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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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.

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Cyclostationarity of Direct-Sequence Spread-Spectrum Signals

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.

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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 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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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 up 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 has a 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].

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The Cycle Detectors

Let’s take a look at a class of signal-presence detectors that exploit cyclostationarity and in doing so illustrate the good things that can happen with CSP whenever cochannel interference is present, or noise models deviate from simple additive white Gaussian noise (AWGN). I’m referring to the cycle detectors, the first CSP algorithms I ever studied.

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