4.2 Signal-to-noise ratio and the Fisher matrix

The detectability of the signal h(t;𝜃) is determined by the signal-to-noise ratio ρ. In general it depends on all the signal’s parameters 𝜃 and can be computed from [see Eq. (47View Equation)]
∘ --------------- ρ(𝜃) = (h(t;𝜃)|h(t;𝜃)). (63 )
The signal-to-noise ratio for the signal (32View Equation) can be written as
∘ ------------ ρ(a,ξ) = aT ⋅ M (ξ) ⋅ a, (64 )
where the components of the matrix M (ξ ) are defined in Eq. (60View Equation).

The accuracy of estimation of the signal’s parameters is determined by Fisher information matrix Γ. The components of Γ in the case of the Gaussian noise can be computed from Eq. (58View Equation). For the signal given in Eq. (32View Equation) the signal’s parameters (collected into the vector 𝜃) split into extrinsic and intrinsic parameters: 𝜃 = (a,ξ), where a = (a1,...,an) and ξ = (ξ1,...,ξm ). It is convenient to distinguish between extrinsic and intrinsic parameter indices. Therefore, we use calligraphic lettering to denote the intrinsic parameter indices: ξ𝒜, 𝒜 = 1,...,m. The matrix Γ has dimension (n + m ) × (n + m ) and it can be written in terms of four block matrices for the two sets of the parameters a and ξ,

( ) Γ (a,ξ) = Γ aa(ξ) Γ aξ(a,ξ) , (65 ) Γ aξ(a,ξ)T Γ ξξ(a,ξ )
where Γ aa is an n × n matrix with components (∂h ∕∂ai|∂h∕∂aj ) (i,j = 1,...,n ), Γ aξ is an n × m matrix with components (∂h∕∂ai |∂h ∕∂ξ𝒜 ) (i = 1,...,n, 𝒜 = 1, ...,m ), and finally Γ ξξ is m × m matrix with components (∂h∕∂ ξ |∂h∕∂ ξ ) 𝒜 ℬ (𝒜, ℬ = 1,...,m ).

We introduce two families of the auxiliary n × n square matrices F (𝒜 ) and S(𝒜 ℬ) (𝒜, ℬ = 1,...,m), which depend on the intrinsic parameters ξ only (the indexes 𝒜, ℬ within parentheses mean that they serve here as the matrix labels). The components of the matrices F (𝒜) and S (𝒜 ℬ) are defined as follows:

( ||∂h (t;ξ)) F(𝒜)ij(ξ) := hi(t;ξ)||---j----- , i,j = 1,...,n, 𝒜 = 1,...,m, (66 ) ∂ ξ𝒜 (∂h (t;ξ )||∂h (t;ξ)) S(𝒜ℬ)ij(ξ) := ---i-----||--j------ , i,j = 1,...,n, 𝒜,ℬ = 1,...,m. (67 ) ∂ξ𝒜 ∂ξℬ
Making use of the definitions (60View Equation) and (66View Equation)–(67View Equation) one can write the more explicit form of the matrices Γ aa, Γ aξ, and Γ ξξ,
Γ (ξ) = M (ξ ), (68 ) aa ( ) Γ aξ(a, ξ) = F(1)(ξ ) ⋅ a ⋅⋅⋅ F(m )(ξ ) ⋅ a , (69 ) ( ) aT ⋅ S (11)(ξ ) ⋅ a ⋅⋅⋅ aT ⋅ S(1m)(ξ) ⋅ a Γ ξξ(a, ξ) = ( ..................................) . (70 ) T T a ⋅ S(m1)(ξ) ⋅ a ⋅⋅⋅ a ⋅ S(mm)(ξ) ⋅ a
The notation introduced above means that the matrix Γ aξ can be thought of as a 1 × m row matrix made of n × 1 column matrices F(𝒜) ⋅ a. Thus, the general formula for the component of the matrix Γ aξ is
n ( ) ( ) ∑ Γ aξ i𝒜 = F(𝒜) ⋅ a i = F(𝒜)ijaj, 𝒜 = 1,...,m, i = 1, ...,n. (71 ) j=1
The general component of the matrix Γ ξξ is given by
∑n ∑n (Γ ) = aT ⋅ S ⋅ a = S a a , 𝒜, ℬ = 1,...,m. (72 ) ξξ 𝒜ℬ (𝒜 ℬ) (𝒜ℬ)ij i j i=1 j=1

