Computational Costs for Spectral Correlation Estimators

Let’s look at the computational costs for spectral-correlation analysis using the three main estimators I’ve previously described on the CSP Blog: the frequency-smoothing method (FSM), the time-smoothing method (TSM), and the strip spectral correlation analyzer (SSCA).

We’ll see that the FSM and TSM are the low-cost options when estimating the spectral correlation function for a few cycle frequencies and that the SSCA is the low-cost option when estimating the spectral correlation function for many cycle frequencies. That is, the TSM and FSM are good options for directed analysis using prior information (values of cycle frequencies) and the SSCA is a good option for exhaustive blind analysis, for which there is no prior information available.

CSP Patent: Tunneling

My colleague Dr. Apurva Mody (of BAE Systems, IEEE 802.22, and the WhiteSpace Alliance) and I have received a patent on a CSP-related invention we call tunneling. The US Patent is 9,755,869 and you can read it here or download it here. We’ve got a journal paper in review and a 2013 MILCOM conference paper (My Papers [38]) that discuss and illustrate the involved ideas. I’m also working on a CSP Blog post on the topic.

Update December 28, 2017: Our Tunneling journal paper has been accepted for publication in the journal IEEE Transactions on Cognitive Communications and Networking. You can download the pre-publication version here.

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

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!

Following the CSP Blog

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More on Pure and Impure Sine Waves

Remember when we derived the cumulant as the solution to the pure $n$th-order sine-wave problem? It sounded good at the time, I hope. But here I describe a curious special case where the interpretation of the cumulant as the pure component of a nonlinearly generated sine wave seems to break down.

In this post we look at direct-sequence spread-spectrum (DSSS) signals, which can be usefully modeled as a kind of PSK signal. DSSS signals are used in a variety of real-world situations, including the familiar CDMA and WCDMA signals, covert signaling, and GPS. My colleague Antonio Napolitano has done some work on a large class of DSSS signals (The Literature [R11, R17, R95]), resulting in formulas for their spectral correlation functions, and I’ve made some remarks about their cyclostationary properties myself here and there (My Papers [16]).

A good thing, from the point of view of modulation recognition, about DSSS signals is that they are easily distinguished from other PSK and QAM signals by their spectral correlation functions. Whereas most PSK/QAM signals have only a single non-conjugate cycle frequency, and no conjugate cycle frequencies, DSSS signals have many non-conjugate cycle frequencies and in some cases also have many conjugate cycle frequencies.

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

In this post I provide some comments on another paper I’ve seen on arxiv.org (I have also received copies of it through email) that relates to modulation classification and cyclostationary signal processing. The paper is by O’Shea et al and is called “Convolutional Radio Modulation Recognition Networks.” You can find it at this link.

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, in general, 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).

Cyclic Polyspectra

In this post we take a first look at the spectral parameters of higher-order cyclostationarity (HOCS). In previous posts, I have introduced the topic of HOCS and have looked at the temporal parameters, such as cyclic cumulants and cyclic moments. Those temporal parameters have proven useful in modulation classification and parameter estimation settings, and will likely be an important part of my ultimate radio-frequency scene analyzer.

The spectral parameters of HOCS have not proven to be as useful as the temporal parameters, unless you include the trivial case where the moment/cumulant order is equal to two. In that case, the spectral parameters reduce to the spectral correlation function, which is extremely useful in CSP (see the TDOA and signal-detection posts for examples).

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 paper in 2015 for a different journal.

I was surprised to see the paper published because I had a lot of criticisms of the original paper, and the other reviewers agreed since the paper was rejected. So I did my job, as did the other reviewers, and we tried to keep a flawed paper from entering the literature, where it would stay forever causing problems for readers.

The editor(s) of the journal Expert Systems with Applications did not ask me to review the paper, so I couldn’t give them the benefit of the work I already put into the manuscript, and apparently the editor(s) did not themselves see sufficient flaws in the paper to merit rejection.

It stings, of course, when you submit a paper that you think is good, and it is rejected. But it also stings when a paper you’ve carefully reviewed, and rejected, is published anyway.

Fortunately I have the CSP Blog, so I’m going on another rant. After all, I already did this the conventional rant-free way.

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.

Cyclostationarity of Digital QAM and PSK

Let’s look into the statistical properties of a class of textbook signals that encompasses digital quadrature amplitude modulation (QAM), phase-shift keying (PSK), and pulse-amplitude modulation (PAM). I’ll call the class simply digital QAM (DQAM), and all of its members have an analytical-signal mathematical representation of the form

$\displaystyle s(t) = \sum_{k=-\infty}^\infty a_k p(t - kT_0 - t_0) e^{i2\pi f_0 t + i \phi_0}. \hfill (1)$

In this model, $k$ is the symbol index, $1/T_0 = f_{sym}$ is the symbol rate, $f_0$ is the carrier frequency (sometimes called the frequency offset), $t_0$ is the symbol-clock phase, and $\phi_0$ is the carrier phase. The finite-energy function $p(t)$ is the pulse function (sometimes called the pulse-shaping function). Finally, the random variable $a_k$ is called the symbol, and has a discrete distribution that is called the constellation.

Model (1) is a textbook signal when the sequence of symbols is independent and identically distributed (IID). This condition rules out real-world communication aids such as periodically transmitted bursts of known symbols, adaptive modulation (where the constellation may change in response to the vagaries of the propagation channel), some forms of coding, etc. Also, when the pulse function $p(t)$ is a rectangle (with width $T_0$), the signal is even less realistic, and therefore more textbook.

We will look at the moments and cumulants of this general model in this post. Although the model is textbook, we could use it as a building block to form more realistic, less textbooky, signal models. Then we could find the cyclostationarity of those models by applying signal-processing transformation rules that define how the cumulants of the output of a signal processor relate to those for the input.

Signal Processing Operations and CSP

It is often useful to know how a signal processing operation affects the probabilistic parameters of a random signal. For example, if I know the power spectral density (PSD) of some signal $x(t)$, and I filter it using a linear time-invariant transformation with impulse response function $h(t)$, producing the output $y(t)$, then what is the PSD of $y(t)$? This input-output relationship is well known and quite useful. The relationship is

$\displaystyle S_y^0(f) = \left| H(f) \right|^2 S_x^0(f). \hfill (1)$

In (1), the function $H(f)$ is the transfer function of the filter, which is the Fourier transform of the impulse-response function $h(t)$.

Because the mathematical models of real-world communication signals can be constructed by subjecting idealized textbook signals to various signal-processing operations, such as filtering, it is of interest to us here at the CSP Blog to know how the spectral correlation function of the output of a signal processor is related to the spectral correlation function for the input. Similarly, we’d like to know such input-output relationships for the cyclic cumulants and the cyclic polyspectra.

Another benefit of knowing these CSP input-output relationships is that they tend to build insight into the meaning of the probabilistic parameters. For example, in the PSD input-output relationship (1), we already know that the transfer function at $f = f_0$ scales the input frequency component at $f_0$ by the complex number $H(f_0)$. So it makes sense that the PSD at $f_0$ is scaled by the squared magnitude of $H(f_0)$. If the filter transfer function is zero at $f_0$, then the density of averaged power at $f_0$ should vanish too.

So, let’s look at this kind of relationship for CSP parameters. All of these results can be found, usually with more mathematical detail, in My Papers [6, 13].