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.
An indispensable tool in CSP and all of signal processing!
Previous SPTK Post: The Fourier Series Next SPTK Post: Linear Systems
This post in the Signal Processing Toolkit series deals with a key mathematical tool in CSP: The Fourier transform. Let’s try to see how the Fourier transform arises from a limiting version of the Fourier series.
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”
A crucial tool for developing the temporal parameters of CSP.
Previous SPTK Post: Signal Representations Next SPTK Post: The Fourier Transform
This installment of the Signal Processing Toolkit shows how the Fourier series arises from a consideration of representing arbitrary signals as vectors in a signal space. We also provide several examples of Fourier-series calculations, interpret the Fourier series, and discuss its relevance to cyclostationary signal processing.
For your delectation.
There are some situations in which the spectral correlation function is not the preferred measure of (second-order) cyclostationarity. In these situations, the cyclic autocorrelation (non-conjugate and conjugate versions) may be much simpler to estimate and work with in terms of detector, classifier, and estimator structures. So in this post, I’m going to provide surface plots of the cyclic autocorrelation for each of the signals in the spectral correlation gallery post. The exceptions are those signals I called feature-rich in the spectral correlation gallery post, such as DSSS, LTE, and radar. Recall that such signals possess a large number of cycle frequencies, and plotting their three-dimensional spectral correlation surface is not helpful as it is difficult to interpret with the human eye. So for the cycle-frequency patterns of feature-rich signals, we’ll rely on the stem-style (cyclic-domain profile) plots that I used in the spectral correlation gallery post.
Using CSP to find the exact values of symbol rate, carrier frequency offset, symbol-clock phase, and carrier phase for PSK/QAM signals.
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.
Learning machine learning for radio-frequency signal-processing problems, continued.
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!
Continue reading “Can a Machine Learn a Power Spectrum Estimator?”
A PSK/QAM/SQPSK data set with randomized symbol rate, inband SNR, carrier-frequency offset, and pulse roll-off.
Update April 2025: All but the first five batch files have been removed. I needed to make space since WordPress has a hard limit on storage. Use CSPB.ML.2018R2 in any case.
Update September 2023: A randomization flaw has been found and fixed for CSPB.ML.2018, resulting in CSPB.ML.2018R2. Use that one going forward.
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 
In this short post, I describe some errors that are produced by MATLAB’s strip spectral correlation analyzer function commP25ssca.m. I don’t recommend that you use it; far better to create your own function.
The statistics-oriented wing of electrical engineering is perpetually dazzled by [insert Revered Person]’s Theorem at the expense of, well, actual engineering.
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.
The machine-learning modulation-recognition community consistently claims vastly superior performance to anything that has come before. Let’s test that.
Update September 2023: A randomization flaw has been found and fixed for CSPB.ML.2018, resulting in CSPB.ML.2018R2. Use that one going forward.
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!
An alternative to the strip spectral correlation analyzer.
Let’s look at another spectral correlation function estimator: the FFT Accumulation Method (FAM). This estimator is in the time-smoothing category, is exhaustive in that it is designed to compute estimates of the spectral correlation function over its entire principal domain, and is efficient, so that it is a competitor to the Strip Spectral Correlation Analyzer (SSCA) method. I implemented my version of the FAM by using the paper by Roberts et al (The Literature [R4]). If you follow the equations closely, you can successfully implement the estimator from that paper. The tricky part, as with the SSCA, is correctly associating the outputs of the coded equations to their proper 
Continue reading “CSP Estimators: The FFT Accumulation Method”
Unlike conventional spectrum analysis for stationary signals, CSP has three kinds of resolutions that must be considered in all CSP applications, not just two.
In this post, we look at the ability of various CSP estimators to distinguish cycle frequencies, temporal changes in cyclostationarity, and spectral features. These abilities are quantified by the resolution properties of CSP estimators.
Consider performing some CSP estimation task, such as using the frequency-smoothing method, time-smoothing method, or strip spectral correlation analyzer method of estimating the spectral correlation function. The estimate employs 
Then the temporal resolution 
Continue reading “Resolution in Time, Frequency, and Cycle Frequency for CSP Estimators”
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
Spread-spectrum signals are used to enable shared-bandwidth communication systems (CDMA), precision position estimation (GPS), and secure wireless data transmission.
In this post we look at direct-sequence spread-spectrum (DSSS) signals, which can be usefully modeled as a kind of PSK signal. DSSS signals are used in a variety of real-world situations, including the familiar CDMA and WCDMA signals, covert signaling, and GPS. My colleague Antonio Napolitano has done some work on a large class of DSSS signals (The Literature [R11, R17, R95]), resulting in formulas for their spectral correlation functions, and I’ve made some remarks about their cyclostationary properties myself here and there (My Papers [16]).
A good thing, from the point of view of modulation recognition, about DSSS signals is that they are easily distinguished from other PSK and QAM signals by their spectral correlation functions. Whereas most PSK/QAM signals have only a single non-conjugate cycle frequency, and no conjugate cycle frequencies, DSSS signals have many non-conjugate cycle frequencies and in some cases also have many conjugate cycle frequencies.
Continue reading “Cyclostationarity of Direct-Sequence Spread-Spectrum Signals”
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 is 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 
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 [16,17,25,26,28,30,32,33,38,43,44,50-52,54-56,58-59]. 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.
We are all susceptible to using bad mathematics to get us where we want to go. Here is an example.
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 harder for people trying to learn about cyclostationarity, which eventually leads to the need for something like the CSP Blog as a corrective.
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 
The periodogram is the scaled magnitude-squared finite-time Fourier transform of something. It is as random as its input–it never converges to the power spectrum.
I’ve been reviewing a lot of technical papers lately and I’m noticing that it is becoming common to assert that the limiting form of the periodogram is the power spectral density or that the limiting form of the cyclic periodogram is the spectral correlation function. This isn’t true. These functions do not become, in general, less random (erratic) as the amount of data that is processed increases without limit. On the contrary, they always have large variance. Some form of averaging (temporal or spectral) is needed to permit the periodogram to converge to the power spectrum or the cyclic periodogram to converge to the spectral correlation function (SCF).
In particular, I’ve been seeing things like this:
where 
![t \in [-T/2, T/2]](https://s0.wp.com/latex.php?latex=t+%5Cin+%5B-T%2F2%2C+T%2F2%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
