Overview

Let x = [ x ₁ x ₂ x _M] ^T be a column vector containing M unknown parameters that are to be estimated and y = [ y ₁ y ₂ y _N] ^T be a set of noisy measurements that are linearly related to x as described by the expression:

where n = [ n ₁ n ₂ n _N] ^T is a vector describing the errors corrupting the N measurements, and H is an N M matrix describing the connection between the measurements and x.

The maximum likelihood estimate of x, denoted as , is defined as (see, for example, [1]):

where p( y/ x) is the probability density function of the measurement y for a fixed value of x.

If the measurement errors, { n _i}, for i = 1, , N, are identically Gaussian distributed with zero-mean and variance ? ², and furthermore if errors for different measurements are statistically independent, then (A.2) becomes:

The solution to (A.3) can readily be found by first differentiating y ? H ² with respect to :

and then setting this quantity equal to zero to obtain:

where it is assumed that the matrix inverse involved exists (i.e., that H ^T H is not singular).