The Exponential Nature of the Complex Unit Circle
Euler's equation links exponential scaling and rotation by translating a distance along the unit-circle circumference into a complex value. Cedron Dawg develops an intuitive geometric view, using integer and fractional powers of i to show how points, roots of unity, and multiplication behave as additive moves along that circumference. The article also connects this picture to radians and the conventional Taylor-series proof for broader perspective.
Sum of Two Equal-Frequency Sinusoids
Rick Lyons exposes a frequent trig mistake and delivers complete closed-form expressions for collapsing two equal-frequency sinusoids into a single sinusoid. Using complex-exponential phasor addition and equating real and imaginary parts, he compiles easy-to-use tables for cosine+cosine, sine+sine, and cosine+sine cases and shows how to derive each form. Engineers get corrected identities and compact derivations useful for analysis and communications.
Constrained Integer Behavior
Overflow and underflow are not always bugs, they can be useful in DSP when fixed-width integers wrap during processing. Christopher Felton demonstrates with moving-average (recursive-windowed-averager) and CIC filter examples how 2's complement wraparound in MyHDL's modbv cancels between an integrator and a comb via pole-zero cancellation. He also covers fixed-point resizing choices, saturation versus wrap, and how rounding error can accumulate.
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.
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.
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.
Is It True That j is Equal to the Square Root of -1 ?
A viral YouTube video claimed that saying j equals the square root of negative one is wrong. Rick Lyons shows the apparent paradox comes from misusing square-root identities with negative arguments, not from the usual definition of j. He argues it is safer to define j by j^2 = -1 and illustrates how careless root operations produce contradictions in two appendices.
Python scipy.signal IIR Filtering: An Example
Christopher Felton walks through using scipy.signal IIR filters to demodulate PWM signals, using spectrum and spectrogram analysis to show what works and what does not. He demonstrates using filtfilt to avoid phase delay, compares a single narrow IIR to a very high order FIR, and shows how staged IIR filtering and multirate ideas give much better attenuation. Includes an FPGA-ready MyHDL PWM model.
A Quadrature Signals Tutorial: Complex, But Not Complicated
Quadrature signals are essential in modern communications, yet complex numbers and the j operator intimidate many engineers. In this tutorial Rick Lyons uses phasor geometry, three-dimensional time and frequency plots, and practical I/Q sampling examples to demystify complex exponentials, negative frequency, and how to generate baseband complex signals. Read to get physical intuition and hands-on rules you can apply to modulation, demodulation, and DSP implementations.
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.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins
Cedron presents exact, closed-form formulas to extract the phase and amplitude of a pure complex tone from multiple DFT bin values, using a compact vector formulation. The derivation introduces a delta variable to simplify the sinusoidal bin expression, stacks neighboring bins into a basis vector, and solves for the complex amplitude q by projection. The phase and magnitude follow directly from q, and extra bins reduce leakage when the tone falls between bins.
Feedback Controllers - Making Hardware with Firmware. Part 7. Turbo-charged DSP Oscillators
You can extract high-quality, high-sample-rate sine waves from FPGAs even when floating-point units are constrained by latency. This article compares Intel's NCO IP (multiplier option) with floating-point recursive biquads on Cyclone V and Cyclone 10 GX, and explains a boosted-sample-rate technique that pushes performance toward a 48Msps DAC target. Practical measurement results, spectral data, and resource/cost trade-offs are highlighted.
A Two Bin Solution
Cedron Dawg shows how a real sinusoid's frequency, amplitude and phase can be recovered from only two adjacent DFT bins. The article derives exact two-bin formulas, gives a clear Gambas reference implementation, and demonstrates that accurate parameters can be obtained with very few samples when the tone lies between the bins. It also explains when the method breaks down and how the real-valued unfurling improves robustness.
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.
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.
Feedback Controllers - Making Hardware with Firmware. Part I. Introduction
This first post kicks off a series on using DSP and feedback control with mixed-signal electronics and FPGAs to emulate two-terminal circuits and create low latency controllers. It frames circuit emulation as a feedback problem, highlights latency as the key practical constraint, and outlines the planned evaluation hardware, target devices, and software tools that will be used in later MATLAB/Simulink and FPGA work.
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...
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.
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.
Deesspee #5
Peter Kootsookos's Deesspee #5 is a very short micro-post simply titled "Computers". It acts as a minimalist flag in the Deesspee series pointing readers toward the computing topic on DSPRelated; click through to view the original entry and any context or discussion. This compact post is useful if you track the author's brief topic markers or short-format updates.
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.
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 Solution
Cedron Dawg shows how a real sinusoid's frequency, amplitude and phase can be recovered from only two adjacent DFT bins. The article derives exact two-bin formulas, gives a clear Gambas reference implementation, and demonstrates that accurate parameters can be obtained with very few samples when the tone lies between the bins. It also explains when the method breaks down and how the real-valued unfurling improves robustness.
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.
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.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT
Cedron Dawg derives compact, exact formulas to recover the phase and amplitude of a single complex tone from a DFT bin when the tone frequency is known. The paper turns the complex bin value into closed-form expressions using a sine-fraction amplitude correction and a simple phase shift, and includes working code plus a numeric example for direct implementation.
Polar Coding Notes: Channel Combining and Channel Splitting
Lyons Zhang walks through the core algebra of polar coding, showing how channel combining builds the vector channel W_N from N copies of a binary-input DMC using the polar transform G_N = B_N F^{⊗n}. The notes then define channel splitting, derive the coordinate-channel transition probabilities from the chain rule, and present the recursive formulas that let you compute W_{2N}^{(2i-1)} and W_{2N}^{(2i)} from W_N^{(i)}.
There's No End to It -- Matlab Code Plots Frequency Response above the Unit Circle
If you want a fresh way to inspect a digital filter, this post introduces plotfil3d, a compact MATLAB function that wraps the magnitude response around the unit circle in the Z-plane so you can view it in 3D. It uses freqz to compute H(z) in dB for N points and accepts an optional azimuth to change the viewing angle; the code is provided in the appendix.
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.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins
Cedron presents exact, closed-form formulas to extract the phase and amplitude of a pure complex tone from multiple DFT bin values, using a compact vector formulation. The derivation introduces a delta variable to simplify the sinusoidal bin expression, stacks neighboring bins into a basis vector, and solves for the complex amplitude q by projection. The phase and magnitude follow directly from q, and extra bins reduce leakage when the tone falls between bins.
