The cyclostationarity of frequency-shift-keyed signals depends strongly on the way the carrier phase evolves over time. Many distinct cycle-frequency patterns and spectral correlation shapes are possible.
Let’s get back to basics by looking at a large class of signals known as frequency-shift-keyed (FSK) signals. We will leave to the side, for the most part, the very large class of signals that goes by the name of continuous-phase modulation (CPM), which includes continuous-phase FSK (CPFSK), MSK, GMSK, and many more (The Literature [R188]-[R190]). Those are treated in My Papers , and in a future CSP Blog post.
Here we want to look at more conventional forms of FSK. These signal types don’t necessarily have a continuous phase function. They are generally easier to demodulate and are more robust to noise and interference than the more complicated CPM signal types, but generally have much lower spectral efficiency. They are like the rectangular-pulse PSK of the FSK/CPM world. But they are still used.
By the pricking of my thumbs, something wicked this way comes …
I attended a conference on dynamic spectrum access in 2017 and participated in a session on automatic modulation recognition. The session was connected to a live competition within the conference where participants would attempt to apply their modulation-recognition system to signals transmitted in the conference center by the conference organizers. Like a grand modulation-recognition challenge but confined to the temporal, spectral, and spatial constraints imposed by the short-duration conference.
What I didn’t know going in was the level of frustration on the part of the machine-learner organizers regarding the seemingly inability of signal-processing and machine-learning researchers to solve the radio-frequency scene analysis problem once and for all. The basic attitude was ‘if the image-processors can have the AlexNet image-recognition solution, and thereby abandon their decades-long attempt at developing serious mathematics-based image-processing theory and practice, why haven’t we solved the RFSA problem yet?’
In this post, we’ll switch gears a bit and look at the problem of waveform estimation. This comes up in two situations for me: single-sensor processing and array (multi-sensor) processing. At some point, I’ll write a post on array processing for waveform estimation (using, say, the SCORE algorithm The Literature [R102]), but here we restrict our attention to the case of waveform estimation using only a single sensor (a single antenna connected to a single receiver). We just have one observed sampled waveform to work with. There are also waveform estimation methods that are multi-sensor but not typically referred to as array processing, such as the blind source separation problem in acoustic scene analysis, which is often solved by principal component analysis (PCA), independent component analysis (ICA), and their variants.
The signal model consists of the noisy sum of two or more modulated waveforms that overlap in both time and frequency. If the signals do not overlap in time, then we can separate them by time gating, and if they do not overlap in frequency, we can separate them using linear time-invariant systems (filters).
The next step in dataset complexity at the CSP Blog: cochannel signals.
I’ve developed another dataset for use in assessing modulation-recognition algorithms (machine-learning-based or otherwise) that is more complex than the original sets I posted for the ML Challenge (CSPB.ML.2018 and CSPB.ML.2022). Half of the new dataset consists of one signal in noise and the other half consists of two signals in noise. In most cases the two signals overlap spectrally, which is a signal condition called cochannel interference.
How can we train a neural network to make use of both IQ data samples and CSP features in the context of weak-signal detection?
I’ve been working with some colleagues at Northeastern University (NEU) in Boston, MA, on ways to combine CSP with machine learning. The work I’m doing with Old Dominion University is focused on basic modulation recognition using neural networks and, in particular, the generalization (dataset-shift) problem that is pervasive in deep learning with convolution neural networks. In contrast, the NEU work is focused on specific signal detection and classification problems and looks at how to use multiple disparate data types as inputs to neural-networks; inputs such as complex-valued samples (IQ data) as well as carefully selected components of spectral correlation and spectral coherence surfaces.
My NEU colleagues and I will be publishing a rather lengthy conference paper on a new multi-input-data neural-network approach called ICARUS at InfoCom 2023 this May (My Papers ). You can get a copy of the pre-publication version here or on arxiv.org.
It’s too close to home, and it’s too near the bone …
Park the car at the side of the road You should know Time’s tide will smother you… And I will too
“That Joke Isn’t Funny Anymore” by The Smiths
I applaud the intent behind the paper in this post’s title, which is The Literature [R183], apparently accepted in 2022 for publication in IEEE Access, a peer-reviewed journal. That intent is to list all the found ways in which researchers preprocess radio-frequency data (complex sampled data) prior to applying some sort of modulation classification (recognition) algorithm or system.
The problem is that this attempt at gathering up all of the ‘representations’ gets a lot of the math wrong, and so has a high potential to confuse rather than illuminate.
