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 , :

- sample autocorrelation
- sample power spectral density (PSD)

The PSD is the Fourier transform of the autocorrelation function:

(7.1) |

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

As indicated above, when estimating the true autocorrelation
from observed samples of
, the resulting estimate
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
converges to
the true PSD
as
.

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

- biased autocorrelation
- unbiased autocorrelation

(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 is simply the squared-magnitude of the Fourier transform of :

(7.3) |

This chapter is concerned with noise-like signals that ``last forever'',

Since this gives an unbiased estimator of the true autocorrelation (as will be discussed below), we see that the ``bias'' in 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:

(7.5) |

Since the Fourier transform of a Bartlett window is (§3.5), we find that the DTFT of the biased autocorrelation is a smoothed version of the unbiased PSD (convolved with ).

To avoid terminology confusion below, remember that the
``autocorrelation'' of a signal
is defined here (and in
§2.3.7) to mean the maximally simplified case
, *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
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.

- Introduction to Noise

- Spectral Characteristics of Noise
- White Noise

- Sample Autocorrelation
- Sample Power Spectral Density
- Biased Sample Autocorrelation
- Smoothed Power Spectral Density
- Cyclic Autocorrelation
- Practical Bottom Line
- Why an Impulse is Not White Noise
- The Periodogram

- Welch's Method

- Welch's Method with Windows

- Filtered White Noise
- Example: FIR-Filtered White Noise
- Example: Synthesis of 1/F Noise (Pink Noise)
- Example: Pink Noise Analysis

- Processing Gain
- The Panning Problem

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University