A DSP Quiz Question
A short visual puzzle from Rick Lyons shows how a common plotting trick can fool even experienced DSP engineers. He presents a 3D circular plot that looks like a triangular window but is actually a 32-point hann window, then explains why the circular projection distorts the view. The post highlights the importance of checking equations and 2D plots before naming a window by sight.
The Discrete Fourier Transform and the Need for Window Functions
The FFT alone can mislead: capturing a finite-length signal with a rectangular window smears energy across frequency, producing spectral leakage that hides real components. This post explains the origin of leakage, shows how tapered windows such as the Hanning window suppress sidelobes, and demonstrates the tradeoff between sidelobe suppression and mainlobe widening while covering practical tips on zero-padding and record length.
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.
Modeling Anti-Alias Filters
Modeling anti-alias filters brings textbook aliasing examples to life. This post shows how to build discrete-time models G(z) for analog Butterworth and Chebyshev lowpass anti-alias filters, compares bilinear transform and impulse invariance, and simulates ADC input/output including aliasing of sinusoids and Gaussian noise. It concludes that impulse invariance gives better stopband accuracy and includes Matlab helper functions.
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.
Sampling bandpass signals
Bandpass signals can be sampled at rates below the usual Nyquist limit, and this note shows how the band-limited spectrum appears in baseband after sampling. Using a simple example figure, it defines the center frequency fc = (fmax + fmin)/2 and bandwidth Δf = fmax - fmin, and highlights that choosing fs less than twice the signal's highest frequency violates the sampling theorem.
Digital Filter Instructions from IKEA?
This is a wordless example of a folded FIR filter. Swedish “Bygglek” = build and play.
Simulink-Simulation of SSB demodulation
This post walks through Simulink models that implement SSB demodulation and modulation, using Richard Lyons' phasing method as a foundation. It shows practical models for simple carrier multiplication and for the phasing method with cosine and -sin paths plus Hilbert filtering, and it highlights sampling, decimation, filter choices, and delay alignment to make the techniques work in simulation.
Setting Carrier to Noise Ratio in Simulations
Setting the right Gaussian noise level is easy once you know the math. This post derives simple, practical equations to compute noise density and the rms noise amplitude needed to achieve a target carrier to noise ratio at a receiver output. It shows how to get the noise-equivalent bandwidth from a discrete-time filter, how to compute N0 and sigma, and includes a MATLAB set_cnr function to generate the noise vector.
An Efficient Full-Band Sliding DFT Spectrum Analyzer
Rick Lyons shows two compact sliding DFT networks that compute the 0th bin and all positive-frequency outputs for even and odd N, running sample-by-sample on real input streams. The designs reduce computational workload versus a prior observer-based sliding DFT by using fewer parallel paths, while remaining guaranteed stable and avoiding the traditional comb delay-line. A simple initialization and streaming procedure makes them practical for real-time spectrum analysis.
Simplest Calculation of Half-band Filter Coefficients
Half-band FIR filters put the cutoff at one-quarter of the sampling rate, and nearly half their coefficients are exactly zero, which makes them highly efficient for decimation-by-2 and interpolation-by-2. This post shows the straightforward window-method derivation of half-band coefficients from the ideal sinc impulse response, providing a clear, hands-on explanation for engineers learning filter design. It also points to equiripple options such as Matlab's firhalfband and a later Parks-McClellan implementation.
Linear Feedback Shift Registers for the Uninitiated, Part XVI: Reed-Solomon Error Correction
Jason Sachs demystifies Reed-Solomon codes with hands-on examples and pragmatic tips for embedded engineers. The article shows why RS encoding is just polynomial division in GF(2^m), why decoding is mathematically heavier, and how to implement encoders in Python and in C-friendly form using LFSRs and table-driven methods. Read this for working code, generator-polynomial examples, and an embedded-minded view of RS practicalities.
Music/Audio Signal Processing
Julius Orion Smith III traces his journey from musician to music/audio DSP researcher, sharing the choices that shaped his career and research focus. He recounts work on violin modeling and waveguide synthesis, then highlights modern prototyping tools like Faust and Octave that accelerate experimentation. Read for practical career advice on coursework, publishing, and why free open-source tools matter for rapid audio research.