Back in 2018 I posted a dataset consisting of 112,000 I/Q data files, 32,768 samples in length each, as a part of a challenge to machine learners who had been making strong claims of superiority over signal processing in the area of automatic modulation recognition. One part of the challenge was modulation recognition involving eight digital modulation types, and the other was estimating the carrier frequency offset. That dataset is described here, and I’d like to refer to it as CSPB.ML.2018.
Then in 2022 I posted a companion dataset to CSPB.ML.2018 called CSPB.ML.2022. This new dataset uses the same eight modulation types, similar ranges of SNR, pulse type, and symbol rate, but the random variable that governs the carrier frequency offset is different with respect to the random variable in CSPB.ML.2018. The purpose of the CSPB.ML.2022 dataset is to facilitate studies of the dataset-shift, or generalization, problem in machine learning.
Throughout the past couple of years I’ve been working with some graduate students and a professor at Old Dominion University on merging machine learning and signal processing for problems involving RF signal analysis, such as modulation recognition. We are starting to publish a sequence of papers that describe our efforts. I briefly describe the results of one such paper, My Papers , in this post.
Can we fix peer review in engineering by some form of payment to reviewers?
Let’s talk about another paper about cyclostationarity and correntropy. I’ve critically reviewed two previously, which you can find here and here. When you look at the correntropy as applied to a cyclostationary signal, you get something called cyclic correntropy, which is not particularly useful except if you don’t understand regular cyclostationarity and some aspects of garden-variety signal processing. Then it looks great.
But this isn’t a post that primarily takes the authors of a paper to task, although it does do that. I want to tell the tale to get us thinking about what ‘peer’ could mean, these days, in ‘peer-reviewed paper.’ How do we get the best peers to review our papers?
Let’s take a look at an even faster spectral correlation function estimator. How useful is it for CSP applications in communications?
Reader Gideon pointed out that Antoni had published a paper a year after the paper that I considered in my first Antoni post. This newer paper, The Literature [R172], promises a faster fast spectral correlation estimator, and it delivers on that according to the analysis in the paper. However, I think the faster fast spectral correlation estimator is just as limited as the slower fast spectral correlation estimator when considered in the context of communication-signal processing.
And, to be fair, Antoni doesn’t often consider the context of communication-signal processing. His favored application is fault detection in mechanical systems with rotating parts. But I still don’t think the way he compares his fast and faster estimators to conventional estimators is fair. The reason is that his estimators are both severely limited in the maximum cycle frequency that can be processed, relative to the maximum cycle frequency that is possible.
Another RF-signal dataset to help push along our R&D on modulation recognition.
Update February 2023: A third dataset has been posted to the CSP Blog: CSPB.ML.2023. It features cochannel signals.
Update January 2023: I’m going to put Challenger results in the Comments. I’ve received a Challenger’s decisions and scored them in January 2023. See below.
In this post I provide a second dataset for the Machine-Learning Challenge I issued in 2018 (CSPB.ML.2018). This dataset is similar to the original dataset, but possesses a key difference in that the probability distribution of the carrier-frequency offset parameter, viewed as a random variable, is not the same, but is still realistic.
Blog Note: By WordPress’ count, this is the 100th post on the CSP Blog. Together with a handful of pages (like My Papers and The Literature), these hundred posts have resulted in about 250,000 page views. That’s an average of 2,500 page views per post. However, the variance of the per-post pageviews is quite large. The most popular is The Spectral Correlation Function (> 16,000) while the post More on Pure and Impure Sinewaves, from the same era, has only 316 views. A big Thanks to all my readers!!
What are the ranges of spectral frequency and cycle frequency that we need to consider in a discrete-time/discrete-frequency setting for CSP?
Let’s talk about that diamond-shaped region in the plane we so often see associated with CSP. I’m talking about the principal domain for the discrete-time/discrete-frequency spectral correlation function. Where does it come from? Why do we care? When does it come up?
The Fast Spectral Correlation estimator is a quick way to find small cycle frequencies. However, its restrictions render it inferior to estimators like the SSCA and FAM.
In this post we take a look at an alternative CSP estimator created by J. Antoni et al (The Literature [R152]). The paper describing the estimator can be found here, and you can get some corresponding MATLAB code, posted by the authors, here if you have a Mathworks account.
The merging of conventional probability theory with signal theory leads to random processes, also known as stochastic processes. The ideas involved with random processes are central to cyclostationary signal processing.
