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Sample Power Spectral Density

The Fourier transform of the sample autocorrelation function $ \hat{r}_{v,N}$ (see (5.2)) 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}
$

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\}.
$

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. The PSD $ S_x(\omega)$ can interpreted as a measure of the relative probability that the signal contains energy at frequency $ \omega$ in a window centered about at any given time. An area under the PSD, $ S_x(\omega)\cdot\Delta\omega$, comprises the contribution to the variance of $ x(n)$ from the frequency inverval $ [\omega,\omega+\Delta\omega]$. The total integral of the PSD 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,
$

again assuming $ v(n)$ is zero mean.6.5

Since the sample autocorrelation of white noise approaches an impulse, its PSD approaches a constant, as can be seen in Fig.5.1. 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.


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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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