4.2 Signal-to-noise ratio and the Fisher matrix
The detectability of the signal
is determined by the signal-to-noise ratio
. In
general it depends on all the signal’s parameters
and can be computed from [see Eq. (47)]
The signal-to-noise ratio for the signal (32) can be written as
where the components of the matrix
are defined in Eq. (60).
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 by
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 [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
.