SPTK: Sampling and The Sampling Theorem

The basics of how to convert a continuous-time signal into a discrete-time signal without losing information in the process. Plus, how the choice of sampling rate influences CSP.

Previous SPTK Post: Random Processes Next SPTK Post: Echo Detection

In this Signal Processing ToolKit post we take a close look at the basic sampling theorem used daily by signal-processing engineers. Application of the sampling theorem is a way to choose a sampling rate for converting an analog continuous-time signal to a digital discrete-time signal. The former is ubiquitous in the physical world–for example all the radio-frequency signals whizzing around in the air and through your body right now. The latter is ubiquitous in the computing-device world–for example all those digital-audio files on your Discman Itunes Ipod DVD Smartphone Cloud Neuralink Singularity.

So how are those physical real-world analog signals converted to convenient lists of finite-precision numbers that we can apply arithmetic to? For that’s all [digital or cyclostationary] signal processing is at bottom: arithmetic. You might know the basic rule-of-thumb for choosing a sampling rate: Make sure it is at least twice as big as the largest frequency component in the analog signal undergoing the sampling. But why, exactly, and what does ‘largest frequency component’ mean?

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Update on J. Antoni’s Fast Spectral Correlation Estimator

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.

Let’s take a look.

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Shifted Dataset for the Machine-Learning Challenge: How Well Does a Modulation-Recognition DNN Generalize? [Dataset CSPB.ML.2022]

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!!

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The Principal Domain for the Spectral Correlation Function

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 $(f, \alpha)$ 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?

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J. Antoni’s Fast Spectral Correlation Estimator

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.

Update 2023: I continue with critical analysis of Antoni’s work as applied to ‘using and understanding the statistics of communication signals‘ in a follow-on post.

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.

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SPTK (and CSP): Random Processes

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.

Previous SPTK Post: Examples of Random Variables Next SPTK Post: The Sampling Theorem

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.

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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.

Cyclostationarity of DMR Signals

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.

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Comments on “Deep Neural Network Feature Designs for RF Data-Driven Wireless Device Classification,” by B. Hamdaoui et al

Another post-publication review of a paper that is weak on the ‘RF’ in RF machine learning.

Let’s take a look at a recently published paper (The Literature [R148]) on machine-learning-based modulation-recognition to get a data point on how some electrical engineers–these are more on the side of computer science I believe–use mathematics when they turn to radio-frequency problems. You can guess it isn’t pretty, and that I’m not here to exalt their acumen.

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Spectral Correlation and Cyclic Correlation Plots for Real-Valued Signals

Spectral correlation surfaces for real-valued and complex-valued versions of the same signal look quite different.

In the real world, the electromagnetic field is a multi-dimensional time-varying real-valued function (volts/meter or newtons/coulomb). But in mathematical physics and signal processing, we often use complex-valued representations of the field, or of quantities derived from it, to facilitate our mathematics or make the signal processing more compact and efficient.

So throughout the CSP Blog I’ve focused almost exclusively on complex-valued signals and data. However, there is a considerable older literature that uses real-valued signals, such as The Literature [R1, R151]. You can use either real-valued or complex-valued signal representations and data, as you prefer, but there are advantages and disadvantages to each choice. Moreover, an author might not be perfectly clear about which one is used, especially when presenting a spectral correlation surface (as opposed to a sequence of equations, where things are often more clear).

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50,000 Page Views in 2020

And counting …

Last evening the CSP Blog crossed the 50,000 page-view threshold for 2020, a yearly total that has not been achieved previously!

I want to thank each reader, each commenter, and each person that’s clicked the Donate button. You’ve made the CSP Blog the success it is, and I am so grateful for the time you spend here.

On these occasions I put some of the more interesting CSP-Blog statistics below the fold. If you have been wanting to see a post on a particular CSP or Signal Processing ToolKit topic, and it just hasn’t appeared, feel free to leave me a note in the Comments section.

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Stationary Signal Models Versus Cyclostationary Signal Models

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.

DeepSig’s 2018 Dataset: 2018.01.OSC.0001_1024x2M.h5.tar.gz

The third DeepSig dataset I’ve examined. It’s better!

Update February 2021. I added material relating to the DeepSig-claimed variation of the roll-off parameter in a square-root raised-cosine pulse-shaping function. It does not appear that the roll-off was actually varied as stated in Table I of [R137].

DeepSig’s datasets are popular in the machine-learning modulation-recognition community, and in that community there are many claims that the deep neural networks are vastly outperforming any expertly hand-crafted tired old conventional method you care to name (none are usually named though). So I’ve been looking under the hood at these datasets to see what the machine learners think of as high-quality inputs that lead to disruptive upending of the sclerotic mod-rec establishment. In previous posts, I’ve looked at two of the most popular DeepSig datasets from 2016 (here and here). In this post, we’ll look at one more and I will then try to get back to the CSP posts.

Let’s take a look at one more DeepSig dataset: 2018.01.OSC.0001_1024x2M.h5.tar.gz.

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More on DeepSig’s RML Datasets

The second DeepSig data set I analyze: SNR problems and strange PSDs.

I presented an analysis of one of DeepSig’s earlier modulation-recognition datasets (RML2016.10a.tar.bz2) in the post on All BPSK Signals. There we saw several flaws in the dataset as well as curiosities. Most notably, the signals in the dataset labeled as analog amplitude-modulated single sideband (AM-SSB) were absent: these signals were only noise. DeepSig has several other datasets on offer at the time of this writing:

In this post, I’ll present a few thoughts and results for the “Larger Version” of RML2016.10a.tar.bz2, which is called RML2016.10b.tar.bz2. This is a good post to offer because it is coherent with the first RML post, but also because more papers are being published that use the RML 10b dataset, and of course more such papers are in review. Maybe the offered analysis here will help reviewers to better understand and critique the machine-learning papers. The latter do not ever contain any side analysis or validation of the RML datasets (let me know if you find one that does in the Comments below), so we can’t rely on the machine learners to assess their inputs. (Update: I analyze a third DeepSig dataset here. And a fourth and final one here.)

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Blog Notes: New Page with All CSP Blog Posts in Chronological Order

To aid navigating the CSP Blog, I’ve added a new page called “All CSP Blog Posts.” You can find the page link at the top of the home page, or in various lists on the right side of the Blog, such as “Pages” and “Site Navigation.”

Let me know in the Comments if there are other ways that you think I can improve the usability of the site.

All BPSK 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.

Symmetries of Higher-Order Temporal Probabilistic Parameters in CSP

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.

New Look for a New Year and New Decade

2020 is the fifth full year of existence for the CSP Blog, and the beginning of a new decade that will be full of CSP explorations. I thought I’d freshen up the look of the Blog, so I’ve switched the theme. It is a cleaner look with fewer colors and no more hexagons. I’m not completely happy with it, so I might change it yet again. Let me know if you have problems viewing the content or posting a comment (cmspooner at ieee dot org).

Happy New Year to all my readers!

The Ambiguity Function and the Cyclic Autocorrelation Function: Are They the Same Thing?

To-may-to, to-mah-to?

Let’s talk about ambiguity and correlation. The ambiguity function is a core component of radar signal processing practice and theory. The autocorrelation function and the cyclic autocorrelation function, are key elements of generic signal processing and cyclostationary signal processing, respectively. Ambiguity and correlation both apply a quadratic functional to the data or signal of interest, and they both weight that quadratic functional by a complex exponential (sine wave) prior to integration or summation.

Are they the same thing? Well, my answer is both yes and no.