Sinusoidal Frequency Estimation Based on Time-Domain Samples
Rick Lyons presents three time-domain algorithms for estimating the frequency of real and complex sinusoids from samples. He shows that the Real 3-Sample and Real 4-Sample estimators, while mathematically exact, fail in the presence of noise and can produce biased or invalid outputs. The Complex 2-Sample (Lank-Reed-Pollon) estimator is more robust but can be biased at low SNR and near 0 or Fs/2, so narrowband filtering is recommended.
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.
Launch of Youtube Channel: My First Videos - Embedded World 2017
Stephane Boucher turned his Embedded World 2017 trip into a debut YouTube series of short booth highlight videos. He walks through the steep learning curve of trade-show filming, the specific gear he bought and rented to cope with low light and noise, and the practical mistakes he plans to fix. The post lists filmed vendors and asks readers for feedback to improve future episodes.
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.
DFT Bin Value Formulas for Pure Complex Tones
Cedron Dawg derives closed-form DFT bin formulas for single complex exponentials, eliminating the need for brute-force summation and showing how phase acts as a uniform rotation of all bins. He also gives a Dirichlet-kernel form that yields the magnitude as (M/N)|sin(δN/2)/sin(δ/2)|, explains the large-N sinc limit, and includes C code to verify the results.
Multi-Decimation Stage Filtering for Sigma Delta ADCs: Design and Optimization
A Matlab toolbox streamlines the design and optimization of multi-stage decimation filters for sigma-delta ADCs. MSD-toolbox automates stage-count and decimation-factor selection, generates Parks-McClellan equiripple FIR coefficients, and iteratively selects coefficient quantization to meet in-band noise constraints. It accepts sigma-delta bitstream stimuli for spectral and intra-stage analysis, includes cost estimation routines, and is published open-source on MathWorks with examples and a dissertation reference.
Canonic Signed Digit (CSD) Representation of Integers
Canonic Signed Digit (CSD) encoding slashes the number of nonzero bits in integer coefficients, enabling multiplierless FIR filters implemented with shifts and adds. This post uses MATLAB code to demonstrate CSD rules, show how negative values work, and plot the distribution of signed digits as bit width changes. It finishes with practical techniques to minimize signed digits per coefficient for area and power efficient filter designs.
Frequency Translation by Way of Lowpass FIR Filtering
Rick Lyons shows how you can translate a signal down in frequency and lowpass filter it in a single operation by embedding cosine mixing values into FIR coefficients. The post explains how to build the translating FIR, how to choose the number of coefficient sets, and how decimation can dramatically reduce storage needs while noting practical constraints like the requirement that ft be an integer submultiple of fs.
Minimum Shift Keying (MSK) - A Tutorial
How does MSK achieve both excellent spectral efficiency and a constant-envelope signal suitable for nonlinear amplifiers? This tutorial builds MSK step‑by‑step from binary FSK, shows the minimum frequency spacing and continuous‑phase construction, and then recasts MSK as an OQPSK (pseudo‑symbol) representation. It finishes by generalizing MSK into CP‑FSK and the wider CPM family so you can connect practical pulse shapes and modulation indices to performance.
New Video: Parametric Oscillations
Tim Wescott just posted a short new video titled "Parametric Oscillations." It’s a little off-topic for the channel, but he used the project as an excuse to break a months-long posting drought. If you follow his work, this quick update shows how small builds can rekindle momentum and prompt informal explorations of oscillation behavior.
A Direct Digital Synthesizer with Arbitrary Modulus
Need exact sampled tones on a coarse grid without a huge sine table? This post shows how to build a Direct Digital Synthesizer with an arbitrary modulus so the output frequency is exactly k·fs/L, using a look-up table as small as 20 entries for the 10 MHz/0.5 MHz-step example. It also explains fixed-point LUT rounding, accumulator bit sizing, and how to produce quadrature outputs when L is multiple of 4.
Model Signal Impairments at Complex Baseband
Neil Robertson presents compact complex-baseband channel models for common signal impairments, implemented as short Matlab functions of up to seven lines. Using QAM examples and constellation plots, he demonstrates how interfering carriers, two-path multipath, sinusoidal phase noise, and Gaussian noise distort constellations and affect MER. The examples are lightweight and practical, making it easy to test receiver diagnostics and prototype adaptive-equalizer scenarios.
Amplitude modulation and the sampling theorem
I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College. He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.
Most of the code for the chapter is done: you can check it out in this IPython notebook. I haven't written the text yet, but I'll outline it here, and paste in the key figures.
Convolution...
The 2024 DSP Online Conference
The DSP Online Conference returns for a fourth year, running October 29–31, with a program designed for students, engineers, hobbyists, and experts. Organized into four tracks—general DSP theory, communications, audio, and DSP with deep learning—the event accepts short MicroTalks through two-hour Workshops and offers early-bird registration plus immediate archive access for registrants.
Ancient History
The other day I was downloading an IDE for a new (to me) OS. When I went to compile some sample code, it failed. I went onto a forum, where I was told "if you read the release notes you'd know that the peripheral libraries are in a legacy download". Well damn! Looking back at my previous versions I realized I must have done that and forgotten about it. Everything changes, and keeping up with it takes time and effort.
When I first started with microprocessors we...
