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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P25:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former eras. This is especially true of certain nonlinear circuits and sys- tems that engineering students and experimenters may not yet be com- fortable with. | 106 DISCRETE-SIGNAL ANALYSIS AND DESIGN Equation 6-12 is assumed as usual to be one record of a steady-state repetitive sequence. Note that the flip of x n does not occur as it did in Eq. 5-4 for convolution. We only want to compare the sequence with an exact time-shifted replica. Note also the division by N because Ca t is by definition a time-averaged value for each t and convolution is not. As such it measures the average power commonality of the two sequences as a function of their separation in time. When the shift t 0 Ca t Ca 0 and Eq. 6-12 reduces to Eq. 6-5 which is by definition the average power for x ex n. Figure 6-4 is an example of the autocorrelation of a sequence in part a no noise and the identical shifted t 13 sequence in part b c. There are three overlaps and the values of the autocorrelation vs overlap which is the sum of partial products polynomial multiplication are shown in part c . The correlation value for t 13 is 1 0.1875 0.9375 0.125 0.875 0.0625 EA 13 ------------------------ ------------------------ 0.0225 16 This value is indicated in part c third from the left and also third from the right. This procedure is repeated for each value of t. At t 0 parts a and b are fully overlapping and the value shown in part c is 0.365. For these two identical sequences the maximum autocorrelation occurs at t 0 and the value 0.365 is the average power in the sequence. Compare Fig. 6-4 with Fig. 5-4 to see how circular autocorrelation is performed. We can also see that x 1 n and x2 n have 16 positions and the autocorrelation sequence has 33 16 16 1 positions which demonstrates the same smoothing and stretching effect in auto correlation that we saw in convolution. As we decided in Chapter 5 the extra effort in circular correlation is not usually necessary and we can work around it. Cross-Correlation Two different waveforms can be completely or partially dependent or completely independent. In each of these cases the two noise-contaminated waveforms are .
