Practical Control Engineering by David M. Koenig

Author:David M. Koenig
Language: eng
Format: epub
Publisher: McGraw-Hill Education
Published: 2009-03-02T16:00:00+00:00

FIGURE 8-4 Autocorrelation of a white noise sequence.

If the data stream symbolized by ω is unautocorrelated, rω(n) will be small for all n. On the other hand, if there is a periodic component in data stream then rω(n) will have a significant value for the value of n (and multiples of it) corresponding to the period of the oscillating component in the data stream.

The rw(n) of the white noise sequence plotted in Fig. 8-1 is shown in Fig. 8-4. Notice that the autocorrelation for a lag index of zero is unity because the ith sample is completely autocorrelated with itself. For the other lag indices the rω(n) bounces insignificantly around zero.

After adding a sine wave to the noisy data in Fig. 8-1, a new signal is created that also looks like white noise. This new signal is shown in Fig. 8-5. The histogram of this sequence is shown in Fig. 8-6. The autocorrelation of this second data sequence is shown in Fig. 8-7. The peaks show that there is a periodic component that appears to have a period of approximately six or seven samples. That is, samples spaced apart by 6 or 7 samples are autocorrelated. In fact, the sine wave buried in the white noise has a period of 6.5 sample intervals. The time domain plot of the data in Fig. 8-5 gives no hint as to the presence of a periodic component because of the background noise. However, the autocorrelation plot shows peaks because the averages of the lagged products tend to allow the noise to cancel out.

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