Linear Feedback Shift Registers for the Uninitiated, Part V: Difficult Discrete Logarithms and Pollard's Kangaroo Method
Most discrete-log problems are hopeless by brute force, but clever algorithms cut that cost to feasible levels. This installment walks through baby-step giant-step, Pollard’s rho and kangaroo methods, and how Silver-Pohlig-Hellman and index calculus leverage group structure to speed attacks on GF(2^n) fields. Jason Sachs includes Python examples, heuristics, and complexity nuggets so you can see when each method is practical.
Linear Feedback Shift Registers for the Uninitiated, Part IV: Easy Discrete Logarithms and the Silver-Pohlig-Hellman Algorithm
Discrete logarithms can be either trivial or infeasible depending on how group elements are represented, and Jason Sachs shows a practical route when they are intentionally easy. This article walks through using LFSRs as fast counters, why a smooth group order matters, and how the Silver-Pohlig-Hellman algorithm plus the Chinese Remainder Theorem recovers exponents in GF(2) with small prime factors.
Linear Feedback Shift Registers for the Uninitiated, Part III: Multiplicative Inverse, and Blankinship's Algorithm
Jason Sachs walks through Blankinship's constant-space variant of the Extended Euclidean Algorithm and shows how to compute multiplicative inverses both modulo an integer and in GF(2)[x]. The article uses clear numeric and polynomial examples, Python snippets, and an LFSR finite-field example to show how the algorithm yields Bézout coefficients and inverses useful for discrete-log tricks and cryptographic contexts. Readers get a practical recipe for inverse computation.
Linear Feedback Shift Registers for the Uninitiated, Part II: libgf2 and Primitive Polynomials
Jason Sachs digs into practical finite-field arithmetic for LFSRs, using his libgf2 Python library as the hands-on guide. He shows how to test whether a polynomial is primitive, why that matters for maximal-length sequences, and how the library implements addition, multiplication, exponentiation, and shifts over GF(2). The post is both a math refresher and a code walkthrough for engineers who want to compute with LFSRs instead of just talk about them.
Linear Feedback Shift Registers for the Uninitiated, Part I: Ex-Pralite Monks and Finite Fields
Jason Sachs demystifies linear feedback shift registers with a practical, bitwise view and the algebra that explains why they work. Readable examples compare Fibonacci and Galois implementations, show a simple software implementation, and reveal the correspondence between N-bit Galois LFSRs and GF(2^N) so you can pick taps and reason about maximal-length pseudorandom sequences.
Ten Little Algorithms, Part 6: Green’s Theorem and Swept-Area Detection
Jason shows how Green's Theorem becomes a practical, low-cost method to detect real-time rotation from two orthogonal sensors by accumulating swept area. The post derives a compact discrete integrator S[n] = S[n-1] + (x[n]*(y[n]-y[n-1]) - y[n]*(x[n]-x[n-1]))/2, compares integer and floating implementations, and analyzes noise scaling and sampling rate tradeoffs. Includes Python demos and threshold guidance.
Donald Knuth Is the Root of All Premature Optimization
Knuth's famous line "premature optimization is the root of all evil" has turned into a blunt rule on forums, Jason Sachs argues, and that overuse masks important nuance. He walks through concrete embedded examples, from dsPIC33E floating-point timings to an ROI analysis in the Kittens Game and a continuous optimization toy problem, to show when to measure, when to speculate, and why profilers can mislead.
Zebras Hate You For No Reason: Why Amdahl's Law is Misleading in a World of Cats (And Maybe in Ours Too)
Amdahl’s Law is a useful warning, but Jason Sachs argues it can be misleading if you stop at the equation. Using the Kittens Game as a playful model, he shows how Gustafson’s perspective, positive feedback loops, and system-level synergy can turn modest component speedups into big real-world wins. The article closes with concrete embedded-systems examples like ISR timing and developer productivity.
The Other Kind of Bypass Capacitor
Most engineers treat bypass capacitors as supply decoupling, but Jason Sachs digs into the other kind: a capacitor placed in the feedback path to tame unpredictable high-frequency plant behavior. He walks through real examples, Bode plots, and a simple RC model to show how the cap forces unity-gain feedback at high frequency, stabilizing switching regulators and wideband amplifiers while revealing the speed versus stability tradeoff.
How to Succeed in Motor Control: Olaus Magnus, Donald Rumsfeld, and YouTube
Jason Sachs turned frustration with algorithm-heavy motor-control app notes into a practical MASTERs class, now available on YouTube. He walks through building a fifteen-minute field-oriented control refresher, the hazards teams commonly miss, and the months of prep required to make a polished technical lecture. Read for a candid behind-the-scenes look at teaching motor control to engineers and tips you can apply to your next drive project.