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.
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.
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.
Waveforms that are their own Fourier Transform
Steve Smith admits a long-standing mistake and overturns the claim that only Gaussians are their own Fourier transform. He gives trivial and nontrivial examples, explains why infinitely many such waveforms exist, and shows a quick discrete construction using the DFT with a 1/sqrt(N) normalization. Engineers get an intuitive 30-second argument plus a practical recipe to build self-Fourier signals.
The Beginning of a New Chapter
After years of hesitation, Stephane Boucher and Jacob Beningo finally turned their virtual events into an in-person reality with the inaugural Signal Processing Summit and Embedded Systems Summit at the Sonesta Silicon Valley. The post captures the logistics, a last-minute travel scare during a US government shutdown, the joy of meeting speakers like Fred Harris, and practical lessons for future technical events. It closes by inviting community feedback and venue suggestions.
Fractional Delay FIR Filters
You can realize arbitrary fractional-sample delays with standard FIR filters by shifting a sinc impulse response and removing symmetry, then windowing the result. This post shows a practical window-method implementation using Chebyshev windows, gives Matlab functions (frac_delay_fir.m and frac_delay_lpf.m) in the appendix, and walks through examples that demonstrate the delay, magnitude trade-offs, and how increasing taps widens the flat-delay bandwidth.
Evaluate Window Functions for the Discrete Fourier Transform
Spectral leakage makes DFTs of continuous sinewaves misleading, and windowing is the practical workaround. This post supplies Matlab code to plot spectra of windowed sinewaves and compute figures of merit, so you can compare windows such as flattop and Chebyshev. See how sidelobe level, mainlobe bandwidth, processing loss, noise bandwidth, and scallop loss trade off to guide your window choice.
Linear-phase DC Removal Filter
Rick Lyons presents a practical, multiplier-free way to remove DC while preserving linear phase by cascading D-point moving-average filters. He shows how choosing D as a power of two gives bit-shift scaling, how a dual-MA yields a narrow transition band with modest ripple, and how a quad-MA drives ripple down to near inaudible levels while noting the fixed-point accumulator sizing required.
Linear Feedback Shift Registers for the Uninitiated, Part XVI: Reed-Solomon Error Correction
Jason Sachs demystifies Reed-Solomon codes with hands-on examples and pragmatic tips for embedded engineers. The article shows why RS encoding is just polynomial division in GF(2^m), why decoding is mathematically heavier, and how to implement encoders in Python and in C-friendly form using LFSRs and table-driven methods. Read this for working code, generator-polynomial examples, and an embedded-minded view of RS practicalities.
An s-Plane to z-Plane Mapping Example
A misleading online diagram prompted Rick Lyons to reexamine how s-plane points map to the z-plane. He spotted apparent errors in the original figure, drew a corrected mapping, and invites readers to inspect both diagrams and point out any remaining mistakes. The short post is a quick visual primer for engineers who rely on accurate s-plane to z-plane mappings in analysis and design.
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.
The Power Spectrum
You can get absolute power from a DFT, not just relative spectra. In this post Neil Robertson shows how to convert FFT outputs into watts per bin using Parseval's theorem, how to form one-sided spectra, and how to normalize windows so power is preserved. Matlab examples demonstrate bin-centered and between-bin sinusoids, leakage, scalloping, and how to recover component power by summing bins.
A New Related Site!
The post announces the launch of MLRelated, a new Related site dedicated to machine learning and deep learning. It positions MLRelated as complementary to existing Related sites by highlighting cross-cutting interests: TinyML for embedded developers, machine/deep learning applications in signal processing, and FPGA-based AI/ML implementations. The new site debuts with a modest amount of content and is expected to expand rapidly through contributions from the Related community in the form of blogs, forum threads, and webinars. The author invites readers to report navigation errors, share feedback, and propose ideas to help steer MLRelated into a practical, community-driven resource for researchers and practitioners in ML and adjacent domains.
