## SPTK: Random Variables

Our toolkit expands to include basic probability theory.

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

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 … Continue reading “Worth the Price of a (Fancy) Cup of Coffee?”

## SPTK: The Analytic Signal and Complex Envelope

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.

## Desultory CSP: More Signals from SigIDWiki.com

More real-world data files from SigIDWiki.com. The range of spectral correlation function types exhibited by man-made RF signals is vast.

## SPTK: The Moving-Average Filter

A simple and useful example of a linear time-invariant system. Good for smoothing and discovering trends by averaging away noise.

## Zero-Padding in Spectral Correlation Estimators

Why does zero-padding help in various estimators of the spectral correlation and spectral coherence functions?

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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 … Continue reading “50,000 Page Views in 2020”

## 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.

## Stationary Signal Models Versus Cyclostationary Signal Models

What happens when a cyclostationary time-series is treated as if it were stationary?

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

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

## More on DeepSig’s RML Data Sets

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

## 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 … Continue reading “Blog Notes: New Page with All CSP Blog Posts in Chronological Order”

## 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.

## Professor Jang Again Tortures CSP Mathematics Until it Breaks

In which my life is made a little harder.

## 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.

## 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.

## SPTK: The Fourier Transform

An indispensable tool in CSP and all of signal processing!

## Symmetries of Higher-Order Temporal Probabilistic Parameters in CSP

What are the unique parts of the multidimensional cyclic moments and cyclic cumulants?

## SPTK: The Fourier Series

A crucial tool for developing the temporal parameters of CSP.

## SPTK: Signal Representations

A signal can be written down in many ways. Some of them are more useful than others and can lead to great insights.

## Signal Processing Toolkit: Signals

Introducing the SPTK on the CSP Blog. Basic signal-processing tools with discussions of their connections to and uses in CSP.

## 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 … Continue reading “New Look for a New Year and New Decade”

## Symmetries of Second-Order Probabilistic Parameters in CSP

Do we need to consider all cycle frequencies, both positive and negative? Do we need to consider all delays and frequencies in our second-order CSP parameters?

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

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

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

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 … Continue reading “CSP Resources: The Ultimate Guides to Cyclostationary Random Processes by Professor Napolitano”

## On Impulsive Noise, CSP, and Correntropy

And I still don’t understand how a random variable with infinite variance can be a good model for anything physical. So there.

## Sponsoring the CSP Blog

I’ve decided to solicit donations to the CSP Blog through PayPal. For the past four years, I’ve been writing blog posts and doing my best to answer comments at no cost to my readers. And it has turned out very well indeed, thanks to all the people that stop by … Continue reading “Sponsoring the CSP Blog”

## For the Beginner at CSP

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 Introduction to CSP How to Create a Simple Cyclostationary Signal: Rectangular-Pulse BPSK The Cyclic Autocorrelation Function The Spectral … Continue reading “For the Beginner at CSP”

## A Gallery of Cyclic Correlations

For your delectation.

## On The Shoulders

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 … Continue reading “On The Shoulders”

## Simple Synchronization Using CSP

Using CSP to find the exact values of symbol rate, carrier frequency offset, symbol-clock phase, and carrier phase for PSK/QAM signals.

## 100,000 Page Views!

The CSP Blog has reached 100,000 page views! Also, a while back it passed the “20,000 visitors” milestone. All of this for 53 posts and 10 pages. More to come! I started the CSP Blog in late 2015, so it has taken a bit over three years to get to … Continue reading “100,000 Page Views!”

## Can a Machine Learn a Power Spectrum Estimator?

Learning machine learning for radio-frequency signal-processing problems, continued.

## Data Set for the Machine-Learning Challenge

A PSK/QAM/SQPSK data set with randomized symbol rate, inband SNR, carrier-frequency offset, and pulse roll-off.

## MATLAB’s SSCA: commP25ssca.m

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.

## How we Learned CSP

We learned it using abstractions involving various infinite quantities. Can a machine learn it without that advantage?

## Useful Signal Processing Blogs or Websites?

Update November 1, 2018: A site called feedspot (blog.feedspot.com) contacted me to tell me I made their “Top 10 Digital Signal Processing Blogs, Websites & Newsletters in 2018” list. Weirdly, there are only eight blogs in the list. What’s most important for this post is the other signal processing blogs … Continue reading “Useful Signal Processing Blogs or Websites?”

## Comments on “Detection of Almost-Cyclostationarity: An Approach Based on a Multiple Hypothesis Test” by S. Horstmann et al

The statistics-oriented wing of electrical engineering is perpetually dazzled by [insert Revered Person]’s Theorem at the expense of, well, actual engineering.

## A Challenge for the Machine Learners

The machine-learning modulation-recognition community consistently claims vastly superior performance to anything that has come before. Let’s test that.

## CSP Estimators: The FFT Accumulation Method

An alternative to the strip spectral correlation analyzer.

## ‘Can a Machine Learn the Fourier Transform?’ Redux, Plus Relevant Comments on a Machine-Learning Paper by M. Kulin et al.

