**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.

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# Tag: signal processing

## All BPSK Signals

## Professor Jang Again Tortures CSP Mathematics Until it Breaks

## SPTK: Frequency Response of LTI Systems

## SPTK: Linear Time-Invariant Systems

## SPTK: The Fourier Transform

## SPTK: The Fourier Series

## SPTK: Signal Representations

## Signal Processing Toolkit: Signals

## New Look for a New Year and New Decade

## Symmetries of Second-Order Probabilistic Parameters in CSP

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

## CSP Resources: The Ultimate Guides to Cyclostationary Random Processes by Professor Napolitano

## On Impulsive Noise, CSP, and Correntropy

## For the Beginner at CSP

## A Gallery of Cyclic Correlations

## On The Shoulders

## Simple Synchronization Using CSP

## Can a Machine Learn a Power Spectrum Estimator?

## MATLAB’s SSCA: commP25ssca.m

## How we Learned CSP

Cyclostationary Signal Processing

Understanding and Using the Statistics of Communication Signals

**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.

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”

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 .

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.

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.

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.

Previous SPTK Post: Signals Next SPTK Post: Fourier Series

In this Signal Processing ToolKit post, we’ll look at the idea of *signal representations*. This is a branch of signal-processing mathematics that expresses one signal in terms of one or more signals drawn from a special set, such as the set of all sine waves, the set of harmonically related sine waves, a set of wavelets, a set of piecewise constant waveforms, etc.

Signal representations are a key component of understanding stationary-signal processing tools such as convolution and Fourier series and transforms. Since Fourier series and transforms are an integral part of CSP, **signal representations are important for all our discussions at the CSP Blog.**

This is the inaugural post of a new series of posts I’m calling the ** Signal Processing Toolkit** (SPTK). The SPTK posts will cover relatively simple topics in signal processing that are useful in the practice of cyclostationary signal processing. So, they are not CSP posts, but CSP practitioners need to know this material to be successful in CSP. The CSP Blog is branching out! (But don’t worry, there are more CSP posts coming too.)

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!

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 and cycle frequency variables and of the frequency-domain functions in the spectral frequency and cycle frequency variables. Or a generalized version of evenness/oddness, such as , where and are closely related functions. We have to consider the non-conjugate and conjugate functions separately, and we’ll also consider both the auto and cross versions of the parameters. We’ll look at higher-order cyclic moments and cumulants in a future post.

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”

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.

My friend and colleague Antonio Napolitano has just published a new book on cyclostationary signals and cyclostationary signal processing:

*Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations*, Academic Press/Elsevier, 2020, ISBN: 978-0-08-102708-0. The book is a comprehensive guide to the structure of cyclostationary random processes and signals, and it also provides pointers to the literature on many different applications. The book is mathematical in nature; use it to deepen your understanding of the underlying mathematics that make CSP possible.

You can check out the book on amazon.com using the following link:

I’ve seen several published and pre-published (arXiv.org) technical papers over the past couple of years on the topic of cyclic correntropy (The Literature [R123-R127]). I first criticized such a paper ([R123]) here, but the substance of that review was about my problems with the presented mathematics, not impulsive noise and its effects on CSP. Since the papers keep coming, apparently, I’m going to put down some thoughts on impulsive noise and some evidence regarding simple means of mitigation in the context of CSP. Preview: I don’t think we need to go to the trouble of investigating cyclic correntropy as a means of salvaging CSP from the clutches of impulsive noise.

Here is a list of links to CSP Blog posts that I think are suitable for a beginner: read them in the order given.

How to Obtain Help from the CSP Blog

How to Create a Simple Cyclostationary Signal: Rectangular-Pulse BPSK

The Cyclic Autocorrelation Function

The Spectral Correlation Function

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 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 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 in the gallery post.

What modest academic success I’ve had in the area of cyclostationary signal theory and cyclostationary signal processing is largely due to the patient mentorship of my doctoral adviser, William (Bill) Gardner, and the fact that I was able to build on an excellent foundation put in place by Gardner, his advisor Lewis Franks, and key Gardner students such as William (Bill) Brown.

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.

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?”

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.

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). The machine learners want to capitalize on the success of machine learning 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.