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.
Should DSP Undergraduate Students Study z-Transform Regions of Convergence?
Rick Lyons argues z-transform regions of convergence are mostly a classroom abstraction with little practical use for real-world DSP engineers. For all stable LTI impulse responses encountered in practice the ROC includes the unit circle, so DTFT and DFT exist and ROC analysis rarely affects implementation. He notes digital oscillators are a notable exception, and suggests reallocating classroom time to more practical engineering topics.
Implementing Impractical Digital Filters
Some published IIR block diagrams are impossible to implement because they contain delay-less feedback paths, and Rick Lyons shows how simple algebra fixes that. He works through two concrete examples—a bandpass built from a FIR notch and a narrowband notch using a feedback loop—and derives equivalent, implementable second-order IIR transfer functions. The post emphasizes spotting problematic loops and replacing them with practical block diagrams.
An Astounding Digital Filter Design Application
Rick Lyons was astonished by the ASN Filter Designer, a hands-on filter design tool that makes tweaking frequency responses as simple as dragging markers with your mouse. The software updates magnitude plots, z-plane pole/zero locations, and filter coefficients in real time, and it also includes a signal analyzer plus a MATLAB-like scripting language for custom coefficient generation. The post links to a demo and user guides so you can try it yourself.
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.
The Swiss Army Knife of Digital Networks
A single discrete-signal network can masquerade as a comb filter, a recursive section, or something much more versatile. Rick Lyons shows how this seven-coefficient structure can be reconfigured to realize a wide range of DSP functions, with tables of impulse responses, pole-zero plots, and frequency responses to illustrate each case. The full explanations live in the downloadable PDF, but the post gives a strong feel for why this is such a handy building block.
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.
Peak to Average Power Ratio and CCDF
Setting digital modulator levels depends on peak-to-average power ratio, because random signals produce occasional high peaks that cause clipping. This post shows how to compute the CCDF of PAPR from a signal vector, with MATLAB code and examples for a sine wave and Gaussian noise. The examples reveal the fixed 3.01 dB PAPR of a sine and the need for large sample counts to capture rare AWGN peaks.
Filter a Rectangular Pulse with no Ringing
You can filter a rectangular pulse with no ringing simply by using an FIR whose coefficients are all positive, and make them symmetric to get identical leading and trailing edges. This post walks through a MATLAB example that convolves a normalized Hanning window with a 32-sample rectangular pulse, showing that window length controls edge duration and that shorter windows widen the spectrum. It also notes this is not a QAM pulse-shaping solution.
Data Types for Control & DSP
Control engineers often default to double precision, but Tim Wescott shows that choice can waste CPU cycles on embedded targets. He separates numeric representation into floating point, integer, and fixed-point, then walks through the tradeoffs, including quantization, overflow, and performance. A concrete PID example highlights why integrator precision and ADC scaling should drive your choice of data type rather than habit.
Finding the Best Optimum
Optimization is seductive but often misleading, especially when mathematical models don't match messy reality. Tim Wescott shares stories from circuits and communications to show how chasing the theoretical global optimum can waste time and money. He recommends framing 'best' in practical terms, validating models, and optimizing for cost and impact so products ship on time and actually work in the real world.
Analytic Signal
In communication theory and modulation theory we always deal with two phases: In-phase (I) and Quadrature-phase (Q). The question that I will discuss in this blog is that why we use two phases and not more.
Software Defined Radio at SAMOS
At SAMOS, the SDR track drew a strong academic crowd, with groups from UMich, Wisconsin-Madison, Linköping, IMEC, and others presenting their latest ideas. Praveen Raghavan also notes that IMEC finally made its SyncPro architecture public, a vector synchronization processor design. The post gives a quick snapshot of where software defined radio research was active, and which major industry names were noticeably absent.
