CSP Estimators: The FFT Accumulation Method

Let’s look at another spectral correlation function estimator: the FFT Accumulation Method (FAM). This estimator is in the time-smoothing category, is exhaustive in that it is designed to compute estimates of the spectral correlation function over its entire principal domain, and is efficient, so that it is a competitor to the Strip Spectral Correlation Analyzer (SSCA) method. I implemented my version of the FAM by using the paper by Roberts et al (The Literature [R4]). If you follow the equations closely, you can successfully implement the estimator from that paper. The tricky part, as with the SSCA, is correctly associating the outputs of the coded equations to their proper $\displaystyle (f, \alpha)$ values.

‘Can a Machine Learn the Fourier Transform?’ Redux, Plus Relevant Comments on a Machine-Learning Paper by M. Kulin et al.

I first considered whether a machine (neural network) could learn the (64-point, complex-valued)  Fourier transform in this post. I used MATLAB’s Neural Network Toolbox and I failed to get good learning results because I did not properly set the machine’s hyperparameters. A kind reader named Vito Dantona provided a comment to that original post that contained good hyperparameter selections, and I’m going to report the new results here in this post.

Since the Fourier transform is linear, the machine should be set up to do linear processing. It can’t just figure that out for itself. Once I used Vito’s suggested hyperparameters to force the machine to be linear, the results became much better:

Can a Machine Learn the Fourier Transform?

(Update: See Part 2 of this post at this link.)

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?

Creating a Simple CS Signal: Rectangular-Pulse BPSK

To test the correctness of various CSP estimators, we need a sampled signal with known cyclostationary parameters. Additionally, the signal should be easy to create and understand. A good candidate for this kind of signal is the binary phase-shift keyed (BPSK) signal with rectangular pulse function.

PSK signals with rectangular pulse functions have infinite bandwidth because the signal bandwidth is determined by the Fourier transform of the pulse, which is a sinc() function for the rectangular pulse. So the rectangular pulse is not terribly practical–infinite bandwidth is bad for other users of the spectrum. However, it is easy to generate, and its statistical properties are known.

So let’s jump in. The baseband BPSK signal is simply a sequence of binary ($\pm$ 1) symbols convolved with the rectangular pulse. The MATLAB script make_rect_bpsk.m does this and produces the following plot:

The signal alternates between amplitudes of +1 and -1 randomly. After frequency shifting and adding white Gaussian noise, we obtain the power spectrum estimate:

The power spectrum plot shows why the rectangular-pulse BPSK signal is not popular in practice. The range of frequencies for which the signal possesses non-zero average power is infinite, so it will interfere with signals “nearby” in frequency. However, it is a good signal for us to use as a test input in all of our CSP algorithms and estimators.

The MATLAB script that creates the BPSK signal and the plots above is here. It is an m-file but I’ve stored it in a .doc file due to WordPress limitations I can’t yet get around.