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

In this post, we look at the ability of various CSP estimators to distinguish cycle frequencies, temporal changes in cyclostationarity, and spectral features. These abilities are quantified by the resolution properties of CSP estimators.

### Resolution Parameters in CSP: Preview

Consider performing some CSP estimation task, such as using the frequency-smoothing method, time-smoothing method, or strip spectral correlation analyzer method of estimating the spectral correlation function. The estimate employs $T$ seconds of data.

Then the temporal resolution $\Delta t$ of the estimate is approximately $T$, the cycle-frequency resolution $\Delta \alpha$ is about $1/T$, and the spectral resolution $\Delta f$ depends strongly on the particular estimator and its parameters. The resolution product $\Delta f \Delta t$ was discussed in this post. The fundamental result for the resolution product is that it must be very much larger than unity in order to obtain an SCF estimate with low variance.

## CSP Estimators: Cyclic Temporal Moments and Cumulants

In this post we discuss ways of estimating $n$-th order cyclic temporal moment and cumulant functions. Recall that for $n=2$, cyclic moments and cyclic cumulants are usually identical. They differ when the signal contains one or more finite-strength additive sine-wave components. In the common case when such components are absent (as in our recurring numerical example involving rectangular-pulse BPSK), they are equal and they are also equal to the conventional cyclic autocorrelation function provided the delay vector is chosen appropriately.

The more interesting case is when the order $n$ is greater than $2$. Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order $n$ greater than $2$ is $4$. Our estimation task is to estimate $n$-th order temporal moment and cumulant functions for $n \ge 4$ using a sampled-data record of length $T$.

## 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.

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Here are the highlights:

## Automatic Spectral Segmentation

In this post, I discuss a signal-processing algorithm that has almost nothing to do with cyclostationary signal processing. Almost. The topic is automated spectral segmentation, which I also call band-of-interest (BOI) detection. When attempting to perform automatic radio-frequency scene analysis (RFSA), we may be confronted with a data block that contains multiple signals in a large number of distinct frequency subbands. Moreover, these signals may be turning on an off within the data block. To apply our cyclostationary signal processing tools effectively, we would like to isolate these signals in time and frequency to the greatest extent possible using linear time-invariant filtering (for separating in the frequency dimension) and time-gating (for separating in the time dimension). Then the isolated signal components can be processed serially.

It is very important to remember that even perfect spectral and temporal segmentation will not solve the cochannel-signal problem. It is perfectly possible that an isolated subband will contain more that one cochannel signal.

The basics of my BOI-detection approach are published in a 2007 conference paper (My Papers [32]). I’ll describe this basic approach, illustrate it with examples relevant to RFSA, and also provide a few extensions of interest, including one that relates to cyclostationary signal processing.

## 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!

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## 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 that I’ll ask below.

## Cyclic Temporal Cumulants

In this post I continue the development of the theory of higher-order cyclostationarity (My Papers [5,6]) that I began here. It is largely taken from my doctoral work (download my dissertation here).

This is a long post. To make it worthwhile, I’ve placed some movies of cyclic-cumulant estimates at the end. Or just skip to the end now if you’re impatient!

In my work on cyclostationary signal processing (CSP), the most useful tools are those for estimating second-order statistics, such as the cyclic autocorrelation, spectral correlation function, and spectral coherence function. However, as we discussed in the post on Textbook Signals, there are some situations (perhaps only academic; see my question in the Textbook post) for which higher-order cyclostationarity is required. In particular, a probabilistic approach to blind modulation recognition for ideal (textbook) digital QAM, PSK, and CPM requires higher-order cyclostationarity because such signals have similar or identical spectral correlation functions and PSDs. (Other high-SNR non-probabilistic approaches can still work, such as blind constellation extraction.)

Recall that in the post introducing higher-order cyclostationarity, I mentioned that one encounters a bit of a puzzle when attempting to generalize experience with second-order cyclostationarity to higher orders. This is the puzzle of pure sine waves (My Papers [5]). Let’s look at pure and impure sine waves, and see how they lead to the probabilistic parameters widely known as cyclic cumulants.

## 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 for illustrating the cycle-frequency patterns and the behavior of the spectral correlation function over frequency. 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 magnitude over spectral frequency $f$ versus the detected cycle frequency $\alpha$ (as in this post).

A complementary gallery of cyclic autocorrelation functions can be found here.

