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 ,

where is an matrix with components , is an matrix with components , and finally is matrix with components .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:

Making use of the definitions (60) and (66)–(67) one can write the more explicit form of the matrices , , and , The notation introduced above means that the matrix can be thought of as a row matrix made of column matrices . Thus, the general formula for the component of the matrix is The general component of the matrix is given byThe 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 [90] to Eq. (65), one gets

where the matrices , , and can be expressed in terms of the matrices , , and as follows: In Eqs. (74) – (76) we have introduced the matrix: We call the matrix (which is the Schur 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)

where is the signal-to-noise ratio. Making use of the definition (77) and Eqs. (71)–(72) we can show that the components of this matrix can be written in the form where is the matrix defined as From the Rayleigh principle [90] 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 components

expresses the information available about the intrinsic parameters, averaged over the possible values of the extrinsic parameters. Note that the factor is specific to the case of extrinsic parameters. We shall call the reduced 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 normalized projected Fisher matrix is independent of the extrinsic parameters and , and it is equal to the reduced matrix [102]. The components of are given by where is the Fisher matrix for the signal .
Living Rev. Relativity 15, (2012), 4
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