Spectrum Analysis of Noise

Spectrum analysis of noise is generally more advanced than the analysis of ``deterministic'' signals such as sinusoids, because the mathematical model for noise is a so-called stochastic process, which is defined as a sequence of random variables (see §C.1). More broadly, the analysis of signals containing noise falls under the subject of statistical signal processing [121]. Appendix C provides a short tutorial on this topic. In this chapter, we will cover only the most basic practical aspects of spectrum analysis applied to signals regarded as noise.

In particular, we will be concerned with estimating two functions from an observed noise sequence $x(n)$ , $n=0,1,2,\ldots N-1$ :

• sample autocorrelation $\hat{r}_x(l)$
• sample power spectral density (PSD) ${\hat S}_x(\omega)$

When the number $N$ of observed samples of $x(n)$ approaches infinity, we assume that the sample autocorrelation $\hat{r}_x(l)$ approaches the true autocorrelation $r_x(l)$ (defined formally in Appendix C). Note that we do not need to know anything about the true autocorrelation function--only that the sample autocorrelation approaches it in the limit as $N\to\infty$ .

The PSD is the Fourier transform of the autocorrelation function:

$$\zbox {S_x(\omega) = \hbox{\sc DTFT}_\omega(r_x)}$$ (7.1)

We'll accept this as nothing more than the definition of the PSD. When the signal $x$ is real, both $r_x$ and $S_x$ are real and even.

As indicated above, when estimating the true autocorrelation $r_x(l)$ from observed samples of $x$ , the resulting estimate $\hat{r}_x(l)$ will be called a sample autocorrelation. Likewise, the Fourier transform of a sample autocorrelation will be called a sample PSD. It is assumed that the sample PSD ${\hat S}_x(\omega)$ converges to the true PSD $S_x(\omega)$ as $N\to\infty$ .

We will also be concerned with two cases of the autocorrelation function itself:

• biased autocorrelation
• unbiased autocorrelation

The biased autocorrelation,^7.1or simply autocorrelation, will be taken to be the simplest case computationally: If $x(n)$ is a discrete-time signal, where $n$ ranges over all integers, then as described in §2.3.7, the autocorrelation of $x$ at ``lag $l$ '' is given by

$$(x \star x)(l) = \sum_{n=-\infty}^{\infty} \overline{x(n)}x(n+l)$$ (7.2)

Note that this definition of autocorrelation is workable only for signals having finite support (nonzero over a finite number of samples). As shown in §2.3.7, the Fourier transform of the autocorrelation of $x$ is simply the squared-magnitude of the Fourier transform of $x$ :

$$\hbox{\sc DTFT}_\omega(x \star x) = \vert X(\omega)\vert^2$$ (7.3)

This chapter is concerned with noise-like signals $x$ that ``last forever'', i.e., they exhibit infinite support. As a result, we cannot work only with $x\star x$ , and will introduce the unbiased sample autocorrelation function

$$\hat{r}_x(l) \isdef \frac{1}{N-\vert l\vert} \sum_{n=0}^{N-1} \overline{x(n)}x(n+l). \protect$$ (7.4)

Since this gives an unbiased estimator of the true autocorrelation (as will be discussed below), we see that the ``bias'' in $x\star x$ consists of a multiplication of the unbiased sample autocorrelation by a Bartlett (triangular) window (see §3.5). This means we can convert the biased autocorrelation to unbiased form by simply ``dividing out'' this window:

$$\hat{r}_x(l) = \left\{\begin{array}{ll} \frac{(x\star x)(l)}{N-\vert l\vert}, & \vert l\vert<N \\ [5pt] 0, & \vert l\vert\ge N \\ \end{array} \right.$$ (7.5)

Since the Fourier transform of a Bartlett window is $\hbox{asinc}^2$ (§3.5), we find that the DTFT of the biased autocorrelation is a smoothed version of the unbiased PSD (convolved with $\hbox{asinc}^2$ ).

To avoid terminology confusion below, remember that the ``autocorrelation'' of a signal $x$ is defined here (and in §2.3.7) to mean the maximally simplified case $x\star x = \sum_n x(n)x(n+l)$ , i.e., without normalization of any kind. This definition of ``autocorrelation'' is adopted to correspond to everyday practice in digital signal processing. The term ``sample autocorrelation'', on the other hand, will refer to an unbiased autocorrelation estimate. Thus, the ``autocorrelation'' of a signal $x\star x$ can be viewed as a Bartlett-windowed (unbiased-)sample-autocorrelation. In the frequency domain, the autocorrelation transforms to the magnitude-squared Fourier transform, and the sample autocorrelation transforms to the sample power spectral density.