## Textbook Signals

What good is having a blog if you can’t offer a rant every once in a while? In this post I talk about what I call textbook signals, which are mathematical models of communication signals that are used by many researchers in statistical signal processing for communications.

We’ve already encountered, and used frequently, the most common textbook signal of all: rectangular-pulse BPSK with independent and identically distributed (IID) bits. We’ve been using this signal to illustrate the cyclostationary signal processing concepts and estimators as they have been introduced. It’s a good choice from the point of view of consistency of all the posts and it is easy to generate and to understand. However, it is not a good choice from the perspective of realism. It is rare to encounter a textbook BPSK signal in the practice of signal processing for communications.

I use the term textbook because the textbook signals can be found in standard textbooks, such as Proakis (The Literature [R44]). Textbook signals stand in opposition to signals used in the world, such as OFDM in LTE, slotted GMSK in GSM, 8PAM VSB with synchronization bits in ATSC-DTV, etc.

Typical communication signals combine a textbook signal with an access mechanism to yield the final physical-layer signal–the signal that is actually transmitted (My Papers [11], [16]). What is important for us, here on the CSP blog, is that this combination usually results in a signal with radically different cyclostationarity than the textbook component. So it is not enough to understand textbook signals’ cyclostationarity. We must also understand the cyclostationarity of the real-world signal, which may be sufficiently complex to render mathematical modeling and analysis impossible (at least for me).

## SCF Estimate Quality: The Resolution Product

The two non-parametric spectral-correlation estimators we’ve looked at so far–the frequency-smoothing and time-smoothing methods–require the choice of key estimator parameters. These are the total duration of the processed data block, $T$, and the spectral resolution $F$.

For the frequency-smoothing method (FSM), an FFT with length equal to the data-block length $T$ is required, and the spectral resolution is equal to the width $F$ of the smoothing function $g(f)$. For the time-smoothing method (TSM), multiple FFTs with lengths $T_{tsm} = T / K$ are required, and the frequency resolution is $1/T_{tsm}$ (in normalized frequency units).

The choice for the block length $T$ is partially guided by practical concerns, such as computational cost and whether the signal is persistent or transient in nature, and partially by the desire to obtain a reliable (low-variance) spectral correlation estimate. The choice for the frequency (spectral) resolution is typically guided by the desire for a reliable estimate.

## 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 $Y$.

## CSP Estimators: The Time Smoothing Method

In a previous post, we introduced the frequency-smoothing method (FSM) of spectral correlation function (SCF) estimation. The FSM convolves a pulse-like smoothing window $g(f)$ with the cyclic periodogram to form an estimate of the SCF. An advantage of the method is that is allows fine control over the spectral resolution of the SCF estimate through the choice of $g(f)$, but the drawbacks are that it requires a Fourier transform as long as the data-record undergoing processing, and the convolution can be expensive. However, the expense of the convolution can be mitigated by using rectangular $g(f)$.

In this post, we introduce the time-smoothing method (TSM) of SCF estimation. Instead of averaging (smoothing) the cyclic periodogram over spectral frequency, multiple cyclic periodograms are averaged over time. When the non-conjugate cycle frequency of zero is used, this method produces an estimate of the power spectral density, and is essentially the Bartlett spectrum estimation method. The TSM can be found in My Papers [6] (Eq. (54)), and other places in the literature.

## CSP Estimators: The Frequency-Smoothing Method

In this post I describe a basic estimator for the spectral correlation function (SCF): the frequency-smoothing method (FSM). The FSM is a way to estimate the SCF for a single value of cycle frequency. Recall from the basic theory of the cyclic autocorrelation and SCF that the SCF is obtained by infinite-time averaging of the cyclic periodogram or by infinitesimal-resolution frequency averaging of the cyclic periodogram. The FSM is merely a finite-time/finite-resolution approximation to the SCF definition.

One place the FSM can be found is in (My Papers [6]), where I introduce time-smoothed and frequency-smoothed higher-order cyclic periodograms as estimators of the cyclic polyspectrum. When the cyclic polyspectrum order is set to $n = 2$, the cyclic polyspectrum becomes the spectral correlation function, so the FSM discussed in this post is just a special case of the more general estimator in [6, Section VI.B].

