Resolution Analysis – Cyclostationary Signal Processing

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

I recently came across the conference paper in the post title (The Literature [R101]). Let’s take a look.

The paper is concerned with “detect[ing] the presence of ACS signals with unknown cycle period.” In other words, blind cyclostationary-signal detection and cycle-frequency estimation. Of particular importance to the authors is the case in which the “period of cyclostationarity” is not equal to an integer number of samples. They seem to think this is a new and difficult problem. By my lights, it isn’t. But maybe I’m missing something. Let me know in the Comments.

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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.

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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.

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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).

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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.

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