DISCRETE-SIGNAL ANALYSIS AND DESIGN- P25

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P25:Electronic circuit analysis and design projects often involve time-domainand frequency-domain characteristics that are difÞcult to work with usingthe traditional and laborious mathematical pencil-and-paper methods offormer eras. This is especially true of certain nonlinear circuits and sys-tems that engineering students and experimenters may not yet be com-fortable with.
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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P25106 DISCRETE-SIGNAL ANALYSIS AND DESIGN Equation (6-12) is assumed, as usual, to be one record of a steady-staterepetitive sequence. Note that the “ßip” of x (n) does not occur as it did inEq. (5-4) for convolution. We only want to compare the sequence with anexact time-shifted replica. Note also the division by N because C A (τ) isby deÞnition a time-averaged value for each τ and convolution is not. Assuch, it measures the average power commonality of the two sequences asa function of their separation in time. When the shift τ = 0, C A (τ) = C A (0)and Eq. (6-12), reduces to Eq. (6-5), which is by deÞnition the averagepower for (x + εx )n . Figure 6-4 is an example of the autocorrelation of a sequence in part (a)(no noise) and the identical shifted (τ = 13) sequence in part b, c. Thereare three overlaps, and the values of the autocorrelation vs overlap, whichis the sum of partial products (polynomial multiplication), are shown inpart (c). The correlation value for τ = 13 is (1)(0.1875) + (0.9375)(0.125) + (0.875)(0.0625) CA (13) = = 0.0225 16This value is indicated in part (c), third from the left and also third fromthe right. This procedure is repeated for each value of τ. At τ = 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 occursat τ = 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 autocorrelationis performed. We can also see that x 1 (n) and x 2 (n) have 16 positionsand the autocorrelation sequence has 33 = (16 + 16 + 1) positions, whichdemonstrates the same smoothing and stretching effect in auto correla-tion that we saw in convolution. As we decided in Chapter 5, the extraeffort in circular correlation is not usually necessary, and we can workaround it.Cross-CorrelationTwo different waveforms can be completely or partially dependent orcompletely independent. In each of these cases the two noise-contaminatedwaveforms are time-shifted with respect to each other in increments of τ. PROBABILITY AND CORRELATION 107 N := 16 n := −N, −N + 1.. N τ := −N, −N + 1.. N x1(n) := 0 x2(n) := 0 N−1 ∑ (x1(n)⋅x2(τ + n)) n 1 1− if n ≥ 0 z (τ) := 1 − n if n ≥ 0 16 N 16 n=0 0 if n > N 0 if n > N 1 1 1.0 0.9375 0.875 x1(n) 0.5 0 −15 −10 −5 0 n 5 10 15 (a) 1x2(n + 13) 0.5 0.1875 0.125 0.0625 0 −15 −10 −5 0 n 5 10 15 (b) 0.4 0.365 0.3 z(τ) 0.2 0.1 0.0225 0.0225 0 −15 −10 −5 0 5 10 15 τ (c) Figure 6-4 Example of autocorrelation.108 DISCRETE-SIGNAL ANALYSIS AND DESIGNEquation (6-13) is the basic equation for the cross-correlation of twodifferent waves x (n) and y(n): N −1 1 CC (τ) = (x + εx )n (y + εy )(n+τ) (6-13) N n=0 We have pointed out one major difference between the correlation andconvolution equations. In correlation there is no “ßip” of one of the waves,as explained in Chapter 7. This is in agreement with the desire to comparea wave with a time-shifted replica of itself or a replica of two differentwaves, one of which is time-shifted with respect to the other. In the caseof convolution we derived a useful relationship for the Fourier transformof convolution. In Chapter 7, correlation leads to another useful idea inlinear analysis, called the Wiener-Khintchine (see Google, e.g.) principle. Figure 6-5 (with no noise) is an example of cross-correlation. The twotime-domain sequences can have different lengths, different shapes, anddifferent amplitude scale factors. The maximum value of cross-correlationoccurs at τ = − 3 and − 4, which is quite a bit different from Fig. 6-4.At τ ...