4.5 Suboptimal filtering

To extract gravitational-wave signals from the detector’s noise one very often uses filters that are not optimal. We may have to choose an approximate, suboptimal filter because we do not know the exact form of the signal (this is almost always the case in practice) or in order to reduce the computational cost and to simplify the analysis. In the case of the signal of the form given in Eq. (32View Equation) the most natural and simplest way to proceed is to use as detection statistic the ℱ-statistic where the filters h ′k(t;ζ ) (k = 1,...,n) are the approximate ones instead of the optimal ones hk (t;ξ) (k = 1, ...,n) matched to the signal. In general the functions h′(t;ζ) k will be different from the functions h (t;ξ) k used in optimal filtering, and also the set of parameters ζ will be different from the set of parameters ξ in optimal filters. We call this procedure the suboptimal filtering and we denote the suboptimal statistic by ℱs. It is defined as [see Eq. (62View Equation)]
ℱ [x;ζ ] := 1N [x; ζ]T ⋅ M (ζ)−1 ⋅ N [x;ζ], (111 ) s 2 s s s
where the data-dependent n × 1 column matrix N s and the square n × n matrix M s have components [see Eq. (60View Equation)]
′ ′ ′ Nsi[x;ζ] := (x (t)|hi(t;ζ)), Msij(ζ) := (hi(t;ζ)|hj(t;ζ)), i,j = 1,...,n. (112 )

We need a measure of how well a given suboptimal filter performs. To find such a measure we calculate the expectation value of the suboptimal statistic ℱs in the case where the data contains the gravitational-wave signal, i.e., when x(t;a,ξ) = n(t) + h(t;a,ξ). We get

E [ℱ [x(t;a,ξ);ζ]] = 1(n + aT ⋅ Q (ξ,ζ) ⋅ M (ζ )− 1 ⋅ Q (ξ, ζ)T ⋅ a), (113 ) s 2 s s s
where we have introduced the matrix Qs with components
Q (ξ,ζ) := (h (t;ξ)|h′(t;ζ)), i,j = 1,...,n. (114 ) sij i j
Let us rewrite the expectation value (113View Equation) in the following form,
( T −1 T ) [ ] 1- 2a--⋅ Qs(ξ,ζ-) ⋅ Ms-(ζ)-⋅ Qs(ξ,ζ-)-⋅ a E ℱs[x(t;a,ξ);ζ] = 2 n + ρ(a,ξ) aT ⋅ M (ξ) ⋅ a , (115 )
where ρ is the optimal signal-to-noise ratio [given in Eq. (64View Equation)]. This expectation value reaches its maximum equal to (n + ρ2)∕2 when the filter is perfectly matched to the signal. Therefore, a natural measure of the performance of a suboptimal filter is the quantity FF defined by
----------------------------------- ∘ T −1 T FF (ξ) := max a-⋅-Qs(ξ,ζ)-⋅ Ms-(ζ-)-⋅ Qs-(ξ,ζ)-⋅ a. (116 ) (a,ζ) aT ⋅ M (ξ) ⋅ a
We call the quantity FF the generalized fitting factor. From the Rayleigh principle, it follows that the generalized fitting factor is the maximum of the largest eigenvalue of the matrix M (ξ)−1 ⋅ Qs(ξ,ζ ) ⋅ Ms (ζ)−1 ⋅ Qs(ξ,ζ )T over the intrinsic parameters of the signal.

In the case of a gravitational-wave signal given by

s(t;A0,ξ ) = A0h (t;ξ), (117 )
the generalized fitting factor defined above reduces to the fitting factor introduced by Apostolatos [13]:
(h(t;ξ)|h ′(t;ζ)) FF (ξ) = maxζ ∘----------------∘---′-------′-----. (118 ) (h(t;ξ)|h(t;ξ)) (h (t;ζ )|h (t;ζ))
The fitting factor is the ratio of the maximal signal-to-noise ratio that can be achieved with suboptimal filtering to the signal-to-noise ratio obtained when we use a perfectly matched, optimal filter. We note that for the signal given by Eq. (117View Equation), FF is independent of the value of the amplitude A0.

For the case of a signal of the form

s(t;A0,ϕ0,ξ ) = A0 cos(ϕ(t;ξ) + ϕ0), (119 )
where ϕ0 is a constant phase, the maximum over ϕ0 in Eq. (118View Equation) can be obtained analytically. Moreover, assuming that over the bandwidth of the signal the spectral density of the noise is constant and that over the observation time cosϕ(t;ξ) oscillates rapidly, the fitting factor is approximately given by
[( ∫ T0 )2 (∫ T0 )2 ]1∕2 FF (ξ) ∼= max cos(ϕ (t;ξ) − ϕ′(t;ζ))dt + sin(ϕ (t;ξ ) − ϕ ′(t;ζ))dt .(120 ) ζ 0 0

In designing suboptimal filters one faces the issue of how small a fitting factor one can accept. A popular rule of thumb is accepting FF = 0.97. Assuming that the amplitude of the signal and consequently the signal-to-noise ratio decreases inversely proportionally to the distance from the source this corresponds to 10% loss of the signals that would be detected by a matched filter.

Proposals for good suboptimal (search) templates for the case of coalescing binaries are given in [35, 134Jump To The Next Citation Point] and for the case-spinning neutron stars in [65Jump To The Next Citation Point, 18].

  Go to previous page Go up Go to next page