Zero-Padding in Spectral Correlation Estimators

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.

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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, 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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SPTK: Ideal Filters

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.

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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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SPTK: Convolution and the Convolution Theorem

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.

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SPTK: Interconnection of Linear Systems

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.

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

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.

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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.

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

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.

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More on DeepSig’s RML Data Sets

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.)

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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.

h/t: Reader Clint.

All BPSK Signals

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.

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Professor Jang Again Tortures CSP Mathematics Until it Breaks

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.

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SPTK: Frequency Response of LTI Systems

The frequency response of a filter tells you how it scales each and every input sine-wave or spectral component.

Previous SPTK Post: LTI Systems             Next SPTK Post: Interconnection of LTI Systems

We continue our progression of Signal-Processing ToolKit posts by looking at the frequency-domain behavior of linear time-invariant (LTI) systems. In the previous post, we established that the time-domain output of an LTI system is completely determined by the input and by the response of the system to an impulse input applied at time zero. This response is called the impulse response and is typically denoted by h(t).

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SPTK: Linear Time-Invariant Systems

LTI systems, or filters, are everywhere in signal processing. They allow us to adjust the amplitudes and phases of spectral components of the input.

Previous SPTK Post: The Fourier Transform         Next SPTK Post: Frequency Response

In this Signal Processing Toolkit post, we’ll take a first look at arguably the most important class of system models: linear time-invariant (LTI) systems.

What do signal processors and engineers mean by system? Most generally, a system is a rule or mapping that associates one or more input signals to one or more output signals. As we did with signals, we discuss here various useful dichotomies that break up the set of all systems into different subsets with important properties–important to mathematical analysis as well as to design and implementation. Then we’ll look at time-domain input/output relationships for linear systems. In a future post we’ll look at the properties of linear systems in the frequency domain.

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SPTK: The Fourier Transform

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.

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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.

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SPTK: The Fourier Series

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.

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