find a link in the main menu of the site at the top of the page.

Blogs Section Now Online!

Stephane Boucher September 18, 2007

I am happy to announce that the blog section is now online.

Last week, I sent an email to all the members of to ask for embedded systems experts who would be interested in blogging on the site. The response was very positive and I have selected 10 highly qualified individuals who will soon be writing here about all sorts of embedded systems related subjects. I am currently in the process of receiving their info (bio, photo, username, etc) and creating their bloggers'...