Am I out of a job?
Update January 31, 2023: I’ve added numbers in square brackets next to the worst of the wrong things. I’ll document the errors at the bottom of the post.
Of course I have to see what ChatGPT has to say about CSP. Including definitions, which I don’t expect it to get too wrong, and code for estimators, which I expect it to get very wrong.
Let’s take a look.
Continue reading “ChatGPT and CSP”It’s too close to home, and it’s too near the bone …
Park the car at the side of the road
“That Joke Isn’t Funny Anymore” by The Smiths
You should know
Time’s tide will smother you…
And I will too
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.
There’s only one thing to do: correct the record.
Continue reading “Correcting the Record: Comments On “Wireless Signal Representation Techniques for Automatic Modulation Classification,” by X. Liu et al”Neural networks with I/Q data as input do not generalize in the modulation-recognition problem setting.
Update May 20, 2022: Here is the arxiv.org link.
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 [51], in this post.
Continue reading “Some Concrete Results on Generalization in Modulation Recognition using Machine Learning”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!!
Continue reading “Shifted Dataset for the Machine-Learning Challenge: How Well Does a Modulation-Recognition DNN Generalize? [Dataset CSPB.ML.2022]”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.
Continue reading “Are Probability Density Functions “Engineered” or “Hand-Crafted” Features?”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.
Continue reading “Stationary Signal Models Versus Cyclostationary Signal Models”
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.
Continue reading “Symmetries of Higher-Order Temporal Probabilistic Parameters in CSP”
Do we need to consider all cycle frequencies, both positive and negative? Do we need to consider all delays and frequencies in our second-order CSP parameters?
As you progress through the various stages of learning CSP (intimidation, frustration, elucidation, puzzlement, and finally smooth operation), the symmetries of the various functions come up over and over again. Exploiting symmetries can result in lower computational costs, quicker debugging, and easier mathematical development.
What exactly do we mean by ‘symmetries of parameters?’ I’m talking primarily about the evenness or oddness of the time-domain functions in the delay 
You can use this post as a resource for mathematical development because I present the symmetry equations. But also each symmetry result is illustrated using estimated parameters via the frequency smoothing method (FSM) of spectral correlation function estimation. The time-domain parameters are obtained from the inverse transforms of the FSM parameters. So you can also use this post as an extension of the second-order verification guide to ensure that your estimator works for a wide variety of input parameters.
Continue reading “Symmetries of Second-Order Probabilistic Parameters in CSP”
A PSK/QAM/SQPSK data set with randomized symbol rate, inband SNR, carrier-frequency offset, and pulse roll-off.
Update February 2023: I’ve posted a third challenge dataset here. It is CSPB.ML.2023 and features cochannel signals.
Update April 2022. I’ve also posted a second dataset here. This new dataset is similar to the original ML Challenge dataset except the random variable representing the carrier frequency offset has a slightly different distribution.
If you refer to either of the posted datasets in a published paper, please use the following designators, which I am also using in papers I’m attempting to publish:
Original ML Challenge Dataset: CSPB.ML.2018.
Shifted ML Challenge Dataset: CSPB.ML.2022.
Update September 2020. I made a mistake when I created the signal-parameter “truth” files signal_record.txt and signal_record_first_20000.txt. Like the DeepSig RML datasets that I analyzed on the CSP Blog here and here, the SNR parameter in the truth files did not match the actual SNR of the signals in the data files. I’ve updated the truth files and the links below. You can still use the original files for all other signal parameters, but the SNR parameter was in error.
Update July 2020. I originally posted 
We learned it using abstractions involving various infinite quantities. Can a machine learn it without that advantage?
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 and here and here). The machine learners want to capitalize on the success of machine learning as 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.
The machine-learning modulation-recognition community consistently claims vastly superior performance to anything that has come before. Let’s test that.
Update February 2023: A third dataset has been posted here. This new dataset, CSPB.ML.2023, features cochannel signals.
Update April 2022: I’ve also posted a second dataset here. This new dataset is similar to the original ML Challenge dataset except the random variable representing the carrier frequency offset has a slightly different distribution.
If you refer to any of the posted datasets in a published paper, please use the following designators, which I am also using in papers I’m attempting to publish:
Original ML Challenge Dataset: CSPB.ML.2018.
