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.
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.
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.
A Fixed-Point Introduction by Example
Christopher Felton walks through binary fixed-point representation with clear examples and a simple W=(wl,iwl,fwl) notation. He argues for designing to range and resolution rather than bit counts, then shows how multiplication and addition affect bit growth and alignment. These concrete examples make it easy to see why rounding, resizing, and radix-point bookkeeping are essential in DSP implementations.
Candan's Tweaks of Jacobsen's Frequency Approximation
Cedron Dawg shows how small tweaks to Jacobsen's three-bin frequency estimator turn a popular approximation into an exact formula, and how a modest cubic correction yields a near-exact, low-cost alternative. The article derives an arctan/tan exact recovery, relates it to Candan's 2011/2013 tweaks, and supplies reference C code and numerical tables so engineers can see when each formula is sufficient.
Approximating the area of a chirp by fitting a polynomial
Once in a while we need to estimate the area of a dataset in which we are interested. This area could give us, for example, force (mass vs acceleration) or electric power (electric current vs charge).
New Video: Parametric Oscillations
Tim Wescott just posted a short new video titled "Parametric Oscillations." It’s a little off-topic for the channel, but he used the project as an excuse to break a months-long posting drought. If you follow his work, this quick update shows how small builds can rekindle momentum and prompt informal explorations of oscillation behavior.
GPS - some terminology!
GPS looks simple on the surface, but Vivek's post breaks out the core terminology behind how a receiver actually locks on and figures out where it is. Using a bar-room analogy, he maps acquisition, tracking, ephemeris, and almanac to the steps a GPS receiver follows before solving for position from satellite signals.
Pentagon Construction Using Complex Numbers
A method for constructing a pentagon using a straight edge and a compass is deduced from the complex values of the Fifth Roots of Unity. Analytic values for the points are also derived.
New Video: Parametric Oscillations
Tim Wescott just posted a short new video titled "Parametric Oscillations." It’s a little off-topic for the channel, but he used the project as an excuse to break a months-long posting drought. If you follow his work, this quick update shows how small builds can rekindle momentum and prompt informal explorations of oscillation behavior.
GPS - some terminology!
GPS looks simple on the surface, but Vivek's post breaks out the core terminology behind how a receiver actually locks on and figures out where it is. Using a bar-room analogy, he maps acquisition, tracking, ephemeris, and almanac to the steps a GPS receiver follows before solving for position from satellite signals.
A Lesson in Statistics Using Random Sequences
Statistics may come naturally to some people, but it is a difficult subject for many of us. Using simulations helps to remove some of the mystery of statistics. Luckily, it is trivial to produce a pseudo-random Gaussian or uniform sequence: a single command in Matlab (or Python) does it. In this article, I try to explain a few concepts using Gaussian and uniform random sequences generated in Matlab. First, we’ll generate a Gaussian sequence, calculate its variance, and plot a histogram and probability density function (PDF). Next, we’ll simulate the roll of a single die to illustrate a uniform distribution. Then we’ll simulate rolling two dice and finally, as an illustration of the Central Limit Theorem, we’ll sum the total when rolling several dice to obtain an approximately Gaussian distribution.
