# Can a Machine Learn a Power Spectrum Estimator?

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!

# Data Set for the Machine-Learning Challenge

I’ve posted $20000$ PSK/QAM signals to the CSP Blog. These are the signals I refer to in the post I wrote challenging the machine-learners. In this brief post, I provide links to the data and describe how to interpret the text file containing the signal-type labels and signal parameters.

# How we Learned CSP

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 techniques used there to automatic recognition (classification) of the type of a captured man-made electromagnetic wave.

# 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 on the list. So check it out if you are looking for other sources of online signal processing information. Enjoy!  blog.feedspot.com/digital_signal_processing_blogs

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Some of my CSP posts get a lot of comments asking for help, and that’s a good thing. I continue to try to help readers to help themselves. Throughout my posts, I link terms and methods to webpages that provide tutorial or advanced information, and most of the time that means wikipedia.

But I’d like to be able to refer readers to good websites that discuss related aspects of signal processing and communication signals, such as filtering, spectrum estimation, mathematical models, Fourier analysis, etc. I’ve had little success with the Google searches I’ve tried.

# CSP Estimators: The FFT Accumulation Method

Let’s look at another spectral correlation function estimator: the FFT Accumulation Method (FAM). This estimator is in the time-smoothing category, is exhaustive in that it is designed to compute estimates of the spectral correlation function over its entire principal domain, and is efficient, so that it is a competitor to the Strip Spectral Correlation Analyzer (SSCA) method. I implemented my version of the FAM by using the paper by Roberts et al (The Literature [R4]). If you follow the equations closely, you can successfully implement the estimator from that paper. The tricky part, as with the SSCA, is correctly associating the outputs of the coded equations to their proper $\displaystyle (f, \alpha)$ values.

# Can a Machine Learn the Fourier Transform?

Update: See Part 2 of this post at this link. If you want to leave on comment, leave it on Part 2. Comments closed on this Part 1 post.

# 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.

# Second-Order Estimator Verification Guide

In this post I provide some tools for the do-it-yourself CSP practitioner. One of the goals of this blog is to help new CSP researchers and students to write their own estimators and algorithms. This post contains some spectral correlation function estimates and numerically evaluated formulas that can be compared to those produced by anybody’s code.

The signal of interest is, of course, our rectangular-pulse BPSK signal with symbol rate $0.1$ (normalized frequency units) and carrier offset $0.05$. You can download a MATLAB script for creating such a signal here.

The formula for the SCF for a textbook BPSK signal is published in several places (The Literature [R47], My Papers [6]) and depends mainly on the Fourier transform of the pulse function used by the textbook signal.

We’ll compare the numerically evaluated formula with estimates produced by my version of the frequency-smoothing method (FSM). The FSM estimates and the theoretical functions are contained in a MATLAB mat file here. (I had to change the extension of the mat file from .mat to .doc to allow posting it to WordPress.) In all the results shown here and that you can download, the processed data-block length is $65536$ samples and the FSM smoothing width is $0.02$ Hz. A rectangular smoothing window is used. For all cycle frequencies except zero (non-conjugate), a zero-padding factor of two is used in the FSM.