Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

Industry's highest performing at the lowest power DSPs now as low as $5.00*
Start development today!
*volume pricing for 10ku

Chapters

See Also

Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Beginning Statistical Signal Processing
       White Noise
          Estimator Variance
             Sample-Variance Variance

Chapter Contents:

Search Spectral Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Sample-Variance Variance

Consider now the sample variance estimator

$\displaystyle \hat{\sigma}_x^2(n) \isdefs \frac{1}{M}\sum_{m=0}^{M-1}x^2(n-m) \isdefs \hat{r}_{x(n)}(0) $

where the mean is assumed to be $ \mu_x ={\cal E}\left\{x(n)\right\}=0$, and $ \hat{r}_{x(n)}(l)$ denotes the unbiased sample autocorrelation of $ x$ based on the $ M$ samples leading up to and including time $ n$. Since $ \hat{r}_{x(n)}(0)$ is unbiased, $ {\cal E}\left\{[\hat{\sigma}_x^2(n)]^2\right\} = {\cal E}\left\{\hat{r}_{x(n)}^2(0)\right\} = \sigma_x^2$. The variance of this estimator is then given by

\begin{eqnarray*} \mbox{Var}\left\{\hat{\sigma}_x^2(n)\right\} &\isdef & {\cal E... ...\ &=& {\cal E}\left\{[\hat{\sigma}_x^2(n)]^2-\sigma_x^4\right\} \end{eqnarray*}

where

\begin{eqnarray*} {\cal E}\left\{[\hat{\sigma}_x^2(n)]^2\right\} &=& \frac{1}{M... ...}\sum_{m_1=0}^{M-1}\sum_{m_1=0}^{M-1}r_{x^2}(\vert m_1-m_2\vert) \end{eqnarray*}

The autocorrelation of $ x^2(n)$ need not be simply related to that of $ x(n)$. However, when $ x$ is assumed to be Gaussian white noise, simple relations do exist. For example, when $ m_1\ne m_2$,

$\displaystyle {\cal E}\left\{x^2(n-m_1)x^2(n-m_2)\right\} = {\cal E}\left\{x^2(n-m_1)\right\}{\cal E}\left\{x^2(n-m_2)\right\}=\sigma_x^2\sigma_x^2= \sigma_x^4. $

by the independence of $ x(n-m_1)$ and $ x(n-m_2)$, and when $ m_1=m_2$, the fourth moment is given by $ {\cal E}\left\{x^4(n)\right\} = 3\sigma_x^4$. More generally, we can simply label the $ k$th moment of $ x(n)$ as $ \mu_k = {\cal E}\left\{x^k(n)\right\}$, where $ k=1$ corresponds to the mean, $ k=2$ corresponds to the variance (when the mean is zero), etc.

When $ x(n)$ is assumed to be Gaussian white noise, we have

$\displaystyle {\cal E}\left\{x^2(n-m_1)x^2(n-m_2)\right\} = \left\{\begin{array... ...sigma_x^4, & m_1\ne m_2 \\ [5pt] 3\sigma_x^4, & m_1=m_2 \\ \end{array}\right. $

so that the variance of our estimator for the variance of Gaussian white noise is

   Var$\displaystyle \left\{\hat{\sigma}_x^2(n)\right\} = \frac{M3\sigma_x^4 + (M^2-M)\sigma_x^4}{M^2} - \sigma_x^4 = \zbox {\frac{2}{M}\sigma_x^4} $

Again we see that the variance of the estimator declines as $ 1/M$.

The same basic analysis as above can be used to estimate the variance of the sample autocorrelation estimates for each lag, and/or the variance of the power spectral density estimate at each frequency.

As mentioned above, to obtain a grounding in statistical signal processing, see references such as [191,115,91].

Previous: Sample-Mean Variance
Next: Gaussian Function Properties

Order a Hardcopy of Spectral Audio Signal Processing

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.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )