Linear Feedback Shift Registers for the Uninitiated, Part XV: Error Detection and Correction
CRCs and Hamming codes look a lot less magical when you view them as redundancy with a purpose. Jason Sachs walks from parity bits and checksums into finite-field polynomial arithmetic, then shows how CRCs map cleanly onto LFSRs and how Hamming codes use syndromes to locate single-bit errors. It is a practical tour of error detection and correction, with enough worked examples to make the theory feel usable.
Linear Regression with Evenly-Spaced Abscissae
Jason Sachs cuts through the matrix algebra to show a tiny trick for linear regression when x values are evenly spaced. You can compute the intercept as the mean and the slope as a simple weighted sum with arithmetic weights, using q = 12/(m^3 - m). The post includes Python examples and a compact routine to get least-squares coefficients without matrix solvers.
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.
Linear Feedback Shift Registers for the Uninitiated, Part XIII: System Identification
Jason Sachs shows how the output of a linear feedback shift register can be used for active system identification, not just spread-spectrum testing. The article compares traditional sine-wave probing with LFSR-based PRBS methods, demonstrates a worked Ra-Rb-C example, and unpacks practical issues such as reflected pseudonoise, ADC quantization, sample counts, and noise-shaping tricks to improve estimates.
A Wish for Things That Work
Jason Sachs revisits his long-running gripe with poor user interfaces, cataloguing annoyances from his Toyota Prius dashboard to desktop apps and browsers. He mixes sharp, real-world examples with a short, practical wishlist for 2018 aimed at making embedded displays, update behavior, security cues, and developer tools noticeably less frustrating for engineers and end users alike.
Linear Feedback Shift Registers for the Uninitiated, Part XII: Spread-Spectrum Fundamentals
Jason Sachs shows why LFSR-generated pseudonoise is a natural fit for direct-sequence spread spectrum, then walks through Fourier basics, spectral plots, and runnable Python examples. The article demonstrates how DSSS multiplies a UART bitstream with a chipping sequence to spread energy, how despreading concentrates the desired signal while scrambling narrowband interference, and how multiple transmitters can share bandwidth when using uncorrelated sequences.
Linear Feedback Shift Registers for the Uninitiated, Part XI: Pseudorandom Number Generation
Jason Sachs breaks down when linear feedback shift registers make good pseudorandom sources and when they fail. He shows why LFSR output bits look very different from full-state integer samples, explains their two-valued autocorrelation and quasi-random behavior, and gives practical guidance on when an LFSR is acceptable for fast hardware bit generation and when you should use a proper PRNG instead.
Linear Feedback Shift Registers for the Uninitiated, Part X: Counters and Encoders
Jason Sachs shows how linear feedback shift registers can be practical counters and compact absolute encoders, and why the choice of polynomial matters. He explains using primitive and reducible polynomials to get long but decode-friendly periods, demonstrates a 48-bit example, and lays out a De Bruijn chain-code encoder that turns an extra track into quick absolute resynchronization. Read to learn implementation tradeoffs and decoding strategies.
Linear Feedback Shift Registers for the Uninitiated, Part IX: Decimation, Trace Parity, and Cyclotomic Cosets
Taking every jth bit of a maximal-length LFSR uncovers a surprising algebraic structure. Jason Sachs walks through cyclotomic cosets, shows why decimation by powers of two preserves minimal polynomials, and connects LFSR output to trace parity and simple bitmask parity computations. The article uses hands-on Python with libgf2, Berlekamp-Massey, and state recovery so you can reproduce and automate these analyses.
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.
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.
Analog-to-Digital Confusion: Pitfalls of Driving an ADC
Wayne's thermistor board showed one ADC channel changing when another was heated, a classic case of ADC input cross-coupling. The post walks through how multiplexed ADCs, the small sample-and-hold capacitor, source impedance, sampling time, repeated sampling rates, and added charge reservoirs interact to create errors. Learn practical fixes including increasing sample time, sizing external caps, adding op-amp buffers, and using an RC dampener with PCB layout tips.
Supply Chain Games: What Have We Learned From the Great Semiconductor Shortage of 2021? (Part 1)
Jason Sachs argues the 2021 semiconductor shortage was not a single surprise but a set of structural imbalances exposed by COVID-19. He connects long lead times, constrained 200mm fabs and mature-node economics to why automotive features like heated seats became scarce, and shows how bullwhip dynamics and inventory practices amplified the problem. This first part uses concrete anecdotes and simple games to make the supply-chain lessons tangible.
