My friend and colleague Antonio Napolitano has just published a new book on cyclostationary signals and cyclostationary signal processing:
Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations, Academic Press/Elsevier, 2020, ISBN: 978-0-08-102708-0. The book is a comprehensive guide to the structure of cyclostationary random processes and signals, and it also provides pointers to the literature on many different applications. The book is mathematical in nature; use it to deepen your understanding of the underlying mathematics that make CSP possible.
You can check out the book on amazon.com using the following link:
I’ve seen several published and pre-published (arXiv.org) technical papers over the past couple of years on the topic of cyclic correntropy (The Literature [R123-R127]). I first criticized such a paper ([R123]) here, but the substance of that review was about my problems with the presented mathematics, not impulsive noise and its effects on CSP. Since the papers keep coming, apparently, I’m going to put down some thoughts on impulsive noise and some evidence regarding simple means of mitigation in the context of CSP. Preview: I don’t think we need to go to the trouble of investigating cyclic correntropy as a means of salvaging CSP from the clutches of impulsive noise.
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.
I continue with my foray into machine learning (ML) by considering whether we can use widely available ML tools to create a machine that can output accurate power spectrum estimates. Previously we considered the perhaps simpler problem of learning the Fourier transform. See here and here.
Along the way I’ll expose my ignorance of the intricacies of machine learning and my apparent inability to find the correct hyperparameter settings for any problem I look at. But, that’s where you come in, dear reader. Let me know what to do!
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 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 image-recognition techniques to the problem of automatic type-recognition for human-made electromagnetic waves.
Update November 1, 2018: A site called feedspot (blog.feedspot.com) contacted me to tell me I made their “Top 10 Digital Signal Processing Blogs, Websites & Newsletters in 2018” list. Weirdly, there are only eight blogs in the list. What’s most important for this post is the other signal processing blogs on the list. So check it out if you are looking for other sources of online signal processing information. Enjoy! blog.feedspot.com/digital_signal_processing_blogs
But I’d like to be able to refer readers to good websites that discuss related aspects of signal processing and communication signals, such as filtering, spectrum estimation, mathematical models, Fourier analysis, etc. I’ve had little success with the Google searches I’ve tried.
I recently came across the conference paper in the post title (The Literature [R101]). Let’s take a look.
The paper is concerned with “detect[ing] the presence of ACS signals with unknown cycle period.” In other words, blind cyclostationary-signal detection and cycle-frequency estimation. Of particular importance to the authors is the case in which the “period of cyclostationarity” is not equal to an integer number of samples. They seem to think this is a new and difficult problem. By my lights, it isn’t. But maybe I’m missing something. Let me know in the Comments.
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:
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 cumulant here at the CSP Blog, and in the paper, because the order is and the number of conjugated terms is .
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].
I recently came across the 2014 paper in the title of this post. I mentioned it briefly in the post on the periodogram. But I’m going to talk about it a bit more here because this is the kind of thing that makes things a bit harder for people trying to learn about cyclostationarity, which eventually leads to the need for something like the CSP Blog.
The idea behind the paper is that it would be nice to avoid the need for prior knowledge of cycle frequencies when using cycle detectors or the like. If you could just compute the entire spectral correlation function, then collapse it by integrating (summing) over frequency , then you’d have a one-dimensional function of cycle frequency and you could then process that function inexpensively to perform detection and classification tasks.
I recently came across a published paper with the title Cyclostationary Correntropy: Definition and Application, by Aluisio Fontes et al. It is published in a journal called Expert Systems with Applications (Elsevier). Actually, it wasn’t the first time I’d seen this work by these authors. I had reviewed a similar paper in 2015 for a different journal.
I was surprised to see the paper published because I had a lot of criticisms of the original paper, and the other reviewers agreed since the paper was rejected. So I did my job, as did the other reviewers, and we tried to keep a flawed paper from entering the literature, where it would stay forever causing problems for readers.
The editor(s) of the journal Expert Systems with Applications did not ask me to review the paper, so I couldn’t give them the benefit of the work I already put into the manuscript, and apparently the editor(s) did not themselves see sufficient flaws in the paper to merit rejection.
It stings, of course, when you submit a paper that you think is good, and it is rejected. But it also stings when a paper you’ve carefully reviewed, and rejected, is published anyway.
Fortunately I have the CSP Blog, so I’m going on another rant. After all, I already did this the conventional rant-free way.
I came across a paper by Cohen and Eldar, researchers at the Technion in Israel. You can get the paper on the Arxiv site here. The title is “Sub-Nyquist Cyclostationary Detection for Cognitive Radio,” and the setting is spectrum sensing for cognitive radio. I have a question about the paper that I’ll ask below.
So why do I obsess over cyclostationary signals and cyclostationary signal processing? What’s the big deal, in the end? In this post I discuss my view of the ultimate use of cyclostationary signal processing (CSP): Radio-Frequency Scene Analysis (RFSA). Eventually, I hope to create a kind of Star Trek Tricorder for RFSA.
In this post I provide some tools for the do-it-yourself CSP practitioner. One of the goals of this blog is to help new CSP researchers and students to write their own estimators and algorithms. This post contains some spectral correlation function and cyclic autocorrelation function estimates and numerically evaluated formulas that can be compared to those produced by anybody’s code.
The signal of interest is, of course, our rectangular-pulse BPSK signal with symbol rate (normalized frequency units) and carrier offset . You can download a MATLAB script for creating such a signal here.
The formula for the SCF for a textbook BPSK signal is published in several places (The Literature [R47], My Papers ) and depends mainly on the Fourier transform of the pulse function used by the textbook signal.
We’ll compare the numerically evaluated spectral correlation formula with estimates produced by my version of the frequency-smoothing method (FSM). The FSM estimates and the theoretical functions are contained in a MATLAB mat file here. (I had to change the extension of the mat file from .mat to .doc to allow posting it to WordPress.) In all the results shown here and that you can download, the processed data-block length is samples and the FSM smoothing width is Hz. A rectangular smoothing window is used. For all cycle frequencies except zero (non-conjugate), a zero-padding factor of two is used in the FSM.
For the cyclic autocorrelation, we provide estimates using two methods: inverse Fourier transformation of the spectral correlation estimate and direct averaging of the second-order lag product in the time domain.
What good is having a blog if you can’t offer a rant every once in a while? In this post I talk about what I call textbook signals, which are mathematical models of communication signals that are used by many researchers in statistical signal processing for communications.
We’ve already encountered, and used frequently, the most common textbook signal of all: rectangular-pulse BPSK with independent and identically distributed (IID) bits. We’ve been using this signal to illustrate the cyclostationary signal processing concepts and estimators as they have been introduced. It’s a good choice from the point of view of consistency of all the posts and it is easy to generate and to understand. However, it is not a good choice from the perspective of realism. It is rare to encounter a textbook BPSK signal in the practice of signal processing for communications.
I use the term textbook because the textbook signals can be found in standard textbooks, such as Proakis (The Literature [R44]). Textbook signals stand in opposition to signals used in the world, such as OFDM in LTE, slotted GMSK in GSM, 8PAM VSB with synchronization bits in ATSC-DTV, etc.
Typical communication signals combine a textbook signal with an access mechanism to yield the final physical-layer signal–the signal that is actually transmitted (My Papers , ). What is important for us, here on the CSP blog, is that this combination usually results in a signal with radically different cyclostationarity than the textbook component. So it is not enough to understand textbook signals’ cyclostationarity. We must also understand the cyclostationarity of the real-world signal, which may be sufficiently complex to render mathematical modeling and analysis impossible (at least for me).