PID Without a PhD
You do not need control theory to implement useful PID loops in embedded projects. Tim Wescott walks through simple, ready-to-use C code, clear explanations of P, I and D terms, and a practical tuning recipe you can apply to motors, precision actuators, and heaters. The article highlights anti-windup, sampling-rate guidance, and when to call in a control expert.
Digital Envelope Detection: The Good, the Bad, and the Ugly
Envelope detection sounds simple, but implementation choices change everything. Rick Lyons gathers common digital detectors, including half-wave, full-wave, square-law, Hilbert-based complex, and synchronous coherent designs, and explains how harmonics, filtering, and carrier recovery change results. He ranks detectors by output SNR from a representative simulation and offers practical tips on filter cutoff, Hilbert transformer bandwidth, and when a simple detector is good enough.
Harmonic Notch Filter
A practical, DSP-friendly recipe for scrubbing 60 Hz power-line hum and its harmonics from noisy ECG and EEG recordings is presented, using IIR notch filters built from second-order all-pass sections. The post derives how to set all-pass phase to place notches and compute biquad coefficients by solving a simple 2x2 system, then shows C code and precomputed coefficients for cascading the first eight odd harmonics at a 2 kHz sample rate. Engineers get a compact, editable implementation with explicit control over notch bandwidth.
A Useful Source of Signal Processing Information
A surprisingly handy web tool turned up for finding signal processing material in PDF and PowerPoint form. Rick Lyons shows how a plain-looking site can surface lots of topic-specific documents, using FM demodulation as the example. If you often hunt for reference slides and papers, this is a quick source worth bookmarking.
3 Good News
Stephane Boucher reports three quick wins for the EmbeddedRelated community: two sponsors have seeded a $1,000 rewards pool, the site now serves all pages over HTTPS, and the new forums have their first active discussions. If you want a share of the sponsor-funded rewards, jump into the forums and check the Vendors Directory for opportunities. Stay tuned for more updates.
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 New Forum is LIVE!
The EmbeddedRelated forum just got a major interface refresh, and Stephane Boucher is rolling it out in beta. The new editor makes it easier to drop in images and files, add LaTeX equations with MathJax, and publish highlighted code snippets with highlight.js. Access is gated by approval for now, mainly to keep trolls, spammers, and bots out.
Autocorrelation and the case of the missing fundamental
A short hands-on exploration shows why we perceive the fundamental pitch even when it's absent from the spectrum. Using saxophone recordings, high-pass filtering, and autocorrelation plots, the post demonstrates that the highest ACF peak often predicts perceived pitch rather than the strongest spectral line. The experiments also show that removing high harmonics eliminates the effect, and that autocorrelation is a useful but incomplete model of pitch perception.
Generating pink noise
This post implements a stochastic Voss-McCartney pink-noise generator in Python, tackling why incremental per-sample algorithms do not map well to NumPy batch operations. It presents a practical NumPy/Pandas approach that uses geometric-distributed update events and pandas' fillna for column-wise zero-order hold to make batch generation efficient. The generated noise shows a power-spectrum slope near -1, matching expected 1/f behavior.
Ancient History
Technology moves fast, and the tools, platforms, and assumptions you rely on can become outdated almost overnight. In this reflective post, the author contrasts the rapid evolution of embedded development with the much slower pace of social change, from programming turnaround times to the underrepresentation of women in engineering. It is a reminder to keep learning, but also to think about how we work and who gets included.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins
Cedron presents exact, closed-form formulas to extract the phase and amplitude of a pure complex tone from multiple DFT bin values, using a compact vector formulation. The derivation introduces a delta variable to simplify the sinusoidal bin expression, stacks neighboring bins into a basis vector, and solves for the complex amplitude q by projection. The phase and magnitude follow directly from q, and extra bins reduce leakage when the tone falls between bins.
