How we Learned CSP

This post is just a blog post. Just some guy on the internet thinking out loud. If you have relevant thoughts or arguments you’d like to advance, please leave them in the Comments section at the end of the post.

How did we, as people not machines, learn to do cyclostationary signal processing? We’ve successfully applied it to many real-world problems, such as weak-signal detection, interference-tolerant detection, interference-tolerant time-delay estimation, modulation recognition, joint multiple-cochannel-signal modulation recognition (My Papers [25,26,28,38,43]), synchronization (The Literature [R7]), beamforming (The Literature [R102,R103]), direction-finding (The Literature [R104-R106]), detection of imminent mechanical failures (The Literature [R017-R109]), linear time-invariant system identification (The Literature [R110-R115]), and linear periodically time-variant filtering for cochannel signal separation (FRESH filtering) (My Papers [45], The Literature [R6]).

How did this come about? Is it even interesting to ask the question? Well, it is to me. I ask it because of the current hot topic in signal processing: machine learning. And in particular, machine learning applied to modulation recognition (see here and here). The machine learners want to capitalize on the success of machine learning applied to image recognition by directly applying the same sorts of techniques used there to automatic recognition (classification) of the type of a captured man-made electromagnetic wave.

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A Challenge for the Machine Learners

A while back I was working with some machine-learning researchers on the problem of carrier-frequency-offset (CFO) estimation. The CFO is the residual carrier frequency exhibited by an imperfectly downconverted radio-frequency signal. I’ll describe it in more detail below. The idea behind the collaboration was to find the SNR, SINR, block-length, etc., ranges for which machine-learning algorithms outperform more traditional approaches, such as those involving exploitation of cyclostationarity. If we’re going to get rid of the feature-based approaches used by experts, then we’d better make sure that the machines can do at least as well as those approaches for the problems typically considered by the experts.

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‘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:

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CSP Estimators: Cyclic Temporal Moments and Cumulants

In this post we discuss ways of estimating n-th order cyclic temporal moment and cumulant functions. Recall that for n=2, cyclic moments and cyclic cumulants are usually identical. They differ when the signal contains one or more finite-strength additive sine-wave components. In the common case when such components are absent (as in our recurring numerical example involving rectangular-pulse BPSK), they are equal and they are also equal to the conventional cyclic autocorrelation function provided the delay vector is chosen appropriately.

The more interesting case is when the order n is greater than 2. Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order n greater than 2 is 4. Our estimation task is to estimate n-th order temporal moment and cumulant functions for n \ge 4 using a sampled-data record of length T.

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More on Pure and Impure Sine Waves

Remember when we derived the cumulant as the solution to the pure nth-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.

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Machine Learning and Modulation Recognition: Comments on “Convolutional Radio Modulation Recognition Networks” by T. O’Shea, J. Corgan, and T. Clancy

In this post I provide some comments on another paper I’ve seen on arxiv.org (I have also received copies of it through email) that relates to modulation classification and cyclostationary signal processing. The paper is by O’Shea et al and is called “Convolutional Radio Modulation Recognition Networks.” You can find it at this link.

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