SAVE THE DATE – DSPRelated’s First Ever In-Person Conference!
After 25 years running DSPRelated and co-organizing the DSP Online Conference, the author announces DSPRelated’s first in-person conference. The event is scheduled in Silicon Valley for October 14–16, 2025 and is organized by engineers for engineers, emphasizing empowering, practical, hands-on sessions designed to leave attendees energized and inspired. Several familiar speakers from the online events — including fred harris, Dan Boschen, and Hilmar Lehnert — have already shown strong interest in presenting. Attendance will be limited by venue capacity, so readers are encouraged to mark their calendars and coordinate with employers to secure travel and passes while awaiting forthcoming registration and program details.
A Matlab Function for FIR Half-Band Filter Design
FIR Half-band filters are not difficult to design. In an earlier post [1], I showed how to design them using the window method. Here, I provide a short Matlab function halfband_synth that uses the Parks-McClellan algorithm (Matlab function firpm [2]) to synthesize half-band filters. Compared to the window method, this method uses fewer taps to achieve a given performance.
Off Topic: The True Gravitational Geodesic
The third of my off topic Physics series resulting in the true gravitational geodesic equation and some surprising results about gravity.
The Discrete Fourier Transform of Symmetric Sequences
Symmetric sequences arise often in digital signal processing. Examples include symmetric pulses, window functions, and the coefficients of most finite-impulse response (FIR) filters, not to mention the cosine function. Examining symmetric sequences can give us some insights into the Discrete Fourier Transform (DFT). An even-symmetric sequence is centered at n = 0 and xeven(n) = xeven(-n). The DFT of xeven(n) is real. Most often, signals we encounter start at n = 0, so they are not strictly speaking even-symmetric. We’ll look at the relationship between the DFT’s of such sequences and those of true even-symmetric sequences.
How the Cooley-Tukey FFT Algorithm Works | Part 4 - Twiddle Factors
The beauty of the FFT algorithm is that it does the same thing over and over again. It treats every stage of the calculation in exactly the same way. However, this. “one-size-fits-all” approach, although elegant and simple, causes a problem. It misaligns samples and introduces phase distortions during each stage of the algorithm. To overcome this, we need Twiddle Factors, little phase correction factors that push things back into their correct positions before continuing onto the next stage.
How the Cooley-Tukey FFT Algorithm Works | Part 3 - The Inner Butterfly
At the heart of the Cooley-Tukey FFT algorithm lies a butterfly, a simple yet powerful image that captures the recursive nature of how the FFT works. In this article we discover the butterfly’s role in transforming complex signals into their frequency components with efficiency and elegance. Starting with the 2-point DFT, we reveal how the FFT reuses repeated calculations to save time and resources. Using a divide-and-conquer approach, the algorithm breaks signals into smaller groups, processes them through interleaving butterfly diagrams, and reassembles the results step by step.
How the Cooley-Tukey FFT Algorithm Works | Part 2 - Divide & Conquer
The Fast Fourier Transform revolutionized the Discrete Fourier Transform by making it much more efficient. In part 1, we saw that if you run the DFT on a power-of-2 number of samples, the calculations of different groups of samples repeat themselves at different frequencies. By leveraging the repeating patterns of sine and cosine values, the algorithm enables us to calculate the full DFT more efficiently. However, the calculations of certain groups of samples repeat more often than others. In this article, we’re going to explore how the divide-and-conquer method prepares the ground for the next stage of the algorithm by grouping the samples into specially ordered pairs.
How the Cooley-Tukey FFT Algorithm Works | Part 1 - Repeating Calculations
The Fourier Transform is a powerful tool, used in many technologies, from audio processing to wireless communication. However, calculating the FT can be computationally expensive. The Cooley-Tukey Fast Fourier Transform (FFT) algorithm provides a significant speedup. It exploits the repetitive nature of calculations within the Discrete Fourier Transform (DFT), the mathematical foundation of the FT. By recognizing patterns in the DFT calculations and reusing intermediate results, the FFT vastly reduces the number of operations required. In this series of articles, we will look at how the Cooley-Tukey FFT algorithm works.
The 2024 DSP Online Conference
The post announces the fifth annual DSP Online Conference, marking the event’s 5th anniversary and featuring renowned DSP practitioners including fred harris, Rick Lyons, Julius Orion Smith III, and Dan Boschen. It outlines access options—purchased passes provide on-demand viewing of all sessions through September 2025—and explains the daily release structure, with new sessions posted at 6 AM EDT and a chat/forum for each presentation. The article describes select live Q&A interactions hosted via Zoom (informal, 30-minute sessions) and lists three scheduled live presentations: Dan Boschen’s workshop on October 30 at 11 AM EDT and Fred Harris’s talks on October 31 at 10 AM and noon EDT. Recordings of live presentations are promised to appear on-demand shortly after they conclude.
Learn to Use the Discrete Fourier Transform
Discrete-time sequences arise in many ways: a sequence could be a signal captured by an analog-to-digital converter; a series of measurements; a signal generated by a digital modulator; or simply the coefficients of a digital filter. We may wish to know the frequency spectrum of any of these sequences. The most-used tool to accomplish this is the Discrete Fourier Transform (DFT), which computes the discrete frequency spectrum of a discrete-time sequence. The DFT is easily calculated using software, but applying it successfully can be challenging. This article provides Matlab examples of some techniques you can use to obtain useful DFT’s.
Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter
Jason Sachs shows how a single-pole IIR low-pass filter, implementable in one line y += alpha * (x - y), tames noise in embedded signals without floating point. The post explains how to compute alpha from tau and delta-t, practical tradeoffs like phase lag and oversampling, and fixed-point pitfalls including how many extra state bits you need to avoid quantization. Short, practical, and code-ready.
Understanding and Relating Eb/No, SNR, and other Power Efficiency Metrics
Eric Jacobsen untangles the common confusion around Eb/N0, SNR, Es/No and related power-efficiency metrics, showing when each metric applies and how to convert between them. He covers practical measurement techniques including spectrum-analyzer and slicer-based estimates, the impact of symbol rate, modulation order and FEC code rate, and offers simple sanity checks to catch common dB and factor-of-two errors. Engineers get a concise toolkit for accurate comparisons.
An Interesting Fourier Transform - 1/f Noise
Power-law signals have a neat Fourier trick: their transforms are power laws too, but with important caveats. Steve Smith walks through the t^α ↔ ω^{-(α+1)} relation, shows how the unit step, the Gamma scaling and a nontrivial phase change the picture, and highlights the special α = -0.5 case that links to 1/f noise. The post frames why phase and physical interpretation keep 1/f noise mysterious.
How the Cooley-Tukey FFT Algorithm Works | Part 1 - Repeating Calculations
The Fourier Transform is a powerful tool, used in many technologies, from audio processing to wireless communication. However, calculating the FT can be computationally expensive. The Cooley-Tukey Fast Fourier Transform (FFT) algorithm provides a significant speedup. It exploits the repetitive nature of calculations within the Discrete Fourier Transform (DFT), the mathematical foundation of the FT. By recognizing patterns in the DFT calculations and reusing intermediate results, the FFT vastly reduces the number of operations required. In this series of articles, we will look at how the Cooley-Tukey FFT algorithm works.
Digital PLL's -- Part 1
A hands-on introduction to time-domain digital phase-locked loops, Neil Robertson builds a simple DPLL model in MATLAB and walks through the NCO, phase detector, and PI loop filter implementations. The post uses phase-in-cycles arithmetic to show how the phase accumulator, detector wrapping, and loop filter interact, and it contrasts linear steady-state behavior with the nonlinear acquisition seen when initial frequency error is large. Part 2 will cover frequency-domain tuning of the loop gains.
Digital PLL's -- Part 2
Neil Robertson builds a Z-domain model of a second-order digital PLL with a proportional-plus-integral loop filter, then derives closed-form formulas for KL and KI from the desired loop natural frequency and damping. The post explains the s → (z - 1)/Ts approximation, shows how to form the closed-loop IIR CL(z) for step and frequency responses, and highlights when the linear Z-domain model falls short of nonlinear acquisition behavior.
Use Matlab Function pwelch to Find Power Spectral Density -- or Do It Yourself
Neil Robertson walks through using Matlab's pwelch and shows how to implement PSD estimation yourself with fft. The post uses concrete examples and complete m-files to demonstrate window selection, converting pxx (W/Hz) to W/bin, Welch DFT averaging, and a worked C/N0 calculation. Readers get practical, runnable recipes for accurate spectrum units, variance reduction with averaging, and peak-power extraction.
Optimizing the Half-band Filters in Multistage Decimation and Interpolation
Multistage decimation and interpolation by powers of two get a lot cheaper if you size each half-band filter differently. Rick Lyons walks through spectra for three-stage examples that show why early stages can use narrower filters for decimation while interpolation reverses the order, and how aliasing and images are handled by later stages. Learn a simple rule to cut multipliers without sacrificing performance.
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.
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.
Computing FFT Twiddle Factors
Rick Lyons gives two compact algorithms to compute individual twiddle factors for radix-2 DIF and DIT FFTs, handy when you need only a subset of outputs such as in pruned FFTs. He explains stage indexing, provides closed-form formulas including the bit-reversal step for DIT, and walks through N=8 examples so you can implement the twiddle-angle calculations directly.
Handling Spectral Inversion in Baseband Processing
Spectral inversion often sneaks in during RF and IF mixing chains and can break downstream demodulation. Eric Jacobsen shows that at baseband you can correct inversion with three trivial, equivalent operations: invert Q, swap I and Q, or invert I, and he explains the math and geometric intuition behind each. The fixes work in modulators or demodulators and tolerate arbitrary phase offsets.
Design IIR Butterworth Filters Using 12 Lines of Code
Build a working lowpass IIR Butterworth filter from first principles in just 12 lines of Matlab using Neil Robertson's butter_synth.m. The post walks through the analog prototype poles, frequency pre-warping, bilinear transform pole mapping, adding N zeros at z = -1, and gain normalization so the result matches Matlab's built-in butter function. It's a compact, hands-on guide with clear formulas and code.
