Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Sample Power Spectral Density

The Fourier transform of the sample autocorrelation function $ \hat{r}_{v,N}$ (see (6.6)) is defined as the sample power spectral density (PSD):

$\displaystyle {\hat S}_{v,N}(\omega) \isdef \hbox{\sc DTFT}_\omega\{\hat{r}_{v,N}\} \isdef \sum_{n=-\infty}^\infty \hat{r}_{v,N}(n)e^{-j\omega n}$ (7.11)

This definition coincides with the classical periodogram when $ \hat{r}_{v,N}$ is weighted differently (by a Bartlett window).

Similarly, the true power spectral density of a stationary stochastic processes $ v(n)$ is given by the Fourier transform of the true autocorrelation function $ r_v(l)$ , i.e.,

$\displaystyle S_{v}(\omega) = \hbox{\sc DTFT}_\omega\{r_v\}.$ (7.12)

For real signals, the autocorrelation function is always real and even, and therefore the power spectral density is real and even for all real signals.

An area under the PSD expressed as a function of frequency $ f$ in Hz, $ S_x(2\pi f)\cdot\Delta f$ , comprises the contribution to the variance of $ x(n)$ from the frequency interval $ [f,f+\Delta f]$ . The total integral of the PSD then gives the total variance:

$\displaystyle \int_{-\pi}^\pi S_v(\omega) \frac{d\omega}{2\pi} = \int_{-0.5}^{0.5} S_v(2\pi f) df = r_v(0) = \sigma_v^2,$ (7.13)

again assuming $ v(n)$ is zero mean.7.5

Since the sample autocorrelation of white noise approaches an impulse, its sample PSD approaches a constant, as can be seen in Fig.6.1. Its true PSD is precisely constant (variance divided by $ 2\pi$ ). This means that white noise contains all frequencies in equal amounts. Since white light is defined as light of all colors in equal amounts, the term ``white noise'' is seen to be analogous.

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``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