The History of CIC Filters: The Untold Story
Hogenauer's 1981 paper is the canonical CIC reference, but this post uncovers an earlier, practical origin story: engineer Richard Newbold used and documented a CIC decimation filter in late 1979. Rick Lyons recounts how Newbold’s HP-35 calculations produced the now-familiar frequency-response plot that appeared in Hogenauer's paper, why managers feared a pole at DC, and how demonstrations won adoption.
Two Easy Ways To Test Multistage CIC Decimation Filters
Rick Lyons shows that you can validate multistage CIC decimation filters with just two obvious tests, no elaborate spectral setup required. Apply a unit-sample impulse to check a combinatorial yout(1) value when D ≥ S, or feed an all-ones step to confirm an S-sample transient followed by a DS steady state; the Appendix ties both checks to Pascal's triangle and binomial math.
GPGPU DSP
Shehrzad Qureshi kicks off his DSP blog by championing GPGPU, focusing on Nvidia's CUDA and real-product experience. He argues that with CPU clock speeds stalled, large-scale parallelism on GPUs is the practical path forward for many signal-processing tasks. The post traces GPGPU history from shader 'hacks' to modern APIs and previews future posts comparing CUDA vs OpenCL, Intel's Larrabee, and Nvidia Fermi.
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.
Recruiting New Bloggers!
EmbeddedRelated is expanding its blogging team, and Stephane Boucher is inviting engineers, students, hobbyists, and researchers to contribute. He points to the success of earlier contributors and says the community has already read their articles more than 1,250,000 times. If you have knowledge to share, this post explains how to pitch a topic and get started.
Peak to Average Power Ratio and CCDF
Setting digital modulator levels depends on peak-to-average power ratio, because random signals produce occasional high peaks that cause clipping. This post shows how to compute the CCDF of PAPR from a signal vector, with MATLAB code and examples for a sine wave and Gaussian noise. The examples reveal the fixed 3.01 dB PAPR of a sine and the need for large sample counts to capture rare AWGN peaks.
New Comments System (please help me test it)
DSPRelated just got a practical upgrade, Stephane Boucher has released a new comments system built from his earlier forum work. It supports drag-and-drop or Insert Image uploads, MathML, TeX and ASCIImath rendered by MathJax, syntax-highlighted code via highlight.js, and in-place editing and deletion of comments. Improved email notifications alert authors and commenters to replies, and readers are invited to post test comments and report problems.
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.
Complex Down-Conversion Amplitude Loss
Rick Lyons shows why a standard complex down-converter seems to halve amplitudes yet only imposes a -3 dB power loss. He walks through mixing math from an RF cosine to i and q paths, demonstrates that each path has peak A/2 but the complex output has half the average power, and offers practical guidance for software modeling and avoiding spectral interpretation traps.
Computing Translated Frequencies in Digitizing and Downsampling Analog Bandpass Signals
Textbooks rarely give ready formulas for tracking where individual spectral lines land after bandpass sampling or decimation. Rick Lyons provides three concise equations, with Matlab code, that compute translated frequencies for analog bandpass sampling, real digital downsampling, and complex downsampling. Practical examples show how to place the sampled image at fs/4 and how to translate a complex bandpass to baseband for efficient demodulation.
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.
Sonos, Shut Up and Take My Money! - Is Spatial Audio Finally Here?
Stephane bought a Sonos ERA 300 and discovered that spatial audio can finally feel convincing from a single wireless speaker, provided you set it up correctly. The trick is using Dolby Atmos tracks played inside the Sonos app, plus Sonos' calibration and a close listening position. The post shares setup tips, vivid listening impressions, and encouragement for more spatial mixes to come.
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.
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.
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.
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.
Decimators Using Cascaded Multiplierless Half-band Filters
In my last post, I provided coefficients for several multiplierless half-band FIR filters. In the comment section, Rick Lyons mentioned that such filters would be useful in a multi-stage decimator. For such an application, any subsequent multipliers save on resources, since they operate at a fraction of the maximum sample frequency. We’ll examine the frequency response and aliasing of a multiplierless decimate-by-8 cascade in this article, and we’ll also discuss an interpolator cascade using the same half-band filters.