Overview of my Articles
Cedron presents a guided tour of his DSPRelated articles that teach the discrete Fourier transform through derivations, numerical examples, and sample code. The collection centers on novel "bin value" formulas and exact frequency estimators for complex and real tones, with methods for phase and amplitude recovery and iterative multitone resolution. The overview also points to a zeroing-sine window family and an integer pseudo-differentiator for efficient peak and zero-crossing detection.
Number Theory for Codes
If CRCs have felt like black magic, this post peels back the curtain with basic number theory and polynomial arithmetic over GF(2). It shows how fixed-width processor arithmetic becomes arithmetic in a finite field, how bit sequences are treated as polynomials, and why primitive polynomials generate every nonzero element. You also get practical insights on CRC implementation with byte tables and LFSRs.
Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT
Cedron Dawg derives closed-form three-bin frequency estimators for a pure complex tone in a DFT using a linear algebra view that treats three adjacent bins as a vector. He shows any vector K orthogonal to [1 1 1] yields a = (K·Z)/(K·D·Z) and derives practical K choices including a Von Hann (Pascal) kernel and a data-driven projection. The post compares estimators under noise and gives simple selection rules.
In Search of The Fourth Wave
While working on Think DSP the presenter ran into a curious spectral pattern: sawtooth waves have all harmonics with amplitudes that scale like 1/f, square waves keep only odd harmonics with 1/f, and triangle waves keep odd harmonics with 1/f^2. That observation motivates a simple question: is there a basic waveform that has all integer harmonics but a 1/f^2 rolloff? The talk walks through four solution approaches, a fifth idea from the audience, and links to a runnable Colab notebook.
A Narrow Bandpass Filter in Octave or Matlab
Building very narrow FIR bandpass filters at high sample rates often yields extremely long impulse responses. This post shows a practical Octave/Matlab implementation that uses complex downconversion to baseband plus a multistage Matrix IFIR and running-sum cascade to slash computation. With the provided example (48 kHz, 850 Hz center, 10 Hz passband) you get <1 dB ripple and >60 dB stopband while running 20x to 100x faster than a single-stage FIR.
DFT Bin Value Formulas for Pure Real Tones
Cedron Dawg derives a closed-form expression for the DFT bin values produced by a pure real sinusoid, then uses that formula to explain well known DFT behaviors. The post walks through the algebra from Euler identities to a compact computational form, highlights the integer versus non-integer frequency cases, and verifies the result with C code and printed numeric output.
A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT
Cedron Dawg derives an exact two-bin frequency formula for a pure complex tone in the DFT, eliminating amplitude and phase to isolate frequency via a complex quotient and the complex logarithm. He presents an adjacent-bin simplification that replaces a complex multiply with a bin offset plus an atan2 angle, and discusses integer-frequency handling and aliasing. C source and numerical examples show the formula working in practice.
Feedback Controllers - Making Hardware with Firmware. Part 9. Closing the low-latency loop
This article demonstrates combining DSP and feedback-control on an Intel Cyclone floating-point FPGA to build low-latency closed-loop circuit emulators and controllers. Using a single floating-point biquad at 1.6 Msps, an IFFT multi-tone 4.096 ms capture for wideband measurement, and MATLAB references for verification, the author achieves sub-nanosecond timing insight and applies DSP phase compensation to cancel about 100 pF of PCB parasitics.
Compute Images/Aliases of CIC Interpolators/Decimators
CIC filters provide multiplier-free interpolation and decimation for large sample-rate changes, but their images and aliases can trip up designs. This post supplies two concise Matlab functions and hands-on examples to compute interpolator images and decimator aliases, showing spectra and freqz plots. Readers will learn how interpolation ratio and number of stages alter passband, stopband, and aliasing behavior.
Computing an FFT of Complex-Valued Data Using a Real-Only FFT Algorithm
Rick Lyons shows a compact trick to get an N-point complex FFT using only real-input FFT routines by transforming the real and imaginary parts separately and recombining their outputs. The post presents a one-line recombination formula, Xc(m) = real[Xr(m)] - imag[Xi(m)] + j{imag[Xr(m)] + real[Xi(m)]}, and an algebraic derivation based on the two-real-in-one-complex FFT identity. Useful for systems that only provide real-only FFTs.
