What modest academic success I’ve had in the area of cyclostationary signal theory and cyclostationary signal processing is largely due to the patient mentorship of my doctoral adviser, William (Bill) Gardner, and the fact that I was able to build on an excellent foundation put in place by Gardner, his advisor Lewis Franks, and key Gardner students such as William (Bill) Brown.

# Higher-Order Cyclostationarity

# Simple Synchronization Using CSP

In this post I discuss the use of cyclostationary signal processing applied to communication-signal synchronization problems. First, just what are *synchronization problems*? Synchronize and synchronization have multiple meanings, but the meaning of synchronize that is relevant here is something like:

**syn·chro·nize**: *To cause to occur or operate with exact coincidence in time or rate*

If we have an analog amplitude-modulated (AM) signal (such as voice AM used in the AM broadcast bands) at a receiver we want to remove the effects of the carrier sine wave, resulting in an output that is only the original voice or music message. If we have a digital signal such as binary phase-shift keying (BPSK), we want to remove the effects of the carrier but also sample the message signal at the correct instants to optimally recover the transmitted bit sequence.

# Data Set for the Machine-Learning Challenge

I’ve posted PSK/QAM signals to the CSP Blog. These are the signals I refer to in the post I wrote challenging the machine-learners. In this brief post, I provide links to the data and describe how to interpret the text file containing the signal-type labels and signal parameters.

# 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.

# A Challenge for the Machine Learners

## UPDATE

I’ve decided to post the data set I discuss here to the CSP Blog for all interested parties to use. See the new post on the Data Set. If you do use it, please let me and the CSP Blog readers know how you fared with your experiments in the Comments section of either post. Thanks!

# ‘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:

# CSP Estimators: Cyclic Temporal Moments and Cumulants

In this post we discuss ways of estimating -th order cyclic temporal moment and cumulant functions. Recall that for , 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 is greater than . Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order greater than is . Our estimation task is to estimate -th order temporal moment and cumulant functions for using a sampled-data record of length .

# More on Pure and Impure Sine Waves

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

# 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.

# Modulation Recognition Using Cyclic Cumulants, Part I: Problem Description and Variants

In this post, we start a discussion of what I consider the ultimate application of the theory of cyclostationary signals: Automatic Modulation Recognition. My relevant papers are My Papers [16,17,25,26,28,30,32,33,38,43,44].