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4.7 Suboptimal filtering

To extract signals from the 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. The most natural and simplest way to proceed is to use as our statistic the ℱ-statistic where the filters h ′k(t;ζ ) are the approximate ones instead of the optimal ones matched to the signal. In general the functions h′k(t;ζ) will be different from the functions hk (t;ξ ) 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.

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. We get

1( T T ′− 1 ) E [ℱs ] = 2 4 + a ⋅ Q s ⋅ M s ⋅ Qs ⋅ a , (72 )
where
M ′s(k)(l):= (h ′(k)(t;ζ)||h′(l)(t;ζ )), (73 ) (k)(l) ( (k) || ′(l) ) Q s := h (t;ξ) h (t;ζ) .
Let us rewrite the expectation value E [ℱ ] s in the following form,
1( aT ⋅ QT ⋅ M ′− 1⋅ Q ⋅ a ) E [ℱs] = -- 4 + ρ2------s----s-----s--- , (74 ) 2 aT ⋅ M ⋅ a
where ρ is the optimal signal-to-noise ratio. The expectation value E[ℱ ] s reaches its maximum equal to 2 2 + ρ ∕2 when the filter is perfectly matched to the signal. A natural measure of the performance of a suboptimal filter is the quantity FF defined by
∘ --------------------- aT-⋅ QTs-⋅ M-′−s-1⋅ Qs-⋅ a FF := max aT ⋅ M ⋅ a . (75 ) (a,ζ )
We call the quantity FF the generalized fitting factor.

In the case of a signal given by

s(t;A0,ξ) = A0 h(t;ξ), (76 )
the generalized fitting factor defined above reduces to the fitting factor introduced by Apostolatos [9]:
(h (t;ξ)|h′(t;ζ )) FF = max ∘---------------∘------------------. (77 ) ζ (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 Equation (76View Equation), FF is independent of the value of the amplitude A0. For the general signal with 4 constant amplitudes it follows from the Rayleigh principle that the fitting factor is the maximum of the largest eigenvalue of the matrix QT ⋅ M ′− 1 ⋅ Q ⋅ M− 1 over the intrinsic parameters of the signal.

For the case of a signal of the form

s(t;A0,φ0,ξ ) = A0 cos(φ(t;ξ) + φ0), (78 )
where φ0 is a constant phase, the maximum over φ0 in Equation (77View 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 cos(φ(t;ξ)) oscillates rapidly, the fitting factor is approximately given by
[( ∫ ) ( ∫ ) ]1∕2 T0 ′ 2 T0 ′ 2 FF ∼= max cos(φ (t;ξ ) − φ (t;ζ)) dt + sin(φ (t;ξ) − φ (t;ζ)) dt . (79 ) 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 proportional 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 [2691Jump To The Next Citation Point] and for the case spinning neutron stars in [49Jump To The Next Citation Point16].


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