Shifted ML Challenge Dataset: CSPB.ML.2022.
Cochannel ML Dataset: CSPB.ML.2023.
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!
How do we efficiently estimate higher-order cyclic cumulants? The basic answer is first estimate cyclic moments, then combine using the moments-to-cumulants formula.
In this post we discuss ways of estimating 
![\boldsymbol{\tau} = [\tau_1\ \ \tau_2]](https://s0.wp.com/latex.php?latex=%5Cboldsymbol%7B%5Ctau%7D+%3D+%5B%5Ctau_1%5C+%5C+%5Ctau_2%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![[\tau/2\ \ -\tau/2]](https://s0.wp.com/latex.php?latex=%5B%5Ctau%2F2%5C+%5C+-%5Ctau%2F2%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
The more interesting case is when the order 
Continue reading “CSP Estimators: Cyclic Temporal Moments and Cumulants”
Gaussian and binary signals are in some sense at opposite ends of the pure-impure sine-wave spectrum.
Remember when we derived the cumulant as the solution to the pure
Modulation recognition is the process of assigning one or more modulation-class labels to a provided time-series data sequence.
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]. See also my machine-learning modulation-recognition critiques by clicking on Machine Learning in the CSP Blog Categories on the right side of any post or page.
Higher-order statistics in the frequency domain for cyclostationary signals. As complicated as it gets at the CSP Blog.
In this post we take a first look at the spectral parameters of higher-order cyclostationarity (HOCS). In previous posts, I have introduced the topic of HOCS and have looked at the temporal parameters, such as cyclic cumulants and cyclic moments. Those temporal parameters have proven useful in modulation classification and parameter estimation settings, and will likely be an important part of my ultimate radio-frequency scene analyzer.
The spectral parameters of HOCS have not proven to be as useful as the temporal parameters unless you include the trivial case where the moment/cumulant order is equal to two. In that case, the spectral parameters reduce to the spectral correlation function, which is extremely useful in CSP (see the TDOA and signal-detection posts for examples).
PSK and QAM signals form the building blocks for a large number of practical real-world signals. Understanding their probability structure is crucial to understanding those more complicated signals.
Let’s look into the statistical properties of a class of textbook signals that encompasses digital quadrature amplitude modulation (QAM), phase-shift keying (PSK), and pulse-amplitude modulation (PAM). I’ll call the class simply digital QAM (DQAM), and all of its members have an analytical-signal mathematical representation of the form
In this model, 
Model (1) is a textbook signal when the sequence of symbols is independent and identically distributed (IID). This condition rules out real-world communication aids such as periodically transmitted bursts of known symbols, adaptive modulation (where the constellation may change in response to the vagaries of the propagation channel), some forms of coding, etc. Also, when the pulse function 
We will look at the moments and cumulants of this general model in this post. Although the model is textbook, we could use it as a building block to form more realistic, less textbooky, signal models. Then we could find the cyclostationarity of those models by applying signal-processing transformation rules that define how the cumulants of the output of a signal processor relate to those for the input.
How does the cyclostationarity of a signal change when it is subjected to common signal-processing operations like addition, multiplication, and convolution?
It is often useful to know how a signal processing operation affects the probabilistic parameters of a random signal. For example, if I know the power spectral density (PSD) of some signal 
In (1), the function 
Because the mathematical models of real-world communication signals can be constructed by subjecting idealized textbook signals to various signal-processing operations, such as filtering, it is of interest to us here at the CSP Blog to know how the spectral correlation function of the output of a signal processor is related to the spectral correlation function for the input. Similarly, we’d like to know such input-output relationships for the cyclic cumulants and the cyclic polyspectra.
Another benefit of knowing these CSP input-output relationships is that they tend to build insight into the meaning of the probabilistic parameters. For example, in the PSD input-output relationship (1), we already know that the transfer function at 
So, let’s look at this kind of relationship for CSP parameters. All of these results can be found, usually with more mathematical detail, in My Papers [6, 13].
Using complex-valued signal representations is convenient but also has complications: You have to consider all possible choices for conjugating different factors in a moment.