The covariance matrix C, which approximates the expected covariances of the ML estimators of the parameters 𝜃, is defined as −1 Γ. Applying the standard formula for the inverse of a block matrix [90Jump To The Next Citation Point] to Eq. (65View Equation), one gets

( ) Caa(a,ξ) Caξ(a,ξ ) C (a,ξ) = T , (73 ) Caξ(a,ξ) C ξξ(a,ξ )
where the matrices Caa, Caξ, and C ξξ can be expressed in terms of the matrices Γ aa = M, Γ aξ, and Γ ξξ as follows:
-- Caa(a,ξ ) = M (ξ)− 1 + M (ξ)−1 ⋅ Γ aξ(a,ξ) ⋅Γ (a,ξ)− 1 ⋅ Γ aξ(a,ξ)T ⋅ M (ξ)−1, (74 ) −1 -- −1 Caξ(a,ξ ) = − M (ξ) ⋅ Γ aξ(a,ξ) ⋅Γ (a,ξ) , (75 ) C (a,ξ ) = Γ-(a,ξ)−1. (76 ) ξξ
In Eqs. (74View Equation) – (76View Equation) we have introduced the m × m matrix:
-- Γ (a, ξ) := Γ ξξ(a,ξ ) − Γ aξ(a,ξ )T ⋅ M (ξ)− 1 ⋅ Γ aξ(a,ξ). (77 )
We call the matrix -- Γ (which is the Schur complement of the matrix M) the projected Fisher matrix (onto the space of intrinsic parameters). Because the matrix Γ- is the inverse of the intrinsic-parameter submatrix Cξξ of the covariance matrix C, it expresses the information available about the intrinsic parameters that takes into account the correlations with the extrinsic parameters. The matrix -- Γ is still a function of the putative extrinsic parameters.

We next define the normalized projected Fisher matrix (which is the m × m square matrix)

-- -- Γ n(a,ξ) := Γ (a,ξ)-, (78 ) ρ(a,ξ)2
where ρ is the signal-to-noise ratio. Making use of the definition (77View Equation) and Eqs. (71View Equation)–(72View Equation) we can show that the components of this matrix can be written in the form
T (Γ-(a,ξ )) = a--⋅ A-(𝒜-ℬ)(ξ-) ⋅ a, 𝒜,ℬ = 1,...,m, (79 ) n 𝒜ℬ aT ⋅ M (ξ) ⋅ a
where A(𝒜ℬ) is the n × n matrix defined as
A (ξ ) := S (ξ) − F (ξ)T ⋅ M (ξ)−1 ⋅ F (ξ), 𝒜,ℬ = 1,...,m. (80 ) (𝒜ℬ) (𝒜 ℬ) (𝒜) (ℬ )
From the Rayleigh principle [90Jump To The Next Citation Point] it follows that the minimum value of the component -- (Γ n(a,ξ))𝒜ℬ is given by the smallest eigenvalue of the matrix M −1 ⋅ A(𝒜ℬ). Similarly, the maximum value of the component -- (Γ n(a,ξ))𝒜ℬ is given by the largest eigenvalue of that matrix.

Because the trace of a matrix is equal to the sum of its eigenvalues, the m × m square matrix ^Γ with components

( ) 1 ( −1 ) ^Γ (ξ )𝒜 ℬ := -Tr M (ξ) ⋅ A(𝒜ℬ)(ξ) , 𝒜,ℬ = 1,...,m, (81 ) n
expresses the information available about the intrinsic parameters, averaged over the possible values of the extrinsic parameters. Note that the factor 1∕n is specific to the case of n extrinsic parameters. We shall call ^Γ 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

h(t;A0,ϕ0, ξ) = A0g(t;ξ) cos(ϕ(t;ξ) − ϕ0), (82 )
the normalized projected Fisher matrix -- Γ n is independent of the extrinsic parameters A0 and ϕ0, and it is equal to the reduced matrix ^Γ [102Jump To The Next Citation Point]. The components of ^Γ are given by
(Γ 0)ϕ0𝒜 (Γ 0)ϕ0ℬ ^Γ ð’œ ℬ = (Γ 0)𝒜ℬ −--------------, (83 ) (Γ 0)ϕ0ϕ0
where Γ 0 is the Fisher matrix for the signal g(t;ξ)cos (ϕ (t;ξ ) − ϕ0 ).
  Go to previous page Go up Go to next page