Turn It On Again: Modeling Power MOSFET Turn-On Dependence on Source Inductance
This is a short article explaining how to analyze part of the behavior of a power MOSFET during turn-on, and how it is influenced by the parasitic inductance at the source terminal. The brief qualitative reason that source inductance is undesirable is that it uses up voltage when current starts increasing during turn-on (remember, V = L dI/dt), voltage that would otherwise be available to turn the transistor on faster. But I want to show a quantitative approximation to understand the impact of additional source inductance, and I want to compare it to the effects of extra inductance at the gate or drain.
10 Circuit Components You Should Know
Jason Sachs pulls together ten underrated but highly practical circuit components that every embedded engineer should know. From multifunction logic gates that act like a Swiss Army knife for glue logic to TL431 shunt regulators and tiny charge-pump inverters, each item is presented with real-world use cases and caveats. Read this to expand your parts toolbox and simplify future designs.
Linear Feedback Shift Registers for the Uninitiated
Jason Sachs assembled an eighteen-part deep dive into linear feedback shift registers, connecting the simple shift-register circuit to finite-field algebra and practical tools. The series walks through primitive polynomials, Berlekamp-Massey state recovery, libgf2-based analysis, discrete-log methods, and real-world uses from PRNGs and Gold codes to Reed-Solomon and CRC reverse-engineering. It’s a single reference for engineers who want both theory and working code.
How to Analyze a Differential Amplifier
Jason Sachs walks through the algebra and intuition behind the classic four-resistor differential amplifier. He derives the exact output equation, isolates error terms from resistor mismatch and op-amp imperfections, and explains why common-mode gain depends on mismatch not on the differential gain. Read this for clear formulas, modal insight into common-mode versus differential-mode, and practical steps to reduce offsets in real designs.
Someday We’ll Find It, The Kelvin Connection
Low-ohm measurements will fool your multimeter unless you use Kelvin connections. Jason Sachs walks through four-wire sensing using a current-limited supply and two DMMs, explains thermoelectric and connector-related errors, and shows why schematics and PCB layout must reflect Kelvin sense pads to avoid subtle measurement and circuit problems.
Lost Secrets of the H-Bridge, Part III: Practical Issues of Inductor and Capacitor Ripple Current
Jason Sachs cuts through the math to show what ripple current actually does to H-bridge hardware. He explains why peak current is the limiting factor for inductors, why capacitor ESR usually dominates DC-link voltage ripple, and how center-aligned PWM and duty selection reduce harmonics and ripple. Read this if you want practical rules of thumb and calculation templates for real power-electronics designs.
Isolated Sigma-Delta Modulators, Rah Rah Rah!
Analog isolation can blow up DAQ budgets, but isolated sigma-delta modulators let you send a single 1-bit stream and a clock across the barrier, keeping costs down. Jason walks through Avago, TI, and Analog Devices parts, explains sigma-delta noise shaping in plain terms, and calls out the real engineering work: converting a 10–20 MHz bitstream into usable samples with sinc/CIC decimators or FPGA filtering.
In Memoriam: Frederick P. Brooks, Jr. and The Mythical Man-Month
Fred Brooks’ The Mythical Man-Month is still a surprisingly sharp guide to software projects, and Jason Sachs shows why it matters even more than its old mainframe setting suggests. He revisits Brooks’ ideas on surgical teams, conceptual integrity, throwaway prototypes, and schedule estimation, then maps them to modern embedded and software engineering realities. The result is a tribute, a book review, and a practical reminder that roles, architecture, and testing still make or break delivery.
Linear Feedback Shift Registers for the Uninitiated, Part IV: Easy Discrete Logarithms and the Silver-Pohlig-Hellman Algorithm
Discrete logarithms can be either trivial or infeasible depending on how group elements are represented, and Jason Sachs shows a practical route when they are intentionally easy. This article walks through using LFSRs as fast counters, why a smooth group order matters, and how the Silver-Pohlig-Hellman algorithm plus the Chinese Remainder Theorem recovers exponents in GF(2) with small prime factors.
Scorchers, Part 3: Bare-Metal Concurrency With Double-Buffering and the Revolving Fireplace
Jason Sachs presents a practical, low-overhead concurrency pattern for tiny bare-metal systems where an ISR (Speedy) must safely exchange data with a nonreal-time main loop (Poky). He describes the "revolving fireplace", a double-buffering variant that swaps ownership of two shared memory regions, and walks through C examples, atomic/volatile considerations, and testing strategies so you can implement it on RAM-constrained MCUs.