How to Analyze a Three-Op-Amp Instrumentation Amplifier
The three-op-amp instrumentation amplifier gives you high input impedance, improved net bandwidth, and much lower sensitivity to resistor mismatch than a single-op-amp differential stage. Jason M. Sachs walks through the algebra, numeric examples, and historical notes to show how the preamp isolates common-mode, why splitting gain boosts bandwidth, how overall gain can be set with one resistor, and what practical limits to watch.
Lost Secrets of the H-Bridge, Part I: Ripple Current in Inductive Loads
Jason Sachs digs into what PWM switching actually does to current in an H-bridge with an inductive load, and why that ripple matters for motors and power converters. He derives closed-form ripple formulas, shows how to compute a reference current I_R0 = VDC·T/L, and uses Python and sympy to plot and verify results. Read it for practical rules to halve ripple and raise its frequency.
Adventures in Signal Processing with Python
Jason Sachs shows how PyLab (numpy, scipy, matplotlib) can handle many signal-processing and visualization tasks engineers usually reach for MATLAB to do. He walks through practical examples including PWM ripple, two pole RC filters, and symbolic math with SymPy, and shares real-world installation tips and trade-offs. The post closes with pointers to IPython and pandas to speed interactive analysis and data handling.
Ten Little Algorithms, Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
Today we will be drifting back into the topic of numerical methods, and look at an algorithm that takes in a series of discretely-sampled data points, and estimates the maximum value of the waveform they were sampled from.
The Least Interesting Circuit in the World
Jason Sachs pulls apart the humble power-on reset and shows why the common RC-and-Schmitt trick is the least interesting but most dangerous circuit in your design. He walks through voltage thresholds, brown-out reset behavior, and how slow or noisy Vdd ramps can let parts start in indeterminate states. Read this for practical rules on choosing supervisors, comparators, and reset pulse timing to ensure reliable embedded startup.
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.
Ten Little Algorithms, Part 8: Miller-Rabin Primality Test (and Living with Uncertainty)
Part 8 of the Ten Little Algorithms series: A look at the Miller-Rabin primality test, along with Pollard's rho algorithm for factoring, and some perspectives on very low levels of uncertainty.
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.
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.
Oscilloscope Dreams
Jason Sachs walks through practical oscilloscope buying criteria for embedded engineers, focusing on bandwidth, channel count, hi-res acquisition, and probing. He explains why mixed-signal scopes and hi-res mode matter, when a 100 MHz scope is sufficient and when to keep a higher-bandwidth instrument, and how probe grounding and waveform export can ruin measurements. Real-world brand notes and try-before-you-buy advice round out the guidance.
Important Programming Concepts (Even on Embedded Systems) Part II: Immutability
Immutable data can make embedded code easier to reason about, reduce concurrency bugs, and eliminate defensive copies. Jason Sachs walks through practical techniques that work in resource-constrained systems, from using const and pseudo-immutability to separating old and new state, to the limits of fully persistent data structures when you lack dynamic memory. The article also compares register-level state flow and advocates message passing as a concurrency alternative.
Return of the Delta-Sigma Modulators, Part 1: Modulation
Jason Sachs returns to delta-sigma modulators with a hands-on, code-first treatment that focuses on the DAC side of things. Part 1 walks through first- and second-order kernels, linearized analysis, spectra, and practical coefficient choices while illustrating results with Python simulations. Expect clear rules of thumb for A, R, and B, a derivation of noise shaping behavior, and a useful error bound for RC filtering.
Ten Little Algorithms, Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
Today we will be drifting back into the topic of numerical methods, and look at an algorithm that takes in a series of discretely-sampled data points, and estimates the maximum value of the waveform they were sampled from.
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.
Ten Little Algorithms, Part 4: Topological Sort
Jason Sachs detours from signal processing to make topological sort feel practical and even a little funny, using a Martian Stew recipe to illustrate dependencies and cycles. He walks through two canonical algorithms, Kahn’s method and the depth-first-search variant, compares adjacency-list and matrix graph representations, and provides complete Python implementations so you can run and inspect cycle detection and ordering yourself.
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.
Supply Chain Games: What Have We Learned From the Great Semiconductor Shortage of 2021? (Part 5)
Most of the time a wafer spends in a fab is spent waiting in queues, not being processed. Jason Sachs peels back why cycle times stretch weeks to months, covering WIP accounting, die and wafer banks, the days per mask layer metric, and how variability and utilization tradeoffs turned fab behavior into a major factor in the 2021 chip shortage.
