This post in the Signal Processing Toolkit series deals with a key mathematical tool in CSP: The Fourier transform. Let’s try to see how the Fourier transform arises from a limiting version of the Fourier series.
In this post, we continue our study of the symmetries of CSP parameters. The second-order parameters–spectral correlation and cyclic correlation–are covered in detail in the companion post, including the symmetries for ‘auto’ and ‘cross’ versions of those parameters.
Here we tackle the generalizations of cyclic correlation: cyclic temporal moments and cumulants. We’ll deal with the generalization of the spectral correlation function, the cyclic polyspectra, in a subsequent post. It is reasonable to me to focus first on the higher-order temporal parameters, because I consider the temporal parameters to be much more useful in practice than the spectral parameters.
This topic is somewhat harder and more abstract than the second-order topic, but perhaps there are bigger payoffs in algorithm development for exploiting symmetries in higher-order parameters than in second-order parameters because the parameters are multidimensional. So it could be worthwhile to sally forth.
This installment of the Signal Processing Toolkit shows how the Fourier series arises from a consideration of representing arbitrary signals as vectors in a signal space. We also provide several examples of Fourier series calculations, interpret the Fourier series, and discuss its relevance to cyclostationary signal processing.
There are some situations in which the spectral correlation function is not the preferred measure of (second-order) cyclostationarity. In these situations, the cyclic autocorrelation (non-conjugate and conjugate versions) may be much simpler to estimate and work with in terms of detector, classifier, and estimator structures. So in this post, I’m going to provide plots of the cyclic autocorrelation for each of the signals in the spectral correlation gallery post. The exceptions are those signals I called feature-rich in the spectral correlation gallery post, such as LTE and radar. Recall that such signals possess a large number of cycle frequencies, and plotting their three-dimensional spectral correlation surface is not helpful as it is difficult to interpret with the human eye. So for the cycle-frequency patterns of feature-rich signals, we’ll rely on the stem-style (cyclic-domain profile) plots in the gallery post.
In this post I discuss the use of cyclostationary signal processing applied to communication-signal synchronization problems. First, just what are synchronization problems? Synchronize and synchronization have multiple meanings, but the meaning of synchronize that is relevant here is something like:
syn·chro·nize: To cause to occur or operate with exact coincidence in time or rate
If we have an analog amplitude-modulated (AM) signal (such as voice AM used in the AM broadcast bands) at a receiver we want to remove the effects of the carrier sine wave, resulting in an output that is only the original voice or music message. If we have a digital signal such as binary phase-shift keying (BPSK), we want to remove the effects of the carrier but also sample the message signal at the correct instants to optimally recover the transmitted bit sequence.
I continue with my foray into machine learning (ML) by considering whether we can use widely available ML tools to create a machine that can output accurate power spectrum estimates. Previously we considered the perhaps simpler problem of learning the Fourier transform. See here and here.
Along the way I’ll expose my ignorance of the intricacies of machine learning and my apparent inability to find the correct hyperparameter settings for any problem I look at. But, that’s where you come in, dear reader. Let me know what to do!
I’ve posted PSK/QAM signals to the CSP Blog. These are the signals I refer to in the post I wrote challenging the machine-learners. In this brief post, I provide links to the data and describe how to interpret the text file containing the signal-type labels and signal parameters.
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.
I’ve decided to post the data set I discuss here to the CSP Blog for all interested parties to use. See the new post on the Data Set. If you do use it, please let me and the CSP Blog readers know how you fared with your experiments in the Comments section of either post. Thanks!
Let’s look at another spectral correlation function estimator: the FFT Accumulation Method (FAM). This estimator is in the time-smoothing category, is exhaustive in that it is designed to compute estimates of the spectral correlation function over its entire principal domain, and is efficient, so that it is a competitor to the Strip Spectral Correlation Analyzer (SSCA) method. I implemented my version of the FAM by using the paper by Roberts et al (The Literature [R4]). If you follow the equations closely, you can successfully implement the estimator from that paper. The tricky part, as with the SSCA, is correctly associating the outputs of the coded equations to their proper values.
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.
Then the temporal resolution of the estimate is approximately , the cycle-frequency resolution is about , and the spectral resolution depends strongly on the particular estimator and its parameters. The resolution product 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.
In this post we discuss ways of estimating -th order cyclic temporal moment and cumulant functions. Recall that for , 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 is greater than . Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order greater than is . Our estimation task is to estimate -th order temporal moment and cumulant functions for using a sampled-data record of length .
Remember when we derived the cumulant as the solution to the pure 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 ).
Let’s talk about another published paper on signal detection involving cyclostationarity and/or cumulants. This one is called “Energy-Efficient Processor for Blind Signal Classification in Cognitive Radio Networks,” (The Literature [R69]), and is authored by UCLA researchers E. Rebeiz and four colleagues.
My focus on this paper it its idea that broad signal-type classes, such as direct-sequence spread-spectrum (DSSS), QAM, and OFDM can be reliably distinguished by the use of a single number: the fourth-order cumulant with two conjugated terms. This kind of cumulant is referred to as the cumulant here at the CSP Blog, and in the paper, because the order is and the number of conjugated terms is .
In this post, we start a discussion of what I consider the ultimate application of the theory of cyclostationary signals: Automatic Modulation Recognition. My relevant papers are My Papers [16,17,25,26,28,30,32,33,38,43,44].
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 , then you’d have a one-dimensional function of cycle frequency and you could then process that function inexpensively to perform detection and classification tasks.
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:
where is the Fourier transform of on . In other words, the usual cyclic periodogram we talk about here on the CSP blog. See, for example, The Literature [R71], Equation (3).
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).