resolution product – Cyclostationary Signal Processing

Zero-Padding in Spectral Correlation Estimators

Why does zero-padding help in various estimators of the spectral correlation and spectral coherence functions?

Update to the exchange: May 7, 2021.

Reader Clint posed a great question about zero-padding in the frequency-smoothing method (FSM) of spectral correlation function estimation. The question prompted some pondering on my part, and I went ahead and did some experiments with the FSM to illustrate my response to Clint. The exchange with Clint (ongoing!) was deep and detailed enough that I thought it deserved to be seen by other CSP-Blog readers. One of the problems with developing material, or refining explanations, in the Comments sections of the CSP Blog is that these sections are not nearly as visible in the navigation tools featured on the Blog as are the Posts and Pages.

Resolution in Time, Frequency, and Cycle Frequency for CSP Estimators

Unlike conventional spectrum analysis for stationary signals, CSP has three kinds of resolutions that must be considered in all CSP applications, not just two.

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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The Periodogram

The periodogram is the scaled magnitude-squared finite-time Fourier transform of something. It is as random as its input–it never converges to the power spectrum.

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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The Cycle Detectors

CSP shines when the problem involves strong noise or cochannel interference. Here we look at CSP-based signal-presence detection as a function of SNR and SIR.

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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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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