What to See at Embedded World 2019
Skip the overwhelm at Embedded World 2019, Stephane Boucher lays out a practical preview of what to see and how to prioritize your time. The post helps embedded engineers focus on demos, vendor booths, and sessions that matter without getting lost on the show floor. Read it to plan a short, efficient visit that maximizes technical takeaways and networking opportunities.
Is It True That j is Equal to the Square Root of -1 ?
A viral YouTube video claimed that saying j equals the square root of negative one is wrong. Rick Lyons shows the apparent paradox comes from misusing square-root identities with negative arguments, not from the usual definition of j. He argues it is safer to define j by j^2 = -1 and illustrates how careless root operations produce contradictions in two appendices.
Live Streaming from Embedded World!
Stephane Boucher will bring Embedded World to engineers who cannot attend, streaming high-quality HD video from the show floor. He plans to use a professional camera and a device that bonds three internet links to keep the stream stable, and he is coordinating live sessions with vendors and select talks. Read on to learn how to vote for the presentations you want streamed.
Instantaneous Frequency Measurement
Measuring carrier frequency quickly and with minimal data matters in radar and signal characterization. Parth Vakil explains the delay-and-multiply instantaneous frequency measurement technique, shows how analytic signals and multiple delays resolve the 2π ambiguity, and demonstrates noise, phase-wrapping, and interferer effects using MATLAB code. He also outlines practical mitigations like phase unwrapping and channelization.
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.
DSP Related Math: Nice Animated GIFs
Stephane Boucher collected a compact set of animated GIFs that make common DSP math click visually. He spotted popular posts on the ECE subreddit and aggregated DSP-focused GIFs in one place to speed intuition and teaching. Examples include the relationship between sin and cos with right triangles, constructing a square wave from an infinite series, and the continuous Fourier transform pair of the rect and sinc functions.
A Brief Introduction To Romberg Integration
Romberg integration delivers dramatic accuracy gains for definite integrals by combining multiple trapezoidal approximations into a single highly accurate result. Rick Lyons demonstrates how just five samples can achieve 0.0038% error versus a trapezoidal rule needing 100 samples, and a 17-sample example hits 3.6×10−4% error. The post outlines the N-segment procedure, cost scaling, and links to MATLAB code.
The 2021 DSP Online Conference
Packed with practical talks and hands-on workshops, the 2021 DSP Online Conference gives DSP engineers a quick way to refresh skills and learn new techniques. Registering grants full access to talks, workshops, and Q&A at this year's event plus instant access to last year's videos. Highlights include FIR filter design with Python, software-defined radio, convolution reviews, and DSP/ML tools for IoT, with registration discounts on request.
Feedback Controllers - Making Hardware with Firmware. Part I. Introduction
This first post kicks off a series on using DSP and feedback control with mixed-signal electronics and FPGAs to emulate two-terminal circuits and create low latency controllers. It frames circuit emulation as a feedback problem, highlights latency as the key practical constraint, and outlines the planned evaluation hardware, target devices, and software tools that will be used in later MATLAB/Simulink and FPGA work.
Adaptive Beamforming is like Squeezing a Water Balloon
Think of adaptive beamforming as squeezing a water balloon, a simple analogy that reveals how combining multiple antennas creates focused gains and deep nulls. This post walks through the MVDR (Wiener-filter–based) solution, explains steering and scanning vectors, and shows how array geometry and known signal direction control what you can and cannot cancel. Practical tips highlight limits like the N-1 interferer rule.
How Not to Reduce DFT Leakage
Rick Lyons debunks a proposed 'data-flipping' fix for DFT spectral leakage, demonstrating with MATLAB that it can produce higher sidelobes and a troubling mainlobe dip for some input frequencies. He explains that windowing's goal is to reduce amplitude discontinuities in a periodic extension, not merely to force end samples to zero, and concludes the method is frequency-dependent and not recommended.
Model a Sigma-Delta DAC Plus RC Filter
Sigma-delta digital-to-analog converters (SD DAC’s) are often used for discrete-time signals with sample rate much higher than their bandwidth. For the simplest case, the DAC output is a single bit, so the only interface hardware required is a standard digital output buffer. Because of the high sample rate relative to signal bandwidth, a very simple DAC reconstruction filter suffices, often just a one-pole RC lowpass. In this article, I present a simple Matlab function that models the combination of a basic SD DAC and one-pole RC filter. This model allows easy evaluation of the overall performance for a given input signal and choice of sample rate, R, and C.
Who else is going to Sensors Expo in San Jose? Looking for roommate(s)!
This will be my first time attending this show and I must say that I am excited. I am bringing with me my cameras and other video equipment with the intention to capture as much footage as possible and produce a (hopefully) fun to watch 'highlights' video. I will also try to film as many demos as possible and share them with you.
I enjoy going to shows like this one as it gives me the opportunity to get out of my home-office (from where I manage and run the *Related sites) and actually...
Launch of EmbeddedRelated.tv
Stephane Boucher launches EmbeddedRelated.tv to host live broadcasts from Embedded World, starting next week. The site will show a constantly evolving schedule, a Live! tab to find ongoing streams, and ad-hoc demos added from the show floor. Expect schedule conflicts and small hiccups, and plan to refresh the page and join the forum thread for real-time updates and feedback.
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.