## Signal Selectivity

In this post I describe and illustrate the most important property of cyclostationary statistics: signal selectivity. The idea is that the cyclostationary parameters for a single signal can be estimated for that signal even when it is corrupted by strong noise and cochannel interferers. Cochannel means that the interferer occupies a frequency band that partially or completely overlaps the frequency band for the signal of interest.

A mixture of signals, whether cochannel or not, is modeled by the simple sum of the signals, as in

$x(t) = s_1(t) + s_2(t) + \ldots + s_K(t) + w(t), \hfill (1)$

where $w(t)$ is additive noise. We can write this more compactly as

$x(t) = \displaystyle \sum_{k=1}^K s_k(t) + w(t). \hfill (2)$

## Introduction to Higher-Order Cyclostationarity

We’ve seen how to define second-order cyclostationarity in the time- and frequency-domains, and we’ve looked at ideal and estimated spectral correlation functions for a synthetic rectangular-pulse BPSK signal. In future posts, we’ll look at how to create simple spectral correlation estimators, but in this post I want to introduce the topic of higher-order cyclostationarity (HOCS).  This post is more conceptual in nature; for mathematical details about HOCS, see the posts on cyclic cumulants and cyclic polyspectra. Estimators of higher-order parameters, such as cyclic cumulants and cyclic moments, are discussed in this post.

To contrast with HOCS, we’ll refer to second-order parameters such as the cyclic autocorrelation and the spectral correlation function as parameters of second-order cyclostationarity (SOCS).

The first question we might ask is Why do we care about HOCS? And one answer is that SOCS does not provide all the statistical information about a signal that we might need to perform some signal-processing task. There are two main limitations of SOCS that drive us to HOCS.

## The Spectral Correlation Function for Rectangular-Pulse BPSK

In this post, I show the non-conjugate and conjugate spectral correlation functions (SCFs) for the rectangular-pulse BPSK signal we generated in a previous post. The theoretical SCF can be analytically determined for a rectangular-pulse BPSK signal with independent and identically distributed bits (see My Papers [6] for example or The Literature [R1]). The cycle frequencies are, of course, equal to those for the CAF for rectangular-pulse BPSK. In particular, for the non-conjugate SCF, we have cycle frequencies of $\alpha = k f_{bit}$ for all integers $k$, and for the conjugate SCF we have $\alpha = 2f_c \pm k f_{bit}$.

## The Spectral Correlation Function

Spectral correlation is perhaps the most widely used characterization of the cyclostationarity property. The main reason is that the computational efficiency of the FFT can be harnessed to characterize the cyclostationarity of a given signal or data set in an efficient manner. And not just efficient, but with a reasonable total computational cost, so that one doesn’t have to wait too long for the result.

Just as the normal power spectrum is actually the power spectral density, or more accurately, the spectral density of time-averaged power (variance), the spectral correlation function is the spectral density of time-averaged correlation (covariance). What does this mean? Consider the following schematic showing two narrowband spectral components of an arbitrary signal:

The sequence of shaded rectangles on the left are meant to imply a time-series corresponding to the output of a bandpass filter centered at $f-A/2$ with bandwidth $B.$ Similarly, the sequence of shaded rectangles on the right imply a time-series corresponding to the output of a bandpass filter centered at $f+A/2$ with bandwidth $B.$

## The Cyclic Autocorrelation for Rectangular-Pulse BPSK

The cyclic autocorrelation for rectangular-pulse BPSK can be derived as a relatively simple closed-form expression (see My Papers [6] for example or The Literature [R1]). It can be estimated in a variety of ways, which we will discuss in future posts. The non-conjugate cycle frequencies for the signal are harmonics of the bit rate, $k f_{bit}$, and the conjugate cycle frequencies are the non-conjugate cycle frequencies offset by the doubled carrier, or $2f_c + k f_{bit}$.

Recall that the simulated rectangular-pulse BPSK signal has $10$ samples per bit, or a bit rate of $0.1$, and a carrier offset of $0.05$, all in normalized units (meaning the sampling rate is unity). We’ve previously selected a sampling rate of $1.0$ MHz to provide a little physical realism. This means the bit rate is $100$ kHz and the carrier offset frequency is $50$ kHz. From these numbers, we see that the non-conjugate cycle frequencies are $k 100$ kHz, and that the conjugate cycle frequencies are $2(50) + k 100$ kHz, or $100 + k 100$ kHz.