Use Matlab Function pwelch to Find Power Spectral Density -- or Do It Yourself
Neil Robertson walks through using Matlab's pwelch and shows how to implement PSD estimation yourself with fft. The post uses concrete examples and complete m-files to demonstrate window selection, converting pxx (W/Hz) to W/bin, Welch DFT averaging, and a worked C/N0 calculation. Readers get practical, runnable recipes for accurate spectrum units, variance reduction with averaging, and peak-power extraction.
Four Ways to Compute an Inverse FFT Using the Forward FFT Algorithm
Rick Lyons lays out four practical techniques to get an inverse FFT when you only have forward FFT software or FPGA cores available. The post highlights a classic data-reversal trick, a conjugate-symmetry optimized flow, and two methods that avoid reversals using data swapping or complex conjugation plus scaling. Each method notes when it is preferable so engineers can pick the least costly implementation.
Understanding and Relating Eb/No, SNR, and other Power Efficiency Metrics
Eric Jacobsen untangles the common confusion around Eb/N0, SNR, Es/No and related power-efficiency metrics, showing when each metric applies and how to convert between them. He covers practical measurement techniques including spectrum-analyzer and slicer-based estimates, the impact of symbol rate, modulation order and FEC code rate, and offers simple sanity checks to catch common dB and factor-of-two errors. Engineers get a concise toolkit for accurate comparisons.
A Beginner's Guide to OFDM
Orthogonal Frequency Division Multiplexing made modern high-speed wireless practical by turning one fast serial bitstream into many slow parallel streams carried on orthogonal sinusoids. This beginner guide explains, with minimal math, how the iDFT/DFT pair builds OFDM, how spectral slicing makes each subcarrier effectively flat so equalization reduces to simple divisions, and why a cyclic prefix prevents inter-symbol interference.
The DFT Magnitude of a Real-valued Cosine Sequence
Rick Lyons proves a simple but often-missing result: the N-point DFT peak magnitude of a real cosine with an integer number of cycles equals A·N/2. He uses Euler's formula and geometric-series summation, shows a neat shortcut that avoids l'Hôpital's rule, and connects the math to practical fixed-point FFT sizing and overflow prevention on two's-complement hardware. The post also notes conjugate symmetry and the same result for sine inputs.
Already 3000+ Attendees Registered for the Upcoming Embedded Online Conference
More than 3,000 engineers have already signed up for the Embedded Online Conference, and free registration closes at the end of February. Stephane Boucher highlights four practical tracks—DSP and machine learning, FPGA, embedded systems programming, and embedded systems security—and notes that every talk will be available to stream on demand from May 20. If you prefer no-travel learning or want flexible access to world-class talks, register now.
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.
The DSP Online Conference - Right Around the Corner!
Three months after a forum post, Stephane Boucher and Jacob Beningo pulled together the DSP Online Conference, a two-day virtual event featuring 14 talks from leading DSP experts. Most sessions are 30 to 60 minutes with a 30-minute Zoom Q&A, while extended deep dives from speakers like fred harris are included. Registered attendees get one-year on-demand access, and free or reduced passes are available.
Already 3000+ Attendees Registered for the Upcoming Embedded Online Conference
More than 3,000 engineers have already signed up for the Embedded Online Conference, and free registration closes at the end of February. Stephane Boucher highlights four practical tracks—DSP and machine learning, FPGA, embedded systems programming, and embedded systems security—and notes that every talk will be available to stream on demand from May 20. If you prefer no-travel learning or want flexible access to world-class talks, register now.
Free Goodies from Embedded World - Full Inventory and Upcoming Draw Live-Streaming Date
Stephane came back from Embedded World with a massive haul of development kits, tools and swag and decided to give it away to multiple winners. Read the full inventory, learn how to enter by liking or sharing the LinkedIn and Twitter posts, and tune in Friday March 29 at 1pm EST on EmbeddedRelated.tv for the live draw where winners will pick their prizes.
Free Goodies from Embedded World - What to Do Next?
Stephane Boucher went on a hunt for free stuff at Embedded World to assemble a giveaway bundle for a lucky reader. This short update shares that haul and asks the embedded community for ideas on what to do next. It is a conversational call for suggestions, aiming to turn conference swag into a useful prize.
Back from Embedded World 2019 - Funny Stories and Live-Streaming Woes
Stephane Boucher tried live-streaming multiple talks from Embedded World 2019 and turned a chaotic experiment into a useful set of lessons for embedded engineers. Between broken tripods, flaky venue WiFi, tricky German SIM purchases, and audio nightmares, he learned practical fixes for reliable streams and better video quality. Read this if you want candid, tactical advice on streaming hardware, connectivity, and on-site troubleshooting.
Spread the Word and Run a Chance to Win a Bundle of Goodies from Embedded World
EmbeddedRelated is turning Embedded World into a live-streaming experiment, and the more engineers help spread the word, the better the coverage could get. Stephane Boucher is asking readers to follow updates on Twitter and LinkedIn, where every like, share, or repost adds another chance to win a box of vendor goodies. The prize mix includes t-shirts, dev kits, gadgets, and plenty of pens.
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.
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.
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.