Feedback Controllers - Making Hardware with Firmware. Part 3. Sampled Data Aspects
This article digs into practical sampled-data issues you must address when building feedback controllers for circuit emulation. It highlights a common MATLAB versus Simulink discrepancy caused by DAC holding, explains why FOH (ramp-invariant) c2d conversion matters, and surveys latency, bit depth, filter and precision trade-offs. It also lists candidate ADCs, DACs and FPGAs used in a real evaluation platform to guide hardware choices.
Filtering Noise: The Basics (Part 1)
How do you pull signals out of random noise? This post builds intuition from first principles for discrete-time white Gaussian noise and shows how simple linear FIR filtering (averaging) reduces noise. You’ll get derivations for the output mean, variance and autocorrelation, learn why the uniform moving-average minimizes noise under a unity-DC constraint, and why its sinc spectrum can be problematic. Part 1 of a short series.
Multiplierless Half-band Filters and Hilbert Transformers
This article provides coefficients of multiplierless Finite Impulse Response 7-tap, 11-tap, and 15-tap half-band filters and Hilbert Transformers. Since Hilbert transformer coefficients are simply related to half-band coefficients, multiplierless Hilbert transformers are easily derived from multiplierless half-bands.
Feedback Controllers - Making Hardware with Firmware. Part 4. Engineering of Evaluation Hardware
This installment follows the hardware from concept to first power-up for a low-latency feedback controller and arbitrary circuit emulator. It walks through the practical engineering steps, from requirements, block diagrams, and issue tracking to component selection, simulation, PCB planning, purchasing, and staged bring-up. The result is a realistic look at how careful due diligence and a few trade-offs turned a research idea into working evaluation hardware.
Modeling a Continuous-Time System with Matlab
Neil Robertson demonstrates a practical workflow for converting a continuous-time transfer function H(s) into an exact discrete-time H(z) using Matlab's impinvar. He walks through a 3rd-order Butterworth example, shows how to match impulse and step responses, and compares frequency response and group delay so engineers can see where the discrete model stays accurate and when sampling-rate limits cause departure.
Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 1
Misconfigured Code Composer Studio settings cause most 'Can't initialize target CPU' errors on TI C6000 boards, not a faulty silicon. Mike Dunn walks through the practical first steps: confirm your CCS version, identify the exact emulator and board or device part number, and ensure you have the correct emulator driver. The post also shows how to duplicate TI's factory board configuration to avoid common setup mistakes.
Feedback Controllers - Making Hardware with Firmware. Part 9. Closing the low-latency loop
This article demonstrates combining DSP and feedback-control on an Intel Cyclone floating-point FPGA to build low-latency closed-loop circuit emulators and controllers. Using a single floating-point biquad at 1.6 Msps, an IFFT multi-tone 4.096 ms capture for wideband measurement, and MATLAB references for verification, the author achieves sub-nanosecond timing insight and applies DSP phase compensation to cancel about 100 pF of PCB parasitics.
Exponential Smoothing with a Wrinkle
Cedron Dawg shows how pairing forward and backward exponential smoothing produces exact, frequency-dependent dampening for sinusoids while canceling time-domain lag. The average of the two passes scales the tone by a closed-form factor, and their difference acts like a first-derivative with a quarter-cycle phase shift. The post derives the analytic dampening formulas, compares them to the derivative, and includes a Python demo for DFT preprocessing.
Add the Hilbert Transformer to Your DSP Toolkit, Part 2
This post shows a simple practical route to a Hilbert transformer by starting from a half-band FIR filter and tweaking its symmetry. It walks through a 19-tap example synthesized with Matlab's firpm (Parks-McClellan), explains the required frequency scaling, and shows how even-numbered taps become (or can be forced) zero through symmetry and coefficient quantization. Useful design rules are summarized for choosing ntaps.
