The Periodogram

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

White noise is a random signal in which all frequencies are equally present over time. It is analogous to white light which, as Isaac Newton first realized, consists of the superposition of all visible light frequencies. — Click for https://ccrma.stanford.edu/~jos/sasp/White_Noise.html

Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

Welch's method for estimating power spectral densities may be defined as the average of periodograms of successive signal blocks over time. — Click for https://ccrma.stanford.edu/~jos/sasp/Welch_s_Method.html

The amplitude envelope, or relatively slowly-changing outline of a sound waveform, makes for a useful first approximation of the instantaneous loudness of the sound. — Click for http://en.wikipedia.org/wiki/ADSR_envelope

The Fourier Transform (FT) gives the (complex) spectrum of a continuous-time signal of any length. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Transform_FT_Inverse.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

In signal processing, noise is a random signal. Broadband noise contains energy over a wide range of frequencies. — Click for http://en.wikipedia.org/wiki/White_noise

Fast Fourier Transforms (FFT) are fast algorithms for computing the Discrete Fourier Transform (DFT) — Click for https://ccrma.stanford.edu/~jos/mdft/Fast_Fourier_Transform_FFT.html

Click for https://ccrma.stanford.edu/~jos/sasp/Bartlett_Triangular_Window.html#10345

Click for https://ccrma.stanford.edu/~jos/sasp/Autocorrelation_I.html#37540

Noise (in signal processing) is a random signal. In the language of statistical signal processing, a noise signal is typically modeled as 'stochastic process', which is in turn defined as a sequence of random variables. — Click for https://ccrma.stanford.edu/~jos/sasp/What_Noise.html

The Discrete Time Fourier Transform (DTFT) is the appropriate Fourier transform for discrete-time signals of arbitrary length. It can be obtained as the limit of a Discrete Fourier Transform (DFT) as its length goes to infinity. — Click for https://ccrma.stanford.edu/~jos/mdft/Discrete_Time_Fourier_Transform.html

Click for https://ccrma.stanford.edu/~jos/sasp/Sample_Power_Spectral_Density.html

The Periodogram

The periodogram is based on the definition of the power spectral density (PSD) (see Appendix C). Let denote a windowed segment of samples from a random process , where the window function (classically the rectangular window) contains nonzero samples. Then the periodogram is defined as the squared-magnitude DTFT of divided by [120, p. 65]:^7.7

$\displaystyle P_{x,M}(\omega)$	$\displaystyle \isdef$	$\displaystyle \frac{1}{M} \left\vert\hbox{\sc DTFT}(x_w)\right\vert^2 = \frac{1}{M} \left\vert\sum_{n=0}^{M-1} x_w(n)e^{-j\omega n}\right\vert^2 \protect$	(7.23)
	$\displaystyle \longleftrightarrow$	$\displaystyle \frac{1}{M} \sum_{n=0}^{M-1-\vert l\vert} x_w(n)x_w(n+\vert l\vert), \quad l\in\mathbb{Z}.$

$\displaystyle {\cal E}\left\{ \lim_{M\to\infty} P_{x,M}(\omega) \right\} = S_x(\omega)$

(7.24)

$\displaystyle P_{x,M}(\omega) = M \hbox{asinc}_M^2 \ast {\hat S}_{x,M}(\omega).$

(7.25)

In practice, we of course compute a sampled periodogram $S_x(\omega_k)$ , $\omega_k = 2\pi/N$ , replacing the DTFT with the length $N\ge M$ FFT. Essentially, the steps of §6.9 include computation of the periodogram.

As mentioned in §6.9, a problem with the periodogram of noise signals is that it too is random for most purposes. That is, while the noise has been split into bands by a Fourier transform, it has not been averaged in any way that reduces randomness within each band, and each band produces a nearly independent random value. In fact, it can be shown [120] that $P_{x,M}(\omega)$ is a random variable whose standard deviation (square root of its variance) is comparable to its mean. This is not a measure of noise level in each band.

The trick to noise spectrum analysis is that many sample power spectra (squared-magnitude FFTs) must be averaged to obtain a ``stable'' statistical estimate of the noise spectral envelope. This is the essence of Welch's method for spectrum analysis of stochastic processes, as elaborated in §6.12 below. The right column of Fig.6.1 illustrates the effect of this averaging for white noise .

The Periodogram

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University