## 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...

## Linear Feedback Shift Registers for the Uninitiated, Part III: Multiplicative Inverse, and Blankinship's Algorithm

Last time we talked about basic arithmetic operations in the finite field \( GF(2)[x]/p(x) \) — addition, multiplication, raising to a power, shift-left and shift-right — as well as how to determine whether a polynomial \( p(x) \) is primitive. If a polynomial \( p(x) \) is primitive, it can be used to define an LFSR with coefficients that correspond to the 1 terms in \( p(x) \), that has maximal length of \( 2^N-1 \), covering all bit patterns except the all-zero...

## Continuous Integration for Embedded Systems

It is no secret that anyone who wants to streamline project management, reduce risk and improve the quality needs some form of "automation" in SW development processes. What is commonly used in most companies as a tool for such automation is called Continuous Integration (CI). It is a good practice for embedded systems as well even though it is much harder to use CI for embedded systems compared to pure software development because embedded systems mostly depend on...

## Finally got a drone!

As a reader of my blog, you already know that I have been making videos lately and thoroughly enjoying the process. When I was in Germany early this summer (and went 280 km/h in a porsche!) to produce SEGGER's 25th anniversary video, the company bought a drone so we could get an aerial shot of the party (at about the 1:35 mark in this video). Since then, I have been obsessing on buying a drone for myself and finally made the move a few weeks ago - I acquired a used DJI...

## Tenderfoot: Embedded Software and Firmware Specialties

Once upon a time (seven years ago) I answered a question on Stack Overflow. Then Stephane suggested I turn that answer into a blog post. Great idea! This post dives deeper into the original question: “Is it possible to fragment this field (embedded software and firmware) into sub-fields?”

This post represents a detailed and updated response to my original Stack Overflow answer. I hope this post provides guidance and useful information to the “tenderfoots” in the...

## Embedded Toolbox: Source Code Whitespace Cleanup

In this installment of my "Embedded Toolbox" series, I would like to share with you the free source code cleanup utility called QClean for cleaning whitespace in your source files, header files, makefiles, linker scripts, etc.

You probably wonder why you might need such a utility? In fact, the common thinking is that compilers (C, C++, etc.) ignore whitespace anyway, so why bother? But, as a professional software developer you should not ignore whitespace, because it can cause all sorts...

## SEGGER's 25th Anniversary Video

Chances are you will find this video more interesting to watch if you take five minutes to first read the story of the week I spent at SEGGER's headquarters at the end of June.

The video is only a little more than 2 minutes long. If you decide to watch it, make sure to go full screen and I would really love to read your thoughts about it in the comments down bellow. Do you think a video like this succeeds in making the viewer want to learn more about the company?...

## 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 (

## 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...

## 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...

## Slew Rate Limiters: Nonlinear and Proud of It!

I first learned about slew rate limits when I was in college. Usually the subject comes up when talking about the nonideal behavior of op-amps. In order for the op-amp output to swing up and down quickly, it has to charge up an internal capacitor with a transistor circuit that’s limited in its current capability. So the slew rate limit \( \frac{dV}{dt} = \frac{I_{\rm max}}{C} \). And as long as the amplitude and frequency aren’t too high, you won’t notice it. But try to...

## 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

## 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.

## 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...

## 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...

## Introduction to Microcontrollers - More On GPIO

Now that we have our LED Blinky program nailed down, it's time to look more closely at outputs, add button/switch inputs, and work with reading inputs and driving outputs based on those inputs.

It's ON - No, It's OFF - No, It's ON...I have to confess, I cheated. Well, let's say I glossed over something very important. In our LED Blinky program, we never cared about whether an output '1' or an output '0' turned on the LED. Since we were just...

## Using the C language to program the am335x PRU

IntroductionSome weeks ago, I published an article on how we used the PRU to implement a power supply control loop having hard realtime constraints:

//www.embeddedrelated.com/showarticle/586.php

Writing this kind of logic in assembly language is not easy. First the assembly language itself may be difficult to learn depending on your background. Then, fixed and floating point arithmetics require lot of code. While macros help to handle the complexity, they still are error prone as you...

## 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...

## 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...

## C Programming Techniques: Function Call Inlining

IntroductionAbstraction is a key to manage software systems as they increase in size and complexity. As shown in a previous post, abstraction requires a developper to clearly define a software interface for both data and functions, and eventually hide the underlying implementation.When using the C language, the interface is often exposed in a header '.h' file, while the implementation is put in one or more corresponding '.c' files.

First, separating an interface from its...

## 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,...

## Oscilloscope Dreams

My coworkers and I recently needed a new oscilloscope. I thought I would share some of the features I look for when purchasing one.

When I was in college in the early 1990's, our oscilloscopes looked like this:

Now the cathode ray tubes have almost all been replaced by digital storage scopes with color LCD screens, and they look like these:

Oscilloscopes are basically just fancy expensive boxes for graphing voltage vs. time. They span a wide range of features and prices:...

## Cortex-M Exception Handling (Part 1)

This article describes how Cortex-M processors handle interrupts and, more generally, exceptions, a concept that plays a central role in the design and implementation of most embedded systems. The main reason of discussing this topic in detail is that, in the past few years, the degree of sophistication (and complexity) of microcontrollers in handling interrupts steadily increased, bringing them on a par with general-purpose processors.

## Coding - Step 0: Setting Up a Development Environment

Articles in this series:

- Coding Step 0 - Development Environments
- Coding Step 1 - Hello World and Makefiles
- Coding Step 2 - Source Control
- Coding Step 3 - High-Level Requirements
- Coding Step 4 - Design

You can easily find a million articles out there discussing compiler nuances, weighing the pros and cons of various data structures or discussing the optimization of databases. Those sorts of articles are fascinating reads for advanced programmers but...

## Using a RTLSDR dongle to validate NRF905 configuration

I am currently working on a system to monitor the garage door status from my flat. Both places are 7 floors apart, and I need to send the data wirelessly. I chose to operate on the 433MHz carrier, and I ordered 2 PTR8000 modules: http://www.electrodragon.com/w/NRF905_Transceiver_433MHz-Wireless_ModuleThe PTR8000 is based on the dual band sub 1GHz NRF905 chipset from NORDICSEMI: http://www.nordicsemi.com/eng/Products/Sub-1-GHz-RF/nRF905I...## How to Arduino - a video toolbox

I've begun producing a new series of video tutorials for the hobbyist new to the Arduino or microcontrollers in general. My videos are very pragmatic - I prefer to answer the question "what is the quickest, simplest and most affordable way to accomplish this?". The videos are meant to be a quick source of "how to" knowledge for the hobbyist that is using an LCD display, ultrasonic sensor or accelerometer for the first time, for example. I hope you enjoy this series of...

## Back from Embedded World 2019 - Funny Stories and Live-Streaming Woes

When the idea of live-streaming parts of Embedded World came to me, I got so excited that I knew I had to make it happen. I perceived the opportunity as a win-win-win-win.

- win #1 - Engineers who could not make it to Embedded World would be able to sample the huge event,
- win #2 - The organisation behind EW would benefit from the extra exposure
- win #3 - Lecturers and vendors who would be live-streamed would reach a (much) larger audience
- win #4 - I would get...