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In this Signal Processing ToolKit post, we examine the concept of a *random variable*.

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# Author: Chad Spooner

## SPTK: Random Variables

## Worth the Price of a (Fancy) Cup of Coffee?

## SPTK: The Analytic Signal and Complex Envelope

## Desultory CSP: More Signals from SigIDWiki.com

## SPTK: The Moving-Average Filter

## Zero-Padding in Spectral Correlation Estimators

## Cyclostationarity of DMR Signals

## SPTK: Ideal Filters

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

## Spectral Correlation and Cyclic Correlation Plots for Real-Valued Signals

## SPTK: Convolution and the Convolution Theorem

## SPTK: Interconnection of Linear Systems

## 50,000 Page Views in 2020

## Are Probability Density Functions “Engineered” or “Hand-Crafted” Features?

## Stationary Signal Models Versus Cyclostationary Signal Models

## DeepSig’s 2018 Data Set: 2018.01.OSC.0001_1024x2M.h5.tar.gz

## More on DeepSig’s RML Data Sets

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

## All BPSK Signals

## Professor Jang Again Tortures CSP Mathematics Until it Breaks

Cyclostationary Signal Processing

Understanding and Using the Statistics of Communication Signals

I'm a signal processing researcher specializing in cyclostationary signal processing (CSP) for communication signals. I hope to use this blog to help others with their cyclo-projects and to learn more about how CSP is being used and extended worldwide.

Our toolkit expands to include basic probability theory.

Previous SPTK Post: Complex Envelopes Next SPTK Post: Examples of Random Variables

In this Signal Processing ToolKit post, we examine the concept of a *random variable*.

Just a reminder that if you are getting some value out of the CSP Blog, I would appreciate it if you could make a donation to offset my costs: I do pay WordPress to keep ads off the site! I also pay extra for a class of service that allows me to post large data sets like the one for the Machine-Learner Challenge.

If everyone that derived value from the CSP Blog were to donate $5, I’d have enough leftover for at least a couple cups of fancy coffee.

In signal processing, and in CSP, we often have to convert real-valued data into complex-valued data and vice versa. Real-valued data is in the real world, but complex-valued data is easier to process due to the use of a substantially lower sampling rate.

Previous SPTK Post: The Moving-Average Filter Next SPTK Post: Random Variables

In this Signal-Processing Toolkit post, we review the signal-processing steps needed to convert a real-valued sampled-data bandpass signal to a complex-valued sampled-data lowpass signal. The former can arise from sampling a signal that has been downconverted from its radio-frequency spectral band to a much lower intermediate-frequency spectral band. So we want to convert such data to complex samples at zero frequency (‘complex baseband’) so we can decimate them and thereby match the sample rate to the signal’s baseband bandwidth. Subsequent signal-processing algorithms (including CSP of course) can then operate on the relatively low-rate complex-envelope data, which is beneficial because the same number of seconds of data can be processed using fewer samples.

Continue reading “SPTK: The Analytic Signal and Complex Envelope”More real-world data files from SigIDWiki.com. The range of spectral correlation function types exhibited by man-made RF signals is vast.

Let’s look at a few more signals posted to sigidwiki.com. Just for fun.

Continue reading “Desultory CSP: More Signals from SigIDWiki.com”A simple and useful example of a linear time-invariant system. Good for smoothing and discovering trends by averaging away noise.

Previous SPTK Post: Ideal Filters Next SPTK Post: The Complex Envelope

We continue our basic signal-processing posts with one on the moving-average, or smoothing, filter. The moving-average filter is a linear time-invariant operation that is widely used to mitigate the effects of additive noise and other random disturbances from a presumably well-behaved signal. For example, a physical phenomenon may be producing a signal that increases monotonically over time, but our measurement of that signal is corrupted by noise, interference, or flaws in measurement. The moving-average filter can reveal the sought-after trend by suppressing the effects of the unwanted disturbances.

Continue reading “SPTK: The Moving-Average Filter”Why does zero-padding help in various estimators of the spectral correlation and spectral coherence functions?

**Update to the exchange: May 7, 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.

Continue reading “Zero-Padding in Spectral Correlation Estimators”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.

Continue reading “Cyclostationarity of DMR Signals”Ideal filters have rectangular or unit-step-like transfer functions and so are not physical. But they permit much insight into the analysis and design of real-world linear systems.

Previous SPTK Post: Convolution Next SPTK Post: The Moving-Average Filter

We continue with our non-CSP signal-processing tool-kit series with this post on ideal filtering. Ideal filters are those filters with transfer functions that are rectangular, step-function-like, or combinations of rectangles and step functions.

Continue reading “SPTK: Ideal Filters”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.

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

Continue reading “Spectral Correlation and Cyclic Correlation Plots for Real-Valued Signals”Convolution is an essential element in everyone’s signal-processing toolkit. We’ll look at it in detail in this post.

Previous SPTK Post: Interconnection of Linear Systems Next SPTK Post: Ideal Filters

This installment of the Signal Processing Toolkit series of CSP Blog posts deals with the ubiquitous signal-processing operation known as convolution. We originally came across it in the context of linear time-invariant systems. In this post, we focus on the mechanics of computing convolutions and discuss their utility in signal processing and CSP.

Continue reading “SPTK: Convolution and the Convolution Theorem”Real-world signal-processing systems often combine multiple kinds of linear time-invariant systems. We look here at the general kinds of connections.

Previous Post: Frequency Response Next Post: Convolution

It is often the case that linear time invariant (or for discrete-time systems, linear shift invariant) systems are connected together in various ways, so that the output of one may be the input to another, or two or more systems may share the same input. In such cases we can often find an *equivalent system* impulse response that takes into account all the component systems. In this post we focus on the serial and parallel connections of LTI systems in both the time and frequency domains.

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.

Continue reading “50,000 Page Views in 2020”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.

Continue reading “Stationary Signal Models Versus Cyclostationary Signal Models”

The third DeepSig data set 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 data sets 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 data sets 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 data sets 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 data set: 2018.01.OSC.0001_1024x2M.h5.tar.gz.

Continue reading “DeepSig’s 2018 Data Set: 2018.01.OSC.0001_1024x2M.h5.tar.gz”The second DeepSig data set I analyze: SNR problems and strange PSDs.

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

Continue reading “More on DeepSig’s RML Data Sets”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.

h/t: Reader Clint.

An analysis of DeepSig’s 2016.10A data set, used in many published machine-learning papers, and detailed comments on quite a few of those papers.

**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.

In which my life is made a little harder.

We first met Professor Jang in a “Comments on the Literature” type of post from 2016. In that post, I pointed out fundamental mathematical errors contained in a paper the Professor published in the *IEEE Communications Letters* in 2014 (The Literature [R71]).

I have just noticed a new paper by Professor Jang, published in the journal *IEEE Access, *which is a peer-reviewed journal, like the* Communications Letters*. This new paper is titled “*Simultaneous Power Harvesting and Cyclostationary Spectrum Sensing in Cognitive Radios*” (The Literature [R144]). Many of the same errors are present in this paper. In fact, the beginning of the paper, and the exposition on cyclostationary signal processing is nearly the same as in The Literature [R71].

Let’s take a look.

Continue reading “Professor Jang Again Tortures CSP Mathematics Until it Breaks”