Example: FIR-Filtered White Noise

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Let's estimate the autocorrelation and power spectral density of the ``moving average'' (MA) process

$\displaystyle x(n) = v(n) + v(n-1) + \cdots + v(n-8)$

where is unit-variance white noise.

Since ,

$\displaystyle h\star h = [8,7,6,5,4,3,2,1,0,\ldots]$

for nonnegative lags ( $l\ge0$ ). More completely, we can write

$\displaystyle (h\star h)(l) = \left\{\begin{array}{ll} 8-l, & \vert l\vert<8 \\ [5pt] 0, & \vert l\vert\ge 8. \\ \end{array}\right.$

Thus, the autocorrelation of is a triangular pulse centered on lag 0. The true (unbiased) autocorrelation is given by

$\displaystyle r_x(l) \isdef {\cal E}\{x(n)x(n+l)\} = \sigma_v^2 (h\star h)(l)$

The true power spectral density (PSD) is then

$\displaystyle \hbox{\sc DTFT}_\omega(h\star h) = 8^2\cdot\hbox{asinc}^2_{8}(\omega) = \frac{\sin^2(4\omega)}{\sin^2(0.5\omega)}$

Figure 5.3 shows a collection of measured autocorrelations together with their associated smoothed-PSD estimates.

**Figure 5.3:** Averaged sample autocorrelations (biased) and their Fourier transforms (smoothed PSD estimates), for FIR-filtered white noise.
$\includegraphics[width=\twidth]{eps/tcolored}$

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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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