When we considered complex-valued signals and second-order statistics, we ended up with two kinds of parameters: non-conjugate and conjugate. So we have the non-conjugate autocorrelation, which is the expected value of the normal second-order lag product in which only one of the factors is conjugated (consistent with the normal definition of variance for complex-valued random variables),
and the conjugate autocorrelation, which is the expected value of the second-order lag product in which neither factor is conjugated
The complex-valued Fourier-series amplitudes of these functions of time 
The Fourier transforms of the non-conjugate and conjugate cyclic autocorrelation functions are the non-conjugate and conjugate spectral correlation functions, respectively.
I never explained the fundamental reason why both the non-conjugate and conjugate functions are needed. In this post, I rectify that omission. The reason for the many different choices of conjugated factors in higher-order cyclic moments and cumulants is also provided. These choices of conjugation configurations, or conjugation patterns, also appear in the more conventional theory of higher-order statistics as applied to stationary signals.
Cyclic cumulants are the amplitudes of the Fourier-series components of the time-varying cumulant function for a cyclostationary signal. They degenerate to conventional cumulants when the signal is stationary.
In this post I continue the development of the theory of higher-order cyclostationarity (My Papers [5,6]) that I began here. It is largely taken from my doctoral work (download my dissertation here).
This is a long post. To make it worthwhile, I’ve placed some movies of cyclic-cumulant estimates at the end. Or just skip to the end now if you’re impatient!
In my work on cyclostationary signal processing (CSP), the most useful tools are those for estimating second-order statistics, such as the cyclic autocorrelation, spectral correlation function, and spectral coherence function. However, as we discussed in the post on Textbook Signals, there are some situations (perhaps only academic; see my question in the Textbook post) for which higher-order cyclostationarity is required. In particular, a probabilistic approach to blind modulation recognition for ideal (textbook) digital QAM, PSK, and CPM requires higher-order cyclostationarity because such signals have similar or identical spectral correlation functions and PSDs. (Other high-SNR non-probabilistic approaches can still work, such as blind constellation extraction.)
Recall that in the post introducing higher-order cyclostationarity, I mentioned that one encounters a bit of a puzzle when attempting to generalize experience with second-order cyclostationarity to higher orders. This is the puzzle of pure sine waves (My Papers [5]). Let’s look at pure and impure sine waves, and see how they lead to the probabilistic parameters widely known as cyclic cumulants.
Why do we need or care about higher-order cyclostationarity? Because second-order cyclostationarity is insufficient for our signal-processing needs in some important cases.
We’ve seen how to define second-order cyclostationarity in the time- and frequency-domains, and we’ve looked at ideal and estimated spectral correlation functions for a synthetic rectangular-pulse BPSK signal. In future posts, we’ll look at how to create simple spectral correlation estimators, but in this post I want to introduce the topic of higher-order cyclostationarity (HOCS). This post is more conceptual in nature; for mathematical details about HOCS, see the posts on cyclic cumulants and cyclic polyspectra. Estimators of higher-order parameters, such as cyclic cumulants and cyclic moments, are discussed in this post.
To contrast with HOCS, we’ll refer to second-order parameters such as the cyclic autocorrelation and the spectral correlation function as parameters of second-order cyclostationarity (SOCS).
The first question we might ask is Why do we care about HOCS? And one answer is that SOCS does not provide all the statistical information about a signal that we might need to perform some signal-processing task. There are two main limitations of SOCS that drive us to HOCS.
Continue reading “Introduction to Higher-Order Cyclostationarity”
![\displaystyle R_x(t, \boldsymbol{\tau}) = E \left[ x(t+\tau_1)x^*(t+\tau_2) \right] \hfill (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+R_x%28t%2C+%5Cboldsymbol%7B%5Ctau%7D%29+%3D+E+%5Cleft%5B+x%28t%2B%5Ctau_1%29x%5E%2A%28t%2B%5Ctau_2%29+%5Cright%5D+%5Chfill+%281%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![\displaystyle R_{x^*}(t, \boldsymbol{\tau}) = E \left[ x(t+\tau_1)x(t+\tau_2) \right]. \hfill (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+R_%7Bx%5E%2A%7D%28t%2C+%5Cboldsymbol%7B%5Ctau%7D%29+%3D+E+%5Cleft%5B+x%28t%2B%5Ctau_1%29x%28t%2B%5Ctau_2%29+%5Cright%5D.+%5Chfill+%282%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)