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.
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.
Dealing With Fixed Point Fractions
Fixed-point fractional math is easy to botch, and this post lays out pragmatic ways to avoid those mistakes. It clarifies the difference between integer and fractional overflow, shows how Q notation helps track binary-point scaling, and explains why multiplies add sign bits that may require shifting. Read for concrete FPGA strategies: keeping bit growth, selective shifts, or aggressive normalization, plus testing tips.
The DFT Output and Its Dimensions
The DFT gives N outputs for N samples, yet for real-valued signals most of those outputs are redundant. This post explains how conjugate symmetry organizes the output into a real DC bin, N/2-1 complex positive-frequency bins, a real Nyquist bin for even N, and then the conjugate mirror bins. A 64-point example illustrates which bins carry unique information and which can be discarded.
Amplitude modulation and the sampling theorem
Amplitude modulation turns out to be a neat way to build intuition for the Nyquist-Shannon sampling theorem. In this draft chapter from Think DSP, the author shows how multiplying by a carrier shifts spectra, why sampling creates repeated copies in frequency, and how low-pass filtering can recover the original signal when those copies do not overlap.
Exponential Smoothing with a Wrinkle
Cedron Dawg shows how pairing forward and backward exponential smoothing produces exact, frequency-dependent dampening for sinusoids while canceling time-domain lag. The average of the two passes scales the tone by a closed-form factor, and their difference acts like a first-derivative with a quarter-cycle phase shift. The post derives the analytic dampening formulas, compares them to the derivative, and includes a Python demo for DFT preprocessing.
Discrete-Time PLLs, Part 1: Basics
In this series of tutorials on discrete-time PLLs we will be focusing on Phase-Locked Loops that can be implemented in discrete-time signal proessors such as FPGAs, DSPs and of course, MATLAB.
Compressive Sensing - Recovery of Sparse Signals (Part 1)
The amount of data that is generated has been increasing at a substantial rate since the beginning of the digital revolution. The constraints on the sampling and reconstruction of digital signals are derived from the well-known Nyquist-Shannon sampling theorem...
Design of an anti-aliasing filter for a DAC
If you need a practical way to design an anti-aliasing filter for a DAC, this post delivers an Octave/Matlab script that numerically optimizes a Laplace-domain transfer function for linear phase and arbitrary magnitude. The routine models the DAC sample-and-hold sinc response, compensates group delay automatically, and can include an optional multiplierless FIR equalizer. An example shows a 5.4 dB objective improvement and reduced analog Q for easier implementation.
Compute Images/Aliases of CIC Interpolators/Decimators
CIC filters provide multiplier-free interpolation and decimation for large sample-rate changes, but their images and aliases can trip up designs. This post supplies two concise Matlab functions and hands-on examples to compute interpolator images and decimator aliases, showing spectra and freqz plots. Readers will learn how interpolation ratio and number of stages alter passband, stopband, and aliasing behavior.
A Fast Real-Time Trapezoidal Rule Integrator
Rick Lyons presents a compact, recursive real-time Trapezoidal Rule integrator that computes N-sample discrete integration using only four arithmetic operations per input sample. The proposed network yields a finite-length, linear-phase impulse response with constant group delay (N-1)/2 and cuts substantial computation compared with a tapped-delay implementation, making it useful for speeding Romberg-based digital filters.
Fibonacci trick
Tim Wescott shares a compact, surprising trick linking Fibonacci numbers and difference equations. Start with any two consecutive Fibonacci numbers, negate the larger-magnitude one, and iterate the usual recurrence; after a few steps you'll arrive at the standard Fibonacci sequence or its negative. This behavior is specific to the Fibonacci recurrence and makes a great illustrative example for teaching linear recurrences.
Signed serial-/parallel multiplication
Struggling with costly wide adders for signed multiplication on FPGAs? Markus Nentwig unpacks a neat bit-level trick that turns two's-complement signed-signed multiplication into a serial-parallel routine using only a one-bit wider adder. Learn how flipping sign bits and a small, controlled constant cancel lets you avoid full sign-extension, and get a parametrized Verilog RTL plus synthesis notes to try it yourself.
Exponential Smoothing with a Wrinkle
Cedron Dawg shows how pairing forward and backward exponential smoothing produces exact, frequency-dependent dampening for sinusoids while canceling time-domain lag. The average of the two passes scales the tone by a closed-form factor, and their difference acts like a first-derivative with a quarter-cycle phase shift. The post derives the analytic dampening formulas, compares them to the derivative, and includes a Python demo for DFT preprocessing.
