Category: Mathematics

Modulation Recognition Using Cyclic Cumulants, Part II: Algorithms

An example of how a decision-theoretic approach to a mathematical problem leads to a feature-based algorithmic solution that, viewed in isolation, appears to have sprung from the brow of Zeus fully formed.

At long last, I am picking up the modulation-recognition (MR) topic again. My plan is to describe the core MR algorithm I’ve developed using cyclic cumulants in this post, and then in a third and final post present some performance results using simulated and captured signals.

Continue reading “Modulation Recognition Using Cyclic Cumulants, Part II: Algorithms”

CSP Reduction to Sine-Wave Generation

“Five different voices behind him bellowed, “REDUCTO!” Five curses flew in five different directions and the shelves opposite them exploded as they hit; the towering structure swayed as a hundred glass spheres burst apart, pearly-white figures unfurled into the air and floated there, their voices echoing from who knew what long-dead past amid the torrent of crashing glass and splintered wood now raining down upon the floor…”

J. K. Rowling, Harry Potter and the Order of the Phoenix

We know that if we subject a cyclostationary signal to a squaring or delay-and-multiply operation we will obtain finite-strength additive sine-wave components at the output of the operation, where at least one of the sine waves has a non-zero frequency.

But I want to make a conjecture: All of CSP can be reduced to interpretations involving sine-wave generation by nonlinear operations. Let’s see if we can show this conjecture is true. After I make my attempt, I’ll also show what ChatGPT comes up with. Any guesses about how well it does?

Continue reading “CSP Reduction to Sine-Wave Generation”

Infinity, Periodicity, and Frequency: Comments on a Recent Signal-Processing Perspectives Paper ([R195])

If a tool isn’t appropriate for your problem, don’t blame the tool. Find another one.

Let’s take a look at a recent perspectives-style paper published in the IEEE Signal Processing Magazine called “On the Concept of Frequency in Signal Processing: A Discussion [Perspectives],” (The Literature [R195]). While I criticize the paper directly, I’m hoping to use this post to provide my own perspective, and perhaps a bit of a tutorial, on the interrelated concepts of frequency, infinity, sine waves, and signal representations.

I appreciate tutorial papers in the signal-processing literature (see, for example, my positive post on Candan’s article about the Dirac delta [impulse] function), because my jaundiced view of the field is such that I think the basics, both of mathematics and communication-related signal-processing, are neglected in favor of fawning over the research flavor of the month. Over time, everybody–students, researchers, professors–is diminished because of this lack of attention to foundations.

Continue reading “Infinity, Periodicity, and Frequency: Comments on a Recent Signal-Processing Perspectives Paper ([R195])”