Linear Feedback Shift Registers for the Uninitiated, Part VII: LFSR Implementations, Idiomatic C, and Compiler Explorer
Jason Sachs takes LFSR theory back to real hardware, showing multiple C implementations and dsPIC33E assembly to squeeze cycles out of Galois LFSR updates. He digs into idiomatic C pitfalls like rotate idioms, demonstrates tricks using unions and 16/32-bit views, and shows when inline assembly with SL/RLC and conditional-skip instructions pays off. The article also uses Compiler Explorer and supplies an MPLAB X test harness for verification.
Lazy Properties in Python Using Descriptors
Python descriptors let you outsource attribute lookup, and Jason Sachs walks through a practical use: lazy, cached properties. He presents a LazyProperty descriptor that defaults to a WeakKeyDictionary cache so computed results are stored on first access and automatically purged when objects are garbage collected. The post shows how to share caches by value using attrkey or swap cache classes for different use cases.
Linear Feedback Shift Registers for the Uninitiated, Part VI: Sing Along with the Berlekamp-Massey Algorithm
Jason Sachs breaks down the Berlekamp-Massey algorithm and shows how to recover an LFSR's minimal connection polynomial from a stream of output bits. The article mixes intuition, worked examples, and Python code to demonstrate the update rule, visual debugging tables, and when the solution is unique. Expect practical implementation notes, a complexity discussion, and a libgf2 example you can run in an IPython notebook.
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.
Beware of Analog Switch Leakage Current
Leakage currents in analog switches can quietly wreck precision reference circuits at elevated temperature. Jason M. Sachs walks through three switch-topology implementations for a switchable 1.25 V reference and shows which topology gives the smallest worst-case output error using real part specs. He explains why op amp input bias is usually negligible and gives practical fixes: lower resistances, better switches, or limiting temperature range.
Help, My Serial Data Has Been Framed: How To Handle Packets When All You Have Are Streams
Framing byte streams is easier to get wrong than you think, and a bad scheme can leave your embedded device acting on the wrong packet. Jason Sachs walks through common plaintext and binary framing approaches, explains why CRCs alone can still permit false resynchronization, and demonstrates COBS as a simple, low-overhead byte-stuffing method that prevents delimiter collisions and guarantees resynchronization.
Linear Feedback Shift Registers for the Uninitiated, Part VII: LFSR Implementations, Idiomatic C, and Compiler Explorer
Jason Sachs takes LFSR theory back to real hardware, showing multiple C implementations and dsPIC33E assembly to squeeze cycles out of Galois LFSR updates. He digs into idiomatic C pitfalls like rotate idioms, demonstrates tricks using unions and 16/32-bit views, and shows when inline assembly with SL/RLC and conditional-skip instructions pays off. The article also uses Compiler Explorer and supplies an MPLAB X test harness for verification.
Slew Rate Limiters: Nonlinear and Proud of It!
Slew-rate limits are a small nonlinear detail that often decides whether a controller behaves nicely or wrecks hardware. Jason Sachs walks through why slew limits appear in electronics and actuators, then shows two practical digital ways to impose limits: constraining input increments and constraining input around the output. He compares performance on underdamped second-order systems, gives closed-form intuition for overshoot, and demonstrates simulations with scipy and ODE solvers.
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.
Important Programming Concepts (Even on Embedded Systems) Part IV: Singletons
Singletons are convenient but often a modularity killer, especially in embedded firmware. Jason Sachs walks through the many faces of singletons, from static members and globals to hardware registers and user-visible application singletons, and shows practical ways to avoid tight coupling. Read this for concrete embedded examples and pragmatic fixes like passing state explicitly, using interfaces or factories, and isolating unavoidable globals in a HAL.
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.
Padé Delay is Okay Today
High-order Padé approximations for time delays break in surprising ways, but the failure is not magic. Jason Sachs walks through why coefficient-based transfer functions and companion-form state-space are numerically fragile, shows how to compute poles and zeros directly from the hypergeometric form with Newton iteration, and demonstrates building modal or block-diagonal state-space realizations to make high-order Padé delays practical while noting remaining limits.
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.
Linear Feedback Shift Registers for the Uninitiated, Part XIV: Gold Codes
Gold codes solve a practical spread-spectrum problem, sharing one PRBS across many transmitters eventually runs into ugly synchronization and correlation issues. Jason Sachs walks through why shifted copies of a single LFSR sequence are not enough, then shows how preferred pairs of m-sequences create a family of Gold codes with bounded cross-correlation. The post wraps with Python experiments and a UART DSSS demo that decodes multiple overlapping messages cleanly.
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.
Another 10 Circuit Components You Should Know
Jason Sachs walks through ten underrated circuit components every embedded engineer should know, from bus switches and thermocouple signal ICs to PCB stiffeners and opto-FET isolators. He mixes practical part examples, high-current hardware tips, and MCU features like CTMU and Peripheral Pin Select so you can pick the right trick when space, isolation, or precision matter.
Lessons Learned from Embedded Code Reviews (Including Some Surprises)
Jason Sachs recounts a round of motor-controller code reviews and the practical lessons his team learned about quality and tooling. He explains how a simple "ready for review" checklist and automated style checks kept meetings focused on substantive issues, and why choosing the right review tool matters after discovering lost comments in Stash. Read for concrete tips on process, subgit mirroring, vera++, and Upsource.
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.
Linear Feedback Shift Registers for the Uninitiated, Part VIII: Matrix Methods and State Recovery
Matrix methods for LFSRs look intimidating, but Jason Sachs walks through companion-matrix representations and shows why they matter for time shifts and state recovery. He derives lookahead masks from powers of the companion matrix, then translates those matrix insights into efficient bitwise and finite-field algorithms. The article includes two simple state-recovery methods and working Python/libgf2 examples you can run and adapt.
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.
Lost Secrets of the H-Bridge, Part II: Ripple Current in the DC Link Capacitor
DC link capacitor ripple often determines capacitor life, yet the H-bridge waveform has a lot more structure than you might expect. Jason Sachs separates capacitor current into ramp and pulse components, derives closed-form peak and RMS expressions for edge and center PWM, and shows harmonic amplitudes. The post includes Python simulation so you can visualize the waveforms and apply the formulas to capacitor selection and EMI analysis.
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.
