tailieunhanh - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P26

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P26: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. | PROBABILITY AND CORRELATION 111 for it is Eq. 6-16 where we use E expectation to mean the same as averaging assuming that many repetitions have been performed E x n - E x n y n - E y n pxv ----------- ------------------------ 6-16 y VV x n V y ny k 7 E x Iyin ox oy After many repetitions and averaging of pxy the numerator is the expected value of the cross-covariance of x n and y n Eq. 6-15 and the denominator is the square root of the product of the variances of x n and y n or more simply just oxoy. V x n and V y n or ox and oy must both be greater than . This equation can be simplified as E x n y n - E x n E y n Pxy ------ ----- 6-17 M y V x n V y ny k 7 E x n y n E x n E y n ox oy If x n and y n are independent then the numerator of Eq. 6-17 is zero E x n y n E x n E y n 6-18 and pxy . However there are some cases not to be explored here where x n and y n are not independent yet pxy is nevertheless equal to zero. So uncorrelated and independent do not always coincide. Looking at Eq. 6-18 we can guess that this might happen. For further insight about the correlation coefficient see Meyer 1970 Chap. 7 . As an example we will calculate Pxy of Fig. 6-5 using Eq. 6-17 and the time-averaged values instead of expected values because Eq. 6-17 is assumed to be noise-free xy x y Pxy -------------- 6-19 ox oy - nnn ----------------------- The same calculation on Fig. 6-4 produces a value of . 112 DISCRETE-SIGNAL ANALYSIS AND DESIGN This brief introduction to correlation and variance is no more than a get acquainted starting point for these topics and is not intended as a substitute for more advanced study and experience with probability and statistical methods. We are limited to signal sequences that are discrete in both time and frequency domains from 0 to N 1 which makes things a little easier. Mathcad calculates very easily all of the equations in this chapter. REFERENCES Carlson A. B. 1986 Communication .