Introduction to Optimal Estimation by E. W. Kamen PhD, J. K. Su PhD (auth.)

By E. W. Kamen PhD, J. K. Su PhD (auth.)

This e-book, built from a suite of lecture notes via Professor Kamen, and because extended and subtle by way of either authors, is an introductory but complete examine of its box. It includes examples that use MATLAB® and lots of of the issues mentioned require using MATLAB®. the first target is to supply scholars with an intensive assurance of Wiener and Kalman filtering besides the advance of least squares estimation, greatest probability estimation and a posteriori estimation, in keeping with discrete-time measurements. within the research of those estimation ideas there's powerful emphasis on how they interrelate and healthy jointly to shape a scientific improvement of optimum estimation. additionally incorporated within the textual content is a bankruptcy on nonlinear filtering, concentrating on the prolonged Kalman filter out and a recently-developed nonlinear estimator in keeping with a block-form model of the Levenberg-Marquadt Algorithm.

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Introduction to Optimal Estimation

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Note that the autocorrelation is afunction oftwo integer variables i and j, with -00 < i < 00 and -00 < j < 00. The autocorrelation function R x ( i, j) measures the correlation between signal sampies. Two examples are given below. 18 Autocorrelation Function Suppose that x(n + 1) = ax(n), where a is a nonzero real number. Then x(n) = anx(O) for n = 0, ±1, ±2, ... 62) Note that if ais negative, Rx(i,j) is negative for all integers i and j such that i+j is odd. 19 Purely Random Signal Suppose that the random variables comprising the random signal x( n) are independent and all have zero mean.

Such a random signal is sometimes said to be purely random. Suppose that the random variables comprising a random signal x (n) are independent and have means E[x(n)] = cn , where the Cn are arbitrary nonzero 46 Random Signals and Systems with Random Inputs real numbers. Then E[x(i)x(j)] = CiCj i- 0 for i i- j, and thus even though x(i) and x(j) are independent (far i i- j), and therefare are uncorrelated, E[x(i)x (j)] is nonzero. So there appears to be "correlation" between x( i) and x (j). This is a result of the nonzero mean; in fact, a nonzero mean can be interpreted as the existence of a deterministic part of the random signal x(n).

50) 42 Random Signals and Systems with Random Inputs and variance The verification of these expressions is left to a homework problem. Given jointly distributed RVs x and Y with conditional density function fy(Ylx = x), the conditional expectation of Y given x = x, denoted by E[Ylx = x], is defined by E[Ylx Let I]! I: I: = x] = yfy(Ylx = x) dy. (x) = yfy(Ylx = x) dy. 53) and thus the conditional expectation can be viewed as a random variable. 52) of I]! (x) that E [E [Ylx]] = E[y]. 54) is left to the reader.

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