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.
Update to a Narrow Bandpass Filter in Octave or Matlab
Paul Lovell presents an updated, compact Octave/Matlab implementation of a narrow bandpass FIR that runs about four times faster and uses float32 to cut processing cost. The design combines a single matrix IFIR stage with three moving-sum (RRS) stages per baseband, auto-calculates the IFIR expansion factor, and adds easier parameter setup plus WAV I/O and FFT plots. A TensorFlow Colab demo is also provided.
Discrete Wavelet Transform Filter Bank Implementation (part 2)
David Valencia walks through practical differences between the discrete wavelet transform and the discrete wavelet packet transform, showing why DWPT yields symmetric frequency resolution while DWT favors a single high-pass branch. He explains how Noble identities let you collapse multi-branch filter banks into equivalent single convolutions, then compares block convolution matrices with chain-processing and links to MATLAB code for both approaches.
Design IIR Bandpass Filters
Designing Butterworth IIR bandpass filters is easier than it looks when you start from a lowpass prototype. This post walks through the s-domain lowpass-to-bandpass transform, bilinear digital mapping, and the bp_synth.m Matlab implementation that produces scaled numerator and denominator coefficients. Practical pole-zero intuition and Matlab examples help you verify magnitude and group-delay behavior for real sampling rates and bandwidths.
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.
Simultaneously Computing a Forward FFT and an Inverse FFT Using a Single FFT
Rick Lyons presents a compact seven-step algorithm to compute a forward FFT and an inverse FFT at the same time using a single radix-2 complex FFT. The method builds intermediate sequences v(n) and z(n), exploits conjugate symmetry, and requires only one N-point FFT plus about 2N additions or subtractions. A clear MATLAB implementation accompanies the explanation so you can try it immediately.
Polyphase Filters and Filterbanks
Kyle walks through practical polyphase filtering and analysis filterbanks, complete with Python code using numpy, scipy and matplotlib. The post shows how splitting an FIR into M polyphase legs gives identical, more efficient decimation while avoiding aliasing, and it flags the subtle reordering, zero padding and FFT versus IDFT ordering issues that trip many implementers. Includes runnable reference code and links for deeper theory.
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.
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.
Computing Large DFTs Using Small FFTs
Rick Lyons demonstrates a practical trick for computing large N-point DFTs by combining multiple smaller radix-2 FFTs when only limited FFT sizes are available. He walks through 16-point and 24-point examples using two and three 8-point FFTs, shows how to assemble outputs with twiddle factors, and explains a symmetry that reduces twiddle storage to N/4 values. The method supports non-power-of-two DFT lengths.
Frequency Dependence in Free Space Propagation
Free-space propagation of electromagnetic waves is essentially independent of frequency, a counterintuitive conclusion Eric Jacobsen demonstrates step by step. He shows the λ^2 factor in the Friis transmission equation comes from antenna effective area and gain, not from the space between antennas, explaining why dipoles favor lower bands while dishes improve with frequency. The post also reminds engineers that material penetration and atmospheric absorption remain genuine frequency dependent concerns.
A Simple Complex Down-conversion Scheme
Rick Lyons shows a compact way to turn a real bandpass signal centered at ±fs/4 into a complex, zero-centered analytic signal. The trick uses a delay, a Hilbert transform filter, and a 4:1 downsample, with a small compensation filter to widen the usable passband. He also points out a no-multiplier implementation using shift-and-add coefficients, or a higher-attenuation version with two multiplies per output sample.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
