Tag: Spectral Correlation Function

Signal Selectivity

We can estimate the spectral correlation function of one signal in the presence of another with complete temporal and spectral overlap provided the signal has a unique cycle frequency.

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 received RF signals, whether cochannel or not, is accurately 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)

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Introduction to Higher-Order Cyclostationarity

Why do we need or care about higher-order cyclostationarity? Because second-order cyclostationarity is insufficient for our signal-processing needs in some important cases.

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.

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The Spectral Correlation Function for Rectangular-Pulse BPSK

Let’s make the spectral correlation function a little less abstract by showing it for a simple textbook BPSK signal.

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}.

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The Spectral Correlation Function

Spectral correlation in CSP means that distinct narrowband spectral components of a signal are correlated-they contain either identical information or some degree of redundant information.

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 (or simply the variance when the mean is zero), 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:

scf_schematic
Figure 1. Illustration of the concept of spectral correlation. The time series represented by the narrowband spectral components centered at f-A/2 and f+A/2 are downconverted to zero frequency and their correlation is measured. When A=0, the result is the power spectral density function, otherwise it is referred to as the spectral correlation function. It is non-zero only for a countable set of numbers \{A\}, which are equal to the frequencies of sine waves that can be generated by quadratically transforming the data.

Let’s define narrowband spectral component to mean the output of a bandpass filter applied to a signal, where the bandwidth of the filter is much smaller than the bandwidth of the 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.

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Welcome to the CSP Blog!

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Thank you for visiting the CSP blog.

The purpose of this blog is to talk about cyclostationary signals and cyclostationary signal processing (CSP). I’ve been working in the area for nearly thirty years, and over that time I’ve received a lot of requests for help with CSP code and algorithms. I thought it was time to put some of the basics out on the web so everybody could benefit. And I’m hoping to learn from you too.

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In future posts, I’ll be showing how to create simple cyclostationary signals, write code for basic CSP estimators and detectors, and discuss papers in the literature.

What is cyclostationarity? It is a property of a class of mathematical models for a large number of signals in the world, most notably man-made modulated radio-frequency signals, like those used by cell phones, broadcast AM/FM/TV, satellites, WiFi modems, and many more systems. The mathematical models can be quite accurate, so we also say that cyclostationarity is a property of the real-world signals themselves.

The key aspect of the model is that cyclostationary signals have probabilistic parameters that vary periodically with time. Traditionally, signals are treated as stationary, which means their parameters do not vary with time. What are these ‘probabilistic parameters?’ Quantities like the mean value, the variance, and higher-order moments. These quantities are defined for both the time-domain signal and for its frequency-domain representation. So we have ‘temporal moments‘ and ‘spectral moments.’ The second-order spectral moment is also called the spectral correlation function (SCF). The SCF is central to many CSP algorithms; a display of the SCF is shown above for a simple bandlimited binary phase-shift keyed (BPSK) signal.

The well-known noise- and interference-tolerance properties of CSP algorithms follow from the periodically time-varying nature of the signals’ parameters.

The most common difficulty I’ve encountered is when a researcher is developing a CSP estimator and is having trouble applying it to their data set. The researchers almost always skip the step of first applying the estimator to a signal with a perfectly known cyclostationary parameters. So, in the next post I’ll describe how to make the simplest digital CS signal, which has known temporal and spectral moments of all orders, so that we can test CSP estimators by comparing their output to the known correct result.

I encourage readers to point out my errors in the comments of my posts and to suggest topics they would like to see covered in future posts. Also, let me know about your application and interests so I can continue to learn too.

I hope you enjoy your time here at the CSP Blog!

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