Category: Advanced CSP

SCF Estimate Quality: The Resolution Product

What factors influence the quality of a spectral correlation function estimate?

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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The Spectral Coherence Function

Cross correlation functions can be normalized to create correlation coefficients. The spectral correlation function is a cross correlation and its correlation coefficient is called the coherence.

In this post I introduce the spectral coherence function, or just coherence. It deserves its own post because the coherence is a useful detection statistic for blindly determining significant cycle frequencies of arbitrary data records. See the posts on the strip spectral correlation analyzer and the FFT accumulation method for examples.

Let’s start with reviewing the standard correlation coefficient \rho defined for two random variables X and Y as

\rho = \displaystyle \frac{E[(X - m_X)(Y - m_Y)]}{\sigma_X \sigma_Y}, \hfill (1)

where m_X and m_Y are the mean values of X and Y, and \sigma_X and \sigma_Y are the standard deviations of X and Y. That is,

m_X = E[X] \hfill (2)

m_Y = E[Y] \hfill (3)

\sigma_X^2 = E[(X-m_X)^2] \hfill (4)

\sigma_Y^2 = E[(Y-m_Y)^2] \hfill (5)

So the correlation coefficient is the covariance between X and Y divided by the geometric mean of the variances of X and Y.

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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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