Reconsidering my first attempt at teaching a machine the Fourier transform with the help of a CSP Blog reader. Also, the Fourier transform is viewed by Machine Learners as an input data representation, and that representation matters.

## Computational Costs for Spectral Correlation Estimators

The costs strongly depend on whether you have prior cycle-frequency information or not.

## CSP Patent: Tunneling

Tunneling == Purposeful severe undersampling of wideband communication signals. If some of the cyclostationarity property remains, we can exploit it at a lower cost.

## Resolution in Time, Frequency, and Cycle Frequency for CSP Estimators

Unlike conventional spectrum analysis for stationary signals, CSP has three kinds of resolutions that must be considered in all CSP applications, not just two.

## CSP Estimators: Cyclic Temporal Moments and Cumulants

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.

## Can a Machine Learn the Fourier Transform?

Well, can it? I mean, can it REALLY? Or just approximately?

## CSP Blog Highlights

Welcome to the CSP Blog! To help new readers, I’m supplying here links to the posts that have gotten the most attention over the lifetime of the Blog. Omitted from this list are the more esoteric topics as well as most of the posts that comment on the engineering literature. … Continue reading “CSP Blog Highlights”

## Automatic Spectral Segmentation

Radio-frequency scene analysis is much more complex than modulation recognition. A good first step is to blindly identify the frequency intervals for which significant non-noise energy exists.

## Blog Notes and How to Obtain Help with Your CSP Work

The CSP Blog has been getting lots of new visitors these past few months; welcome to all! Following the CSP Blog If you want to receive an email each time I publish a new post, look for the Follow Blog via Email widget on the right side of the Blog … Continue reading “Blog Notes and How to Obtain Help with Your CSP Work”

## More on Pure and Impure Sine Waves

Gaussian and binary signals are in some sense at opposite ends of the pure-impure sine-wave spectrum.

## Cyclostationarity of Direct-Sequence Spread-Spectrum Signals

Spread-spectrum signals are used to enable shared-bandwidth communication systems (CDMA), precision position estimation (GPS), and secure wireless data transmission.

## Cumulant (4, 2) is a Good Discriminator? Comments on “Energy-Efficient ProcessorÂ for Blind Signal Classification in Cognitive Radio Networks,” by E. Rebeiz et al.

Let’s talk about another published paper on signal detection involving cyclostationarity and/or cumulants. This one is called “Energy-Efficient ProcessorÂ 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 … Continue reading “Cumulant (4, 2) is a Good Discriminator? Comments on “Energy-Efficient ProcessorÂ for Blind Signal Classification in Cognitive Radio Networks,” by E. Rebeiz et al.”

## Machine Learning and Modulation Recognition: Comments on “Convolutional Radio Modulation Recognition Networks” by T. O’Shea, J. Corgan, and T. Clancy

Update October 2020: Since I wrote the paper review in this post, I’ve analyzed three of O’Shea’s data sets (O’Shea is with the company DeepSig, so I’ve been referring to the data sets as DeepSig’s in other posts): All BPSK Signals, More on DeepSig’s Data Sets, and DeepSig’s 2018 Data … Continue reading “Machine Learning and Modulation Recognition: Comments on “Convolutional Radio Modulation Recognition Networks” by T. O’Shea, J. Corgan, and T. Clancy”

## Modulation Recognition Using Cyclic Cumulants, Part I: Problem Description and Variants

Modulation recognition is the process of assigning one or more modulation-class labels to a provided time-series data sequence.

## Comments on “Blind Cyclostationary Spectrum Sensing in Cognitive Radios” by W. M. Jang

We are all susceptible to using bad mathematics to get us where we want to go. Here is an example.

## The Periodogram

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.

## Cyclic Polyspectra

Higher-order statistics in the frequency domain for cyclostationary signals. As complicated as it gets at the CSP Blog.

## Comments on “Cyclostationary Correntropy: Definition and Application” by Fontes et al

I recently came across a published paper with the title Cyclostationary Correntropy: Definition and Application, by Aluisio Fontes et al. It is published in a journal called Expert Systems with Applications (Elsevier). Actually, it wasn’t the first time I’d seen this work by these authors. I had reviewed a similar … Continue reading “Comments on “Cyclostationary Correntropy: Definition and Application” by Fontes et al”

## Blog Notes and New-Post Poll

The CSP Blog has been up for about a year, and September 2016 was its best month: record numbers of visitors, page views, and views per visitor. Thanks to all of my readers!

## 100-MHz Amplitude Modulation? Comments on “Sub-Nyquist Cyclostationary Detection for Cognitive Radio” by Cohen and Eldar

I came across a paper by Cohen and Eldar, researchers at the Technion in Israel. You can get the paper on the Arxiv site here. The title is “Sub-Nyquist Cyclostationary Detection for Cognitive Radio,” and the setting is spectrum sensing for cognitive radio. I have a question about the paper … Continue reading “100-MHz Amplitude Modulation? Comments on “Sub-Nyquist Cyclostationary Detection for Cognitive Radio” by Cohen and Eldar”

## Cyclostationarity of Digital QAM and PSK

PSK and QAM signals form the building blocks for a large number of practical real-world signals. Understanding their probability structure is crucial to understanding those more complicated signals.