Embedded Toolbox: Programmer's Calculator
A tiny but powerful cross-platform tool, QCalc evaluates full C-syntax expressions so you can paste results straight into firmware. It handles bitwise ops, mixed hex/decimal/binary constants, and scientific math, and it automatically shows integer results in formatted hex and binary. The post explains key features, variable handling, error messages, and how to run qcalc.tcl with the wish Tk interpreter.
Fractional Delay FIR Filters
You can realize arbitrary fractional-sample delays with standard FIR filters by shifting a sinc impulse response and removing symmetry, then windowing the result. This post shows a practical window-method implementation using Chebyshev windows, gives Matlab functions (frac_delay_fir.m and frac_delay_lpf.m) in the appendix, and walks through examples that demonstrate the delay, magnitude trade-offs, and how increasing taps widens the flat-delay bandwidth.
A Differentiator With a Difference
Rick Lyons presents a compact, practical FIR differentiator that combines central-difference noise attenuation with a much wider linear range. The proposed ydif(n) doubles the usable frequency range to about 0.34π (0.17fs), uses ±1/16 coefficients so multiplications become simple 4-bit right shifts, and has an exact three-sample group delay for easy synchronization with other signals.
Code Snippets Section Now LIVE
A new code-sharing section on DSPRelated is now live, giving engineers a central place to browse and share DSP code snippets. Take a few minutes to rate and comment on snippets you can judge, or apply to become a contributor and upload your own examples. You can also learn about the contributor reward program and send feedback to help the community grow, says Stephane Boucher.
New Code Sharing Section & Reward Program for Contributors!
DSPRelated is launching a new code sharing section and looking for contributors to help seed it with useful DSP snippets. Stephane Boucher also introduces a pageview-based reward program, with payouts tied to unique visits so popular code can earn contributors up to $250. It is a practical push to build a high-quality library for the DSP community from the start.
50,000th Member Announced!
DSPRelated just crossed a major milestone, 50,000 registered members, and Stephane Boucher celebrates the anniversary by spotlighting the lucky winner, Charlie Tsai, an assistant professor in Taiwan. The post also looks back at more than a decade of community growth and thanks the contributors, authors, and sponsors who helped make the site a go-to DSP resource. It closes with a promise of big improvements ahead in 2010.
Almost 50,000 Members!
DSPRelated is about to hit a big milestone, and Stephane Boucher marks the moment with a playful giveaway announcement. The post jokes about a BMW prize for the 50,000th registered member, then reveals the real reward, a copy of Rick Lyons’ Understanding Digital Signal Processing. It is a lighthearted community update built around a membership milestone and a bit of fun.
DSPRelated faster than ever!
Stephane Boucher moved DSPRelated's static assets to Amazon CloudFront to shrink page load times worldwide. Images, JavaScript and CSS are now served from the nearest CloudFront edge server, reducing latency especially for readers in Europe and Asia. If you visit regularly, you should notice the speedup, and the author asks readers to report their load-time experience in the comments.
New Papers / Theses Section
Stephane Boucher launched a Papers & Theses section on DSPRelated to gather DSP dissertations and papers in one spot. Authors can submit already-hosted documents or upload PDFs for optional hosting, provided they have sharing rights, and help is available for PDF conversion. Listing your work boosts visibility and opportunities, and non-English documents are welcomed while the section is in beta.
New Blog Section!
DSPRelated just launched a new blogs section, and it is already starting to take shape. Stephane Boucher says he received around 50 proposals from DSP engineers, chose an initial set of 10 bloggers, and is now setting up their accounts. The section is still in beta, but there is also more on the way, including a future area for sharing quality code in asm, C, and MATLAB.
New Discussion Group: DSP & FPGA
Stephane Boucher has launched a new discussion group for engineers implementing DSP functions on FPGAs. It is meant to become a focused place for sharing ideas, but he notes it may take a few weeks before enough members join for the discussion to really get going. If FPGA-based DSP is your thing, this is an open invitation to get involved early.
