Signal processing system and method for performing a nonlinear transformation

Claim 2

A specialized or programmed signal processing system for approximating a mean and a covariance of a function Y of a random variable X of a physical system having a dimensionality n, based upon a measured signal for the random variable X, comprising: a. means for measuring the random variable X a plurality of times to determine a plurality of measured signals k ; b. means for calculating an n x n covariance matrix F from said plurality of measured signals k ; c. means for measuring X to determine the measured signal ; d. means for calculating a scaled covariance matrix F'=αF wherein α is a positive real number; e. means for calculating a matrix square root √F' for F' f. means for calculating Xχ k for k=1 to 2n+1 as follows: 1. for k=0 χ 0 = 2. for k=1 to n χ k =χ 0 -(√F') k where (√F') k indicates the vector formed from the matrix component of √F' along the kth row, and

Claim 3

for k=n+1 to 2n χ k =χ 0 +(√F') k-n g. means for calculating the values of the function Y k =Y(χ k ) for k=0 to 2n; h. means for calculating the weighted average ##EQU6## wherein κ is a real number; i. means for calculating the weighted outer product WOP (Y(χ i (i=0 to 2n))) as: ##EQU7## where the superscript T indicates the transpose; means for approximating covariance of Y() based upon WOP; and means for transmitting signals representing the weighted average and the approximated covariance of Y() to a response system for responding to the physical system.

Claim 4

for k=n+1 to 2n χ k =χ 0 +(√F') k-n g. means for calculating the values of the function Y k =Y(χ k ) for k=0 to 2n; h. means for calculating the weighted average ##EQU6## wherein κ is a real number; i. means for calculating the weighted outer product WOP (Y(χ i (i=0 to 2n))) as: ##EQU7## where the superscript T indicates the transpose; means for approximating covariance of Y() based upon WOP; and means for transmitting signals representing the weighted average and the approximated covariance of Y() to a response system for responding to the physical system.

Claim 5

A system according to claim 1, wherein said means for approximating comprises means for changing the matrix values of WOP by: 1. maintaining all eigenvectors of WOP remain unchanged 2. maintaining all positive eigenvalues of WOP remain unchanged; and 3. replacing each negative eigenvalue of WOP by a non-negative real number.

Claim 6

A system according to claim 2, wherein the means for changing replaces each negative eigenvalue of WOP by zero.

Claim 7

A system according to claim 2, wherein the means for changing replaces each negative eigenvalue of WOP by its absolute value.

Claim 8

A system according to claim 2, wherein the means for changing further comprises means for determining the eigenvectors and corresponding eigenvalues for WOP.

Claim 9

A system according to claim 1, wherein said χ k have the covariance F and mean χ.

Claim 10

A system according to claim 1 wherein κ is between -n and n.

Claim 11

A system according to claim 1, wherein α is between 0 and 2n.

Claim 12

A system according to claim 1, wherein α equals the dimensionality n of X.

Claim 13

A system according to claim 9, wherein κ equals 3 minus α.

Claim 14

A signal processing system for estimating the expected value and covariance of a nonlinear function of a random variable of a physical system when a predetermined covariance for the random variable of the physical system is available, comprising: (1) means for measuring a value of the random variable thereby providing a signal representing the measured value; (2) means, coupled to the means for measuring, for generating a discrete distribution of a first set of values which has the measured value of the random variable as its mean and the predetermined covariance; (3) means, coupled to the means for generating, for determining a second set of values by applying the nonlinear function to the first set of values; (4) means, coupled to the means for determining, for calculating an estimated expected value and an estimated covariance based upon the second set of values; (5) means, coupled to the means for calculating, for transmitting signals representing the estimated expected value and the estimated covariance; and (6) means, coupled to the means for transmitting, for responding to the physical system based upon the estimated expected value and estimated covariance.

Claim 15

A signal processing system for estimating the expected value and covariance of a nonlinear function of a random variable of a physical system when a predetermined covariance for the random variable of the physical system is available, comprising: (1) means for measuring a value of the random variable thereby providing a signal representing the measured value; (2) means, coupled to the means for measuring, for generating a discrete distribution of a first set of values which has the measured value of the random variable as its mean and the predetermined covariance; (3) means, coupled to the means for generating, for determining a second set of values by applying the nonlinear function to the first set of values; (4) means, coupled to the means for determining, for calculating an estimated expected value and an estimated covariance based upon the second set of values; (5) means, coupled to the means for calculating, for transmitting signals representing the estimated expected value and the estimated covariance; and (6) means, coupled to the means for transmitting, for responding to the physical system based upon the estimated expected value and estimated covariance.

Claim 16

A system according to claim 11, wherein the means for measuring a value of the random variable comprises means for measuring a variable which is a function of one of position an object of the physical system, temperature of an object of the physical system, pressure of a fluid of the physical system, chemical composition of a material of the physical system, and a voltage in the physical system.

Claim 17

A system according to claim 11, wherein said discrete distribution is symmetric about the measured value of the random variable.

Claim 18

A computer readable memory storing instructions for configuring or configuring a signal processing system to perform a method of estimating the expected value and covariance of a nonlinear function of a random variable when a predetermined covariance for the random variable is available, by configuring the signal processing system to provide: (1) means for measuring a value of the random variable, thereby providing a signal representing the measured value; (2) means for generating a discrete distribution of a first set of values which has the measured value of the random variable as its mean and has the predetermined covariance; (3) means for determining a second set of values by applying the nonlinear function to the first set of values; (4) means for calculating an estimated expected value and an estimated covariance based upon the second set of values; (5) means for transmitting signals representing the estimated expected value and the estimated covariance; and (6) means for responding to the physical system based upon the estimated expected value and the estimated covariance.

Claim 19

A computer readable memory storing instructions for configuring or configuring a signal processing system to perform a method of estimating the expected value and covariance of a nonlinear function of a random variable when a predetermined covariance for the random variable is available, by configuring the signal processing system to provide: (1) means for measuring a value of the random variable, thereby providing a signal representing the measured value; (2) means for generating a discrete distribution of a first set of values which has the measured value of the random variable as its mean and has the predetermined covariance; (3) means for determining a second set of values by applying the nonlinear function to the first set of values; (4) means for calculating an estimated expected value and an estimated covariance based upon the second set of values; (5) means for transmitting signals representing the estimated expected value and the estimated covariance; and (6) means for responding to the physical system based upon the estimated expected value and the estimated covariance.

Claim 20

A system according to claim 14, wherein the means for measuring a value of the random variable comprises means for measuring a variable which is a function of one of distance to an object of the physical system, temperature of an object of the physical system, pressure of a fluid of the physical system, chemical composition of a material of the physical system, and voltage of the physical system.

Claim 21

A system according to claim 14, wherein said discrete distribution is symmetric about the measured value of the random variable.