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Example: FIR-Filtered White Noise

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

$\displaystyle x(n) = v(n) + v(n-1) + \cdots + v(n-8)$ (7.34)

where $ v(n)$ is unit-variance white noise.

Since $ h = [1,1,1,1,1,1,1,1]$ ,

$\displaystyle h\star h = [8,7,6,5,4,3,2,1,0,\ldots]$ (7.35)

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

$\displaystyle (h\star h)(l) = \left\{\begin{array}{ll} 8-l, & \vert l\vert<8 \\ [5pt] 0, & \vert l\vert\ge 8. \\ \end{array} \right.$ (7.36)

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

$\displaystyle r_x(l) \isdef {\cal E}\{x(n)x(n+l)\} = \sigma_v^2 (h\star h)(l)$ (7.37)

The true power spectral density (PSD) is then

$\displaystyle \hbox{\sc DTFT}_\omega(h\star h) = 8^2\cdot\hbox{asinc}^2_{8}(\omega) = \frac{\sin^2(4\omega)}{\sin^2(0.5\omega)}$ (7.38)

Figure 6.3 shows a collection of measured autocorrelations together with their associated smoothed-PSD estimates.

Figure 6.3: Averaged sample autocorrelations (biased) and their Fourier transforms (smoothed PSD estimates), for FIR-filtered white noise.
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Matlab for Welch's Method