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.
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.
Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka
For my first blog, I thought I would introduce the reader to Scilab [1] and Weka [2]. In order to illustrate how they work, I will put together a script in Scilab that will sample using the microphone and CODEC on your PC and save the waveform as a CSV file.
A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT
Cedron Dawg derives an exact two-bin frequency formula for a pure complex tone in the DFT, eliminating amplitude and phase to isolate frequency via a complex quotient and the complex logarithm. He presents an adjacent-bin simplification that replaces a complex multiply with a bin offset plus an atan2 angle, and discusses integer-frequency handling and aliasing. C source and numerical examples show the formula working in practice.
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.
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.
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.
DFT Bin Value Formulas for Pure Real Tones
Cedron Dawg derives a closed-form expression for the DFT bin values produced by a pure real sinusoid, then uses that formula to explain well known DFT behaviors. The post walks through the algebra from Euler identities to a compact computational form, highlights the integer versus non-integer frequency cases, and verifies the result with C code and printed numeric output.
Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 2)
Cedron Dawg derives a second family of exact time domain formulas for single-tone frequency estimation that trade a few extra calculations for improved noise robustness. Built from [1+cos]^k binomial weighting of neighbor-pair sums, the closed-form estimators are exact and are best evaluated at signal peaks for real tones, while complex tones do not share the zero-crossing limitation. Coefficients up to k=9 are provided.
Discrete Wavelet Transform Filter Bank Implementation (part 1)
David Valencia walks through a practical implementation of discrete wavelet transform filter banks, focusing on cascading branches and efficient equivalent filters. He contrasts DWT and DFT resolution behavior and shows how cascading the low-pass branch sharpens frequency division while the high-pass path remains unchanged. Code pointers and a preview of formfilters() demonstrate how to compute only the needed samples by combining filters with upsampling.
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.
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 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 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.
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.
Filtering Noise: The Basics (Part 1)
How do you pull signals out of random noise? This post builds intuition from first principles for discrete-time white Gaussian noise and shows how simple linear FIR filtering (averaging) reduces noise. You’ll get derivations for the output mean, variance and autocorrelation, learn why the uniform moving-average minimizes noise under a unity-DC constraint, and why its sinc spectrum can be problematic. Part 1 of a short series.
Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 1
Misconfigured Code Composer Studio settings cause most 'Can't initialize target CPU' errors on TI C6000 boards, not a faulty silicon. Mike Dunn walks through the practical first steps: confirm your CCS version, identify the exact emulator and board or device part number, and ensure you have the correct emulator driver. The post also shows how to duplicate TI's factory board configuration to avoid common setup mistakes.
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.
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.
Generating Partially Correlated Random Variables
Designing signals to match a target covariance is simpler than it sounds. This post shows how to build partially correlated complex signals by hand for the two-signal case, then generalizes to N signals using the Cholesky decomposition. Short MATLAB examples demonstrate the two-line implementation and the article highlights numerical caveats when a covariance is only positive semidefinite.
The First-Order IIR Filter -- More than Meets the Eye
While we might be inclined to disdain the simple first-order infinite impulse response (IIR) filter, it is not so simple that we can’t learn something from it. Studying it can teach DSP math skills, and it is a very useful filter in its own right. In this article, we’ll examine the time response of the filter, compare the first-order IIR filter to the FIR moving average filter, use it to smooth a noisy signal, compute the functional form of the impulse response, and find the frequency response.
Bank-switched Farrow resampler
Markus Nentwig proposes a bank-switched variant of the Farrow resampler that breaks each impulse-response segment into multiple sub-segments, enabling accurate interpolation with lower-order polynomials and fewer multiplications per output. This trades increased total coefficient storage for computational savings. The post explains the concept, connects it to polyphase FIR interpolation, and provides Matlab/Octave and C example code for practical evaluation.