## Signal Processing Operations and CSP

How does the cyclostationarity of a signal change when it is subjected to common signal-processing operations like addition, multiplication, and convolution?

## The Cycle Detectors

CSP shines when the problem involves strong noise or cochannel interference. Here we look at CSP-based signal-presence detection as a function of SNR and SIR.

## Radio-Frequency Scene Analysis

Modulation recognition is one thing, holistic radio-frequency scene analysis is quite another.

## CSP-Based Time-Difference-of-Arrival Estimation

Time-delay estimation can be used to determine the angle-of-arrival of a signal impinging on two spatially separated signals. This estimation problem gets hard when there is cochannel interference present.

## Square-Root Raised-Cosine PSK/QAM

SRRC PSK and QAM signals form important components of more complicated real-world communication signals. Let’s look at their second-order cyclostationarity here.

## Blog Notes

In the near future, I’ll post on two new topics: Time-Delay Estimation andÂ the Cyclic Polyspectrum. The blog is getting good traffic: But not many comments. So, feel free to comment on this post with your suggestions on topics that you’d like to see discussed on the CSP blog. Now is … Continue reading “Blog Notes”

## CSP Estimators: The Strip Spectral Correlation Analyzer

The SSCA is a good tool for blind (no prior information) exhaustive (all cycle frequencies) spectral correlation analysis. An alternative is the FFT accumulation method.

## Conjugation Configurations

Using complex-valued signal representations is convenient but also has complications: You have to consider all possible choices for conjugating different factors in a moment.

## Second-Order Estimator Verification Guide

Use this post to help check the accuracy of your second-order CSP estimators.

## Cyclic Temporal Cumulants

Cyclic cumulants are the amplitudes of the Fourier-series components of the time-varying cumulant function for a cyclostationary signal. They degenerate to conventional cumulants when the signal is stationary.

## A Gallery of Spectral Correlation

Pictures are worth N words, and M equations, where N and M are large integers.

## Textbook Signals

Yes, the CSP Blog uses the simplest idealized cyclostationary digital signal–rectangular-pulse BPSK–to connect all the different aspects of CSP. But don’t mistake these ‘textbook’ signals for the real world.

## SCF Estimate Quality: The Resolution Product

What factors influence the quality of a spectral correlation function estimate?

## The Spectral Coherence Function

Cross correlation functions can be normalized to create correlation coefficients. The spectral correlation function is a cross correlation and its correlation coefficient is called the coherence.

## CSP Estimators: The Time Smoothing Method

The non-blind spectral-correlation estimator called the TSM is favored when one wishes to avoid long FFTs.

## CSP Estimators: The Frequency-Smoothing Method

The non-blind spectral-correlation estimator called the FSM is favored when one wishes to have fine control over frequency resolution and can tolerate long FFTs.

## Signal Selectivity

We can estimate the spectral correlation function of one signal in the presence of another with complete temporal and spectral overlap provided the signal has a unique cycle frequency.

## Introduction to Higher-Order Cyclostationarity

Why do we need or care about higher-order cyclostationarity? Because second-order cyclostationarity is insufficient for our signal-processing needs in some important cases.

## The Spectral Correlation Function for Rectangular-Pulse BPSK

Let’s make the spectral correlation function a little less abstract by showing it for a simple textbook BPSK signal.

## The Spectral Correlation Function

Spectral correlation in CSP means that distinct narrowband spectral components of a signal are correlated-they contain either identical information or some degree of redundant information.

## The Cyclic Autocorrelation for Rectangular-Pulse BPSK

Let’s look at a specific example of the cyclic autocorrelation function: the textbook rectangular-pulse BPSK signal with IID symbols.

## The Cyclic Autocorrelation Function

The cyclic autocorrelation function is the amplitude of a Fourier-series component of the time-varying autocorrelation for a cyclostationary signal.

## Creating a Simple CS Signal: Rectangular-Pulse BPSK

We’ll use this simple textbook signal throughout the CSP Blog to illustrate and tie together all the different aspects of CSP.

## Welcome to the CSP Blog!

Thank you for visiting the CSP blog. The purpose of this blog is to talk about cyclostationary signals and cyclostationary signal processing (CSP). I’ve been working in the area for nearly thirty years, and over that time I’ve received a lot of requests for help with CSP code and algorithms. … Continue reading “Welcome to the CSP Blog!”

Hi Dr Spooner. This page is blank as of today.

Hey Dylan. I tried to fix it. I will have to get some WordPress.com help. Thanks!

It’s a little ugly, but good to go!

WordPress gave me a hint on how to restore the pretty version, which I’ve done, but at the expense of disallowing sharing of the posts and the “Like” button. I don’t care about “Likes,” so that’s fine, and people can always email or text the URLs themselves, so this workaround is fine indefinitely.