Linear Feedback Shift Registers for the Uninitiated, Part XV: Error Detection and Correction
Last time, we talked about Gold codes, a specially-constructed set of pseudorandom bit sequences (PRBS) with low mutual cross-correlation, which are used in many spread-spectrum communications systems, including the Global Positioning System.
This time we are wading into the field of error detection and correction, in particular CRCs and Hamming codes.
Ernie, You Have a Banana in Your EarI have had a really really tough time writing this article. I like the...
Linear Regression with Evenly-Spaced Abscissae
What a boring title. I wish I could come up with something snazzier. One word I learned today is studentization, which is just the normalization of errors in a curve-fitting exercise by the sample standard deviation (e.g. point \( x_i \) is \( 0.3\hat{\sigma} \) from the best-fit linear curve, so \( \frac{x_i - \hat{x}_i}{\hat{\sigma}} = 0.3 \)) — Studentize me! would have been nice, but I couldn’t work it into the topic for today. Oh well.
I needed a little break from...
Linear Feedback Shift Registers for the Uninitiated, Part XI: Pseudorandom Number Generation
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 1983When 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 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
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.
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...
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...
From Baremetal to RTOS: A review of scheduling techniques
Transitioning from bare-metal embedded software development to a real-time operating system (RTOS) can be a difficult endeavor. Many developers struggle with the question of whether they should use an RTOS or simply use a bare-metal scheduler. One of the goals of this series is to walk developers through the transition and decision making process of abandoning bare-metal thinking and getting up to speed quickly with RTOSes. Before diving into the details of RTOSes, the appropriate first step...
Data Types for Control & DSP
There's a lot of information out there on what data types to use for digital signal processing, but there's also a lot of confusion, so the topic bears repeating.
I recently posted an entry on PID control. In that article I glossed over the data types used by showing "double" in all of my example code. Numerically, this should work for most control problems, but it can be an extravagant use of processor resources. There ought to be a better way to determine what precision you need...
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:
Elliptic Curve Cryptography - Basic Math
An introduction to the math of elliptic curves for cryptography. Covers the basic equations of points on an elliptic curve and the concept of point addition as well as multiplication.
Elliptic Curve Cryptography - Key Exchange and Signatures
Elliptic curve mathematics over finite fields helps solve the problem of exchanging secret keys for encrypted messages as well as proving a specific person signed a particular document. This article goes over simple algorithms for key exchange and digital signature using elliptic curve mathematics. These methods are the essence of elliptic curve cryptography (ECC) used in applications such as SSH, TLS and HTTPS.
Ten Little Algorithms, Part 7: Continued Fraction Approximation
In this article we explore the use of continued fractions to approximate any particular real number, with practical applications.
Linear Regression with Evenly-Spaced Abscissae
What a boring title. I wish I could come up with something snazzier. One word I learned today is studentization, which is just the normalization of errors in a curve-fitting exercise by the sample standard deviation (e.g. point \( x_i \) is \( 0.3\hat{\sigma} \) from the best-fit linear curve, so \( \frac{x_i - \hat{x}_i}{\hat{\sigma}} = 0.3 \)) — Studentize me! would have been nice, but I couldn’t work it into the topic for today. Oh well.
I needed a little break from...
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...
Elliptic Curve Cryptography - Extension Fields
An introduction to the pairing of points on elliptic curves. Point pairing normally requires curves over an extension field because the structure of an elliptic curve has two independent sets of points if it is large enough. The rules of pairings are described in a general way to show they can be useful for verification purposes.
Linear Feedback Shift Registers for the Uninitiated, Part XV: Error Detection and Correction
Last time, we talked about Gold codes, a specially-constructed set of pseudorandom bit sequences (PRBS) with low mutual cross-correlation, which are used in many spread-spectrum communications systems, including the Global Positioning System.
This time we are wading into the field of error detection and correction, in particular CRCs and Hamming codes.
Ernie, You Have a Banana in Your EarI have had a really really tough time writing this article. I like the...
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...
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...
Data Types for Control & DSP
There's a lot of information out there on what data types to use for digital signal processing, but there's also a lot of confusion, so the topic bears repeating.
I recently posted an entry on PID control. In that article I glossed over the data types used by showing "double" in all of my example code. Numerically, this should work for most control problems, but it can be an extravagant use of processor resources. There ought to be a better way to determine what precision you need...
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 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
Linear Feedback Shift Registers for the Uninitiated, Part XV: Error Detection and Correction
Last time, we talked about Gold codes, a specially-constructed set of pseudorandom bit sequences (PRBS) with low mutual cross-correlation, which are used in many spread-spectrum communications systems, including the Global Positioning System.
This time we are wading into the field of error detection and correction, in particular CRCs and Hamming codes.
Ernie, You Have a Banana in Your EarI have had a really really tough time writing this article. I like the...
Linear Feedback Shift Registers for the Uninitiated, Part XI: Pseudorandom Number Generation
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 1983When 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 Regression with Evenly-Spaced Abscissae
What a boring title. I wish I could come up with something snazzier. One word I learned today is studentization, which is just the normalization of errors in a curve-fitting exercise by the sample standard deviation (e.g. point \( x_i \) is \( 0.3\hat{\sigma} \) from the best-fit linear curve, so \( \frac{x_i - \hat{x}_i}{\hat{\sigma}} = 0.3 \)) — Studentize me! would have been nice, but I couldn’t work it into the topic for today. Oh well.
I needed a little break from...
Mathematics and Cryptography
The mathematics of number theory and elliptic curves can take a life time to learn because they are very deep subjects. As engineers we don't have time to earn PhD's in math along with all the things we have to learn just to make communications systems work. However, a little learning can go a long way to helping make our communications systems secure - we don't need to know everything. The following articles are broken down into two realms, number theory and elliptic...
Square root in fixed point VHDL
In this blog we will design and implement a fixed point square root function in VHDL. The algorithm is based on the recursive Newton Raphson inverse square root algorithm and the implementation offers parametrizable pipeline depth, word length and the algorithm is built with VHDL records and procedures for easy use.
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...
One Clock Cycle Polynomial Math
Error correction codes and cryptographic computations are most easily performed working with GF(2^n)