Χ2 time-frequency discriminator for gravitational wave detection

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 71, Issue 6, 2005, Pages 1-22

  Allen, Bruce ^a  

^a UNIVERSITY OF WISCONSIN MILWAUKEE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; DISCRIMINANT ANALYSIS; FILTER; FORCE; FOURIER ANALYSIS; FREQUENCY ANALYSIS; GRAVITY; MATHEMATICAL COMPUTING; MOLECULAR MECHANICS; PHYSICS; SIGNAL DETECTION; TECHNIQUE; TIME; WAVEFORM;

EID: 18544371968     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.71.062001     Document Type: Article

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43. lsco lies within or below the highest frequencies for which the detector is sensitive, then the stationary-phase approximation and Eq. (4.5) do not hold.
44. 2〉 = 1/p.
45. k〉 vanishes for j ≠ k.
46. A relationship of this form was obtained independently by Jolien Creighton.
47. s.
48. Even if the templates are not orthogonal in each of the p bands, it is possible to apply the Gram-Schmidt procedure in each of these bands separately. But the subsequent template would no longer be an optimal filter for the desired signal.
49. 2 distribution.
50. s, and T′, the nine possible inner products (X̃, Ỹ) are all real.
51. 2 distribution for Gaussian detector noise was first pointed out by Jolien Creighton.
52. For example the Matlab function ncx2inv(r, 2p - 2, λ), where r is the probability that a signal in Gaussian noise would lie below this threshold, for example, 99.9%.
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55. It is reasonable to make this replacement, since the case of interest is large SNR, and for this case the fractional statistical fluctuations in the SNR are small, so that SNR ≈ 〈SNR〉.
56. p is the corresponding probability threshold for a zero mean unit variance Gaussian.
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58. Stanislav Babak pointed out that this case was of interest, because for practical reasons, it may not be possible to construct "equal SNR" intervals.
59. j.
60. The p(p - 1)/2-dimensional orthogonal group O(p) consists of two disconnected components. In one of these components, the determinants of the matrices M are +1 and in the other component the determinants are -1. The first component contains the identity matrix I. Strictly speaking, only the matrices in the component connected to the identity may be thought of as pure rotations. Matrices in the other component are combinations of pure rotations and reflections that do not preserve the "handedness" or orientation of the basis.
61. 2 = P. This is equivalent to saying that P has one zero eigenvalue and that all its other eigenvalues are equal to one. This implies that an orthogonal (unitary) transformation can be found which puts P into block diagonal form with a one sub-block proportional to the p - 1 dimensional identity matrix and the other sub-block vanishing. It follows that for Gaussian detector noise Χ2 has a classical noncentral Χ2 distribution with p - 1 (2p - 2) degrees of freedom.
62. -1 * T
63. ij = f(i - j) is of Topelitz form and depends only upon i - j. One can then show that for large N the diagonal basis is the frequency basis obtained via a Discrete Fourier Transform. So in effect [37] is done in a frequency basis.
64. 2 in a way that automatically includes only the relevant regions of the time-frequency plane, and makes it invariant under transformations such as oversampling that should leave it invariant.
65. j are in one-to-one and monotonic correspondence with the adjacent regions of time.
66. 2 distribution with p rather than p - 1 degrees of freedom was obtained. This mistake was pointed out by Jolien Creighton.
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