In this Signal Processing ToolKit post, I provide an introduction to the concept and use of random processes (also called stochastic processes). This is my perspective on random processes, so although I’ll introduce and use the conventional concepts of stationarity and ergodicity, I’ll end up focusing on the differences between stationary and cyclostationary random processes. The goal is to illustrate those differences with informative graphics and videos; to build intuition in the reader about how the cyclostationarity property comes about, and about how the property relates to the more abstract mathematical object of a random process on one hand and to the concrete data-centric signal on the other.
So … this is the first SPTK post that is also a CSP post.
Why does zero-padding help in various estimators of the spectral correlation and spectral coherence functions?
Update to the exchange: May 7, 2021.May 14, 2021.
Reader Clint posed a great question about zero-padding in the frequency-smoothing method (FSM) of spectral correlation function estimation. The question prompted some pondering on my part, and I went ahead and did some experiments with the FSM to illustrate my response to Clint. The exchange with Clint (ongoing!) was deep and detailed enough that I thought it deserved to be seen by other CSP-Blog readers. One of the problems with developing material, or refining explanations, in the Comments sections of the CSP Blog is that these sections are not nearly as visible in the navigation tools featured on the Blog as are the Posts and Pages.
Let’s take a brief look at the cyclostationarity of a captured DMR signal. It’s more complicated than one might think.
In this post I look at the cyclostationarity of a digital mobile radio (DMR) signal empirically. That is, I have a captured DMR signal from sigidwiki.com, and I apply blind CSP to it to determine its cycle frequencies and spectral correlation function. The signal is arranged in frames or slots, with gaps between successive slots, so there is the chance that we’ll see cyclostationarity due to the on-burst (or on-frame) signaling and cyclostationarity due to the framing itself.
The Machine Learners think that their “feature engineering” (rooting around in voluminous data) is the same as “features” in mathematically derived signal-processing algorithms. I take a lighthearted look.
One of the things the machine learners never tire of saying is that their neural-network approach to classification is superior to previous methods because, in part, those older methods use hand-crafted features. They put it in different ways, but somewhere in the introductory section of a machine-learning modulation-recognition paper (ML/MR), you’ll likely see the claim. You can look through the ML/MR papers I’ve cited in The Literature ([R133]-[R146]) if you are curious, but I’ll extract a couple here just to illustrate the idea.
What happens when a cyclostationary time-series is treated as if it were stationary?
In this post let’s consider the difference between modeling a communication signal as stationary or as cyclostationary.
There are two contexts for this kind of issue. The first is when someone recognizes that a particular signal model is cyclostationary, and then takes some action to render it stationary (sometimes called ‘stationarizing the signal’). They then proceed with their analysis or algorithm development using the stationary signal model. The second context is when someone applies stationary-signal processing to a cyclostationary signal model, either without knowing that the signal is cyclostationary, or perhaps knowing but not caring.
At the center of this topic is the difference between the mathematical object known as a random process (or stochastic process) and the mathematical object that is a single infinite-time function (or signal or time-series).
A related paper is The Literature [R68], which discusses the pitfalls of applying tools meant for stationary signals to the samples of cyclostationary signals.
An analysis of DeepSig’s 2016.10A dataset, used in many published machine-learning papers, and detailed comments on quite a few of those papers.
Update March 2021
See my analyses of three other DeepSig datasets here, here, and here.
Update June 2020
I’ll be adding new papers to this post as I find them. At the end of the original post there is a sequence of date-labeled updates that briefly describe the relevant aspects of the newly found papers. Some machine-learning modulation-recognition papers deserve their own post, so check back at the CSP Blog from time-to-time for “Comments On …” posts.
What are the unique parts of the multidimensional cyclic moments and cyclic cumulants?
In this post, we continue our study of the symmetries of CSP parameters. The second-order parameters–spectral correlation and cyclic correlation–are covered in detail in the companion post, including the symmetries for ‘auto’ and ‘cross’ versions of those parameters.
Here we tackle the generalizations of cyclic correlation: cyclic temporal moments and cumulants. We’ll deal with the generalization of the spectral correlation function, the cyclic polyspectra, in a subsequent post. It is reasonable to me to focus first on the higher-order temporal parameters, because I consider the temporal parameters to be much more useful in practice than the spectral parameters.
This topic is somewhat harder and more abstract than the second-order topic, but perhaps there are bigger payoffs in algorithm development for exploiting symmetries in higher-order parameters than in second-order parameters because the parameters are multidimensional. So it could be worthwhile to sally forth.