Can a Machine Learn the Fourier Transform?

Or any transform for that matter. Or the power spectrum? Autocorrelation function? Cyclic moment? Cyclic cumulant?

I ask because the Machine Learners want to do away with what they call Expert Features in multiple areas involving classification, such as modulation recognition, image classification, facial recognition, etc. The idea is to train the machine (and by machine they seem to almost always mean an artificial neural network, or just neural network for short) by applying labeled data (supervised learning) where the data is the raw data involved in the classification application area. For us, here at the CSP Blog, that means complex-valued data samples obtained through standard RF signal reception techniques. In other words, the samples that we start with in all of our CSP algorithms, such as the frequency-smoothing method, the time-smoothing method, the strip spectral correlation analyzer, the cycle detectors, the time-delay estimators, automatic spectral segmentation, etc.

This is an interesting and potentially valuable line of inquiry, even if it does lead to the superfluousness of my work and the CSP Blog itself. Oh well, gotta face reality.

So can we start with complex samples (commonly called “I-Q samples”, which is short for “inphase and quadrature samples”) corresponding to labeled examples of the involved classes (BPSK, QPSK, AM, FM, etc.) and end up with a classifier with performance that exceeds that of the best Expert Feature classifier? From my point of view, that means that the machine has to learn cyclic cumulants or something even better. I have a hard time imagining something better (that is just a statement about my mental limitations, not about what might exist in the world), so I shift to asking Can a Machine Learn the Cyclic Cumulant?

More on Pure and Impure Sine Waves

Remember when we derived the cumulant as the solution to the pure $n$th-order sine-wave problem? It sounded good at the time, I hope. But here I describe a curious special case where the interpretation of the cumulant as the pure component of a nonlinearly generated sine wave seems to break down.

Cumulant (4, 2) is a Good Discriminator?

Let’s talk about another published paper on signal detection involving cyclostationarity and/or cumulants. This one is called “Energy-Efficient Processor for Blind Signal Classification in Cognitive Radio Networks,” (The Literature [R69]), and is authored by UCLA researchers E. Rebeiz and four colleagues.

My focus on this paper it its idea that broad signal-type classes, such as direct-sequence spread-spectrum (DSSS), QAM, and OFDM can be reliably distinguished by the use of a single number: the fourth-order cumulant with two conjugated terms. This kind of cumulant is referred to as the $(4, 2)$ cumulant here at the CSP Blog, and in the paper, because the order is $n=4$ and the number of conjugated terms is $m=2$.