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Let's estimate the autocorrelation and power spectral density of the
``moving average'' (MA) process
![$\displaystyle x(n) = v(n) + v(n-1) + \cdots + v(n-8)$](img1194.png) |
(7.34) |
where
is unit-variance white noise.
Since
,
![$\displaystyle h\star h = [8,7,6,5,4,3,2,1,0,\ldots]$](img1196.png) |
(7.35) |
for nonnegative lags (
). More completely, we can write
![$\displaystyle (h\star h)(l) = \left\{\begin{array}{ll} 8-l, & \vert l\vert<8 \\ [5pt] 0, & \vert l\vert\ge 8. \\ \end{array} \right.$](img1198.png) |
(7.36) |
Thus, the autocorrelation of
is a triangular pulse centered on lag 0.
The true (unbiased) autocorrelation is given by
![$\displaystyle r_x(l) \isdef {\cal E}\{x(n)x(n+l)\} = \sigma_v^2 (h\star h)(l)$](img1199.png) |
(7.37) |
The true power spectral density (PSD) is then
![$\displaystyle \hbox{\sc DTFT}_\omega(h\star h) = 8^2\cdot\hbox{asinc}^2_{8}(\omega) = \frac{\sin^2(4\omega)}{\sin^2(0.5\omega)}$](img1200.png) |
(7.38) |
Figure 6.3 shows a collection of measured autocorrelations together
with their associated smoothed-PSD estimates.
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