Linear Feedback Shift Registers for the Uninitiated, Part XVIII: Primitive Polynomial Generation
Jason Sachs walks through how to find primitive polynomials for GF(2) LFSRs, moving from naive exhaustive checks to smarter synthetic constructions. The article compares sieve and constructive methods, shows practical optimizations like parity checks and companion-matrix updates, and demonstrates decimation plus Berlekamp-Massey to generate all primitives from one seed; it also teases a novel Falling Coyote Algorithm for additional speedups.
Someday We’ll Find It, The Kelvin Connection
Low-ohm measurements will fool your multimeter unless you use Kelvin connections. Jason Sachs walks through four-wire sensing using a current-limited supply and two DMMs, explains thermoelectric and connector-related errors, and shows why schematics and PCB layout must reflect Kelvin sense pads to avoid subtle measurement and circuit problems.
The CRC Wild Goose Chase: PPP Does What?!?!?!
Jason Sachs walks through a CRC rabbit hole and explains why ambiguous CRC names and incomplete specs lead to subtle protocol bugs. He demonstrates how XMODEM and KERMIT variants with a zero initial value can miss dropped leading-zero bytes, praises the X.25 standard for providing test vectors and a clear CRC16 definition, and warns that RFCs that ship only sample code are a poor substitute for a proper specification.
Signal Processing Contest in Python (PREVIEW): The Worst Encoder in the World
Jason Sachs previews a hands-on Python contest to find the best velocity estimator for a noisy, low-cost quadrature encoder. The post explains the Estimator API, submission constraints, and a 5 second, 10 kHz evaluation harness that uses a simulated "Lucky Wheel" encoder with realistic manufacturing timing errors. Jason also includes a simple baseline estimator and discusses the practical tradeoff between noise reduction and phase lag in velocity estimation.
First-Order Systems: The Happy Family
Jason Sachs takes the modest topic of first-order systems and makes it useful, showing how RC filters behave in both time and frequency domains and why they all share the same shape. He steps through step, ramp, and sinusoidal responses, explains poles, zeros, and Bode behavior, and uses Python plots to make tracking error and the role of tau easy to visualize.
Donald Knuth Is the Root of All Premature Optimization
Knuth's famous line "premature optimization is the root of all evil" has turned into a blunt rule on forums, Jason Sachs argues, and that overuse masks important nuance. He walks through concrete embedded examples, from dsPIC33E floating-point timings to an ROI analysis in the Kittens Game and a continuous optimization toy problem, to show when to measure, when to speculate, and why profilers can mislead.
Real-time clocks: Does anybody really know what time it is?
Most RTC chips still expose calendar fields rather than seconds-since-epoch, forcing embedded engineers to write ugly conversion code. Jason Sachs makes the case for offset encoding, subseconds, and an explicit snapshot feature to simplify interval math, raise precision, and avoid rare timing bugs. Read this practical take on RTC trade-offs and a short wishlist for chip makers.
Ten Little Algorithms, Part 6: Green’s Theorem and Swept-Area Detection
Jason shows how Green's Theorem becomes a practical, low-cost method to detect real-time rotation from two orthogonal sensors by accumulating swept area. The post derives a compact discrete integrator S[n] = S[n-1] + (x[n]*(y[n]-y[n-1]) - y[n]*(x[n]-x[n-1]))/2, compares integer and floating implementations, and analyzes noise scaling and sampling rate tradeoffs. Includes Python demos and threshold guidance.
Bad Hash Functions and Other Stories: Trapped in a Cage of Irresponsibility and Garden Rakes
A tiny filename decision in MATLAB's publish() can silently swap rendered equations, and Jason Sachs shows why that matters. He reproduces the bug, walks through hash-function basics and collision math, and contrasts safe and unsafe caching strategies. The piece then broadens into practical lessons about software fringes, legacy constraints, and the usability traps that leave engineers repeatedly stumbling over avoidable design choices.
10 More (Obscure) Circuit Components You Should Know
Jason Sachs follows up his earlier primer with ten more underused but practical parts that can simplify embedded hardware designs. From MOSFET-based ideal diode controllers that eliminate diode drops to TAOS light-to-frequency sensors that expand dynamic range, the post explains what each component does, when to choose it, and real-world tradeoffs learned from field use. Ideal for engineers looking to broaden their parts toolbox.
