Example: FIR-Filtered White Noise

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

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

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

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

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

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

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

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

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

The true power spectral density (PSD) is then

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

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.