# 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 the posts that comment on the engineering literature.

You can see a pre-publication version of my latest CSP journal paper, on “tunneling“, here.

### What is Cyclostationarity?

Introductory post.

Spectral correlation.

Cyclic autocorrelation.

Higher-order cyclostationarity.

### Can I Get Help with my CSP Work Through the CSP Blog?

General rules for getting help.

Second-order estimator development guide.

### What is Higher-Order Cyclostationarity and What are Cyclic Cumulants?

Introduction to higher-order cyclostationarity.

Cyclic cumulants and cyclic moments.

Optional conjugations in higher-order parameters.

The cyclic polyspectrum.

### How do You Estimate the Parameters of Second-Order Cyclostationarity?

The frequency-smoothing method for spectral correlation estimation, one cycle frequency at a time.

The time-smoothing method for spectral correlation estimation, one cycle frequency at a time.

Exhaustive efficient spectral correlation estimation, all cycle frequencies.

Spectral coherence and blind estimation of significant cycle frequencies.

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

I recently came across the 2014 paper in the title of this post. I mentioned it briefly in the post on the periodogram. But I’m going to talk about it a bit more here because this is the kind of thing that makes things a bit harder for people trying to learn about cyclostationarity, which eventually leads to the need for something like the CSP Blog.

The idea behind the paper is that it would be nice to avoid the need for prior knowledge of cycle frequencies when using cycle detectors or the like. If you could just compute the entire spectral correlation function, then collapse it by integrating (summing) over frequency $f$, then you’d have a one-dimensional function of cycle frequency $\alpha$ and you could then process that function inexpensively to perform detection and classification tasks.

# The Periodogram

I’ve been reviewing a lot of technical papers lately and I’m noticing that it is becoming common to assert that the limiting form of the periodogram is the power spectral density or that the limiting form of the cyclic periodogram is the spectral correlation function. This isn’t true. These functions do not become less random (erratic) as the amount of data that is processed increases without limit. On the contrary, they always have large variance. Some form of averaging (temporal or spectral) is needed to permit the periodogram to converge to the power spectrum or the cyclic periodogram to converge to the spectral correlation function (SCF).

In particular, I’ve been seeing things like this:

$\displaystyle S_x^\alpha(f) = \lim_{T\rightarrow\infty} \frac{1}{T} X_T(f+\alpha/2) X_T^*(f-\alpha/2), \hfill (1)$

where $X_T(f+\alpha/2)$ is the Fourier transform of $x(t)$ on $t \in [-T/2, T/2]$. In other words, the usual cyclic periodogram we talk about here on the CSP blog. See, for example, The Literature [R71], Equation (3).

# CSP Estimators: The Strip Spectral Correlation Analyzer

In this post I present a very useful blind cycle-frequency estimator known in the literature as the strip spectral correlation analyzer (SSCA) (The Literature [R3-R5]). We’ve covered the basics of the frequency-smoothing method (FSM) and the time-smoothing method (TSM) of estimating the spectral correlation function (SCF) in previous posts. The TSM and FSM are efficient estimators of the SCF when it is desired to estimate it for one or a few cycle frequencies (CFs). The SSCA, on the other hand, is efficient when we want to estimate the SCF for all CFs.

# 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.

# A Gallery of Spectral Correlation

In this post I provide plots of the spectral correlation for a variety of simulated textbook signals and several collected communication signals. The plots show the variety of cycle-frequency patterns that arise from the disparate approaches to digital communication signaling. The distinguishability of these patterns, combined with the inability to distinguish based on the power spectrum, leads to a powerful set of classification (modulation recognition) features (My Papers [16, 25, 26, 28]).

In all cases, the cycle frequencies are blindly estimated by the strip spectral correlation analyzer (The Literature [R3, R4]) and the estimates used by the FSM to compute the spectral correlation function. MATLAB is then used to plot the magnitude of the spectral correlation and conjugate spectral correlation, as specified by the determined non-conjugate and conjugate cycle frequencies.

There are three categories of signal types in this gallery: textbook signals, collected signals, and feature-rich signals. The latter comprises some collected signals (e.g., LTE) and some simulated radar signals. For the first two signal categories, the three-dimensional surface plots I’ve been using will suffice. But for the last category, the number of cycle frequencies is so large that the three-dimensional surface is difficult to interpret–it is a visual mess. For these signals, I’ll plot the maximum spectral correlation over spectral frequency $f$ versus the detected cycle frequency $\alpha$ (as in this post).

# The Spectral Coherence Function

In this post I introduce the spectral coherence function, or just coherence. It deserves its own post because the coherence is a useful detection statistic for blindly determining significant cycle frequencies of arbitrary data records.

Let’s start with reviewing the standard correlation coefficient $\rho$ defined for two random variables $X$ and $Y$ as

$\rho = \displaystyle \frac{E[(X - m_X)(Y - m_Y)]}{\sigma_X \sigma_Y}, \hfill (1)$

where $m_X$ and $m_Y$ are the mean values of $X$ and $Y$, and $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$. That is,

$m_X = E[X] \hfill (2)$

$m_Y = E[Y] \hfill (3)$

$\sigma_X^2 = E[(X-m_X)^2] \hfill (4)$

$\sigma_Y^2 = E[(Y-m_Y)^2] \hfill (5)$

So the correlation coefficient is the covariance between $X$ and $Y$ divided by the geometric mean of the variances of $X$ and $Y$.