### Example: FIR-Filtered White Noise

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

 (7.34)

where is unit-variance white noise.

Since ,

 (7.35)

for nonnegative lags ( ). More completely, we can write

 (7.36)

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

 (7.37)

The true power spectral density (PSD) is then

 (7.38)

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

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Matlab for Welch's Method