The Zeroing Sine Family of Window Functions
A previously unrecognized family of DFT window functions is introduced, built from products of shifted sines that deliberately zero out tail samples and control nonzero support. Cedron Dawg presents recursive and semi-root constructions, runnable code, and numerical examples, and shows that the odd-N member L=(N-1)/2 numerically matches a discrete Hermite-Gaussian DFT eigenvector. The post highlights practical properties, an even-N fix, and applications to spectrograms and tone decomposition.
DFT Graphical Interpretation: Centroids of Weighted Roots of Unity
DFT bin values can be seen as centroids of weighted roots of unity, a geometric picture that makes many DFT properties immediate. Cedron Dawg uses the geometric-series identity and polar plots of integer and fractional tones to show why constants appear only at DC, how wrapping relates to bin index, and how phase, scaling, offsets, and real-signal symmetry affect bin magnitudes and angles.
Deconvolution by least squares (Using the power of linear algebra in signal processing).
When we deal with our normal discrete signal processing operations, like FIR/IIR filtering, convolution, filter design, etc. we normally think of the signals as a constant stream of numbers that we put in a sequence
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.
FIR sideways (interpolator polyphase decomposition)
Markus Nentwig presents a compact way to implement a symmetric FIR interpolator by rethinking the usual tapped delay line. The 1:3 polyphase example uses separate delay lines per coefficient to skip multiplies on known zeros and exploit symmetry, cutting multiplications substantially; a Matlab/Octave demo and notes on ASIC-friendly implementation are included to help evaluate real-world cost tradeoffs.
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.
Feedback Controllers - Making Hardware with Firmware. Part 4. Engineering of Evaluation Hardware
This installment follows the hardware from concept to first power-up for a low-latency feedback controller and arbitrary circuit emulator. It walks through the practical engineering steps, from requirements, block diagrams, and issue tracking to component selection, simulation, PCB planning, purchasing, and staged bring-up. The result is a realistic look at how careful due diligence and a few trade-offs turned a research idea into working evaluation hardware.
Some Thoughts on Sampling
Sampling's 1/Ts amplitude factor is not a paradox but a consequence of axis scaling and impulse density, once you view the units correctly. This post walks through impulse trains in continuous and discrete time, uses DFT examples and Parseval's relation, and shows how downsampling and time scaling produce the familiar spectral replicas and their amplitudes. The geometry of the axes resolves the confusion.
Feedback Controllers - Making Hardware with Firmware. Part 3. Sampled Data Aspects
This article digs into practical sampled-data issues you must address when building feedback controllers for circuit emulation. It highlights a common MATLAB versus Simulink discrepancy caused by DAC holding, explains why FOH (ramp-invariant) c2d conversion matters, and surveys latency, bit depth, filter and precision trade-offs. It also lists candidate ADCs, DACs and FPGAs used in a real evaluation platform to guide hardware choices.
Feedback Controllers - Making Hardware with Firmware. Part 2. Ideal Model Examples
An engineer's guide to building ideal continuous-time models for hardware emulation, using TINA Spice, MATLAB and Simulink to validate controller and circuit behavior. The article shows how a passive R-C network can be emulated by an amplifier, a current measurement and a summer, with Spice, MATLAB and Simulink producing coincident Bode responses. Small phase differences between MATLAB and Simulink are noted, and sampled-data issues are slated for the next installment.
Exponential Smoothing with a Wrinkle
Cedron Dawg shows how pairing forward and backward exponential smoothing produces exact, frequency-dependent dampening for sinusoids while canceling time-domain lag. The average of the two passes scales the tone by a closed-form factor, and their difference acts like a first-derivative with a quarter-cycle phase shift. The post derives the analytic dampening formulas, compares them to the derivative, and includes a Python demo for DFT preprocessing.
A brief look at multipath radio channels
Markus Nentwig walks through a hands-on RF experiment that makes multipath and fading visible using a network analyzer and simple dipole antennas. He shows how reflections produce frequency-domain notches when path differences equal half wavelengths, and how doubling distance increases free-space path loss by roughly 6 dB. The post explains why narrowband signals often see flat fading while wideband links become frequency-selective, motivating OFDM and multi-tap channel models.
Feedback Controllers - Making Hardware with Firmware. Part 6. Self-Calibration Related.
Self-calibration is the missing piece that turns this mixed-signal hardware from a prototype into a usable instrument. In this installment, the author lays out how the board will measure itself, generate reference signals, and verify ADC and DAC behavior before the low-latency control firmware is built. The result is a practical framework for evaluation, production test, and routine self-test.
Hidden Linear Algebra in DSP
Linear algebra is hiding in plain sight inside many DSP techniques, not just abstract theory. By treating linear systems as matrix operators y = A x you reveal Toeplitz structure in LTI systems, connect to covariance matrices, and gain geometric intuition via eigenvalues and eigenvectors. This matrix viewpoint complements convolution-based thinking and offers practical tools for filter and channel analysis.
