The accuracy of estimation of the signal’s parameters is determined by Fisher information matrix . The components of in the case of the Gaussian noise can be computed from Eq. (58). For the signal given in Eq. (32) the signal’s parameters (collected into the vector ) split into extrinsic and intrinsic parameters: , where and . It is convenient to distinguish between extrinsic and intrinsic parameter indices. Therefore, we use calligraphic lettering to denote the intrinsic parameter indices: , . The matrix has dimension and it can be written in terms of four block matrices for the two sets of the parameters and ,
We introduce two families of the auxiliary square matrices and (), which depend on the intrinsic parameters only (the indexes within parentheses mean that they serve here as the matrix labels). The components of the matrices and are defined as follows:
The covariance matrix , which approximates the expected covariances of the ML estimators of the parameters , is defined as . Applying the standard formula for the inverse of a block matrix  to Eq. (65), one getsSchur complement of the matrix ) the projected Fisher matrix (onto the space of intrinsic parameters). Because the matrix is the inverse of the intrinsic-parameter submatrix of the covariance matrix , it expresses the information available about the intrinsic parameters that takes into account the correlations with the extrinsic parameters. The matrix is still a function of the putative extrinsic parameters.
We next define the normalized projected Fisher matrix (which is the square matrix) it follows that the minimum value of the component is given by the smallest eigenvalue of the matrix . Similarly, the maximum value of the component is given by the largest eigenvalue of that matrix.
Because the trace of a matrix is equal to the sum of its eigenvalues, the square matrix with componentsreduced Fisher matrix. This matrix is a function of the intrinsic parameters alone. We shall see that the reduced Fisher matrix plays a key role in the signal processing theory that we present here. It is used in the calculation of the threshold for statistically significant detection and in the formula for the number of templates needed to do a given search.
For the case of the signal. The components of are given by
Living Rev. Relativity 15, (2012), 4
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