Best NET-CC: Extension of BCC search to GW interferometer networks Archana Pai Albert Einstein Institute, Potsdam with Éric Chassande-Mottin AstroParticule et Cosmologie, Paris CNRS, Observatoire de la Côte de Azur, Nice (20th Dec. 2006, GWDAW-11, AEI Berlin)
Talk Outline • Detection problem of arbitrary chirp with known phase : Max LR methods ⇒ Linear Least SQuare Problem can be ill-posed! = Consequence for binary inspiral detection? • Formulate network max LLR with Synthetic Streams ⇒ Implementation of Best Net-CC =
Talk Outline • Detection problem of arbitrary chirp with known phase : Max LR methods ⇒ Linear Least SQuare Problem can be ill-posed! = Consequence for binary inspiral detection? • Formulate network max LLR with Synthetic Streams ⇒ Implementation of Best Net-CC = Best CC (BCC) – Time frequency based detection of unmodelled chirps Look for TF track MLR = Longest Path in TF
Signal Model : Smooth chirp GW polarisations: h + = A (1+cos 2 ǫ ) cos( ϕ − ϕ 0 ) 2 h × = A cos ǫ sin( ϕ − ϕ 0 ) A: amplitude, ϕ 0 : initial phase, ϕ : arbitrary phase
Signal Model : Smooth chirp GW polarisations: h + = A (1+cos 2 ǫ ) cos( ϕ − ϕ 0 ) 2 h × = A cos ǫ sin( ϕ − ϕ 0 ) A: amplitude, ϕ 0 : initial phase, ϕ : arbitrary phase Network Signal, s ∈ ℜ Nd p 1 p 2 � � � � s = 1 Φ ∗ Φ D D ∗ ⊗ ≡ Π P 2 p 3 � �� � � �� � ∅ D p 4 � �� � P
Signal Model: Understand Π What is SVD of Π = U Π Σ Π V H Π ? SVD of ⊗ is ⊗ of SVD Φ and Φ ∗ are orthogonal SVD(Π) = SVD( D ) ⊗ SVD(Φ) � �� � � �� � Σ Π = diag ( σ 1 ,σ 2 ) I 2
Signal Model: Understand Π What is SVD of Π = U Π Σ Π V H Π ? SVD of ⊗ is ⊗ of SVD Φ and Φ ∗ are orthogonal SVD(Π) = SVD( D ) ⊗ SVD(Φ) � �� � � �� � Σ Π = diag ( σ 1 ,σ 2 ) I 2 Condition Number cond(Π) = σ 1 /σ 2 σ 1 > σ 2 σ 2 can be very small = ⇒ F + and F × are collinear. [Klimenko et. al PRD 2005, Rakhmanov CQG 2006]
Detector Networks: cond (Π) − 1 = σ 2 /σ 1 < 0 . 1
Maximum of Network LLR • Maximize Network Likelihood Ratio wrt P : Λ = −� x − Π P � 2 + � x � 2 Solve Linear LSQ:- Pseudo-inverse of Π i.e. ˆ P = V Π Σ − 1 Π U H Π x
Maximum of Network LLR • Maximize Network Likelihood Ratio wrt P : Λ = −� x − Π P � 2 + � x � 2 Solve Linear LSQ:- Pseudo-inverse of Π i.e. ˆ P = V Π Σ − 1 Π U H Π x Π x � 2 = � ( U H Λ(ˆ ∅ ) x � 2 • Network MLR: P ) = � U H D ⊗ U H � � D X T U ∅ U H X = x 1 . . . x d N × d � �� � Project data on to U ∅ first and then combine with weights [Pai, Dhurandhar, Bose, PRD 2001]
Maximum of Network LLR • Maximize Network Likelihood Ratio wrt P : Λ = −� x − Π P � 2 + � x � 2 Solve Linear LSQ:- Pseudo-inverse of Π i.e. ˆ P = V Π Σ − 1 Π U H Π x Π x � 2 = � ( U H Λ(ˆ ∅ ) x � 2 • Network MLR: P ) = � U H D ⊗ U H � � U H D X T � �� � U ∅ X = x 1 . . . x d N × d Project data on to U D and then Matched filtering
Synthetic Streams and Null Streams � � Π x � 2 = � ( U H Λ(ˆ ∅ ) x � 2 Network MLR: P ) = � U H D ⊗ U H U D = d 1 d 2 � � Φ ∗ Φ U H D X T � �� � U ∅ U ∅ = (a) Project data on to U D = construct synthetic streams (b) Matched filtering of synthetic streams
Synthetic Streams and Null Streams � � Π x � 2 = � ( U H Λ(ˆ ∅ ) x � 2 Network MLR: P ) = � U H D ⊗ U H U D = d 1 d 2 � � Φ ∗ Φ U H D X T � �� � U ∅ U ∅ = (a) Project data on to U D = construct synthetic streams (b) Matched filtering of synthetic streams • 2 non-zero singular values ⇒ 2 synthetic streams: Y 1 = Xd 1 and Y 2 = Xd 2 P ) = | Φ H Y 1 | 2 + | Φ H Y 2 | 2 Λ = Λ(ˆ ˆ N
Synthetic Streams and Null Streams � � Π x � 2 = � ( U H Λ(ˆ ∅ ) x � 2 Network MLR: P ) = � U H D ⊗ U H U D = d 1 d 2 � � Φ ∗ Φ U H D X T � �� � U ∅ U ∅ = (a) Project data on to U D = construct synthetic streams (b) Matched filtering of synthetic streams • 2 non-zero singular values ⇒ 2 synthetic streams: Y 1 = Xd 1 and Y 2 = Xd 2 P ) = | Φ H Y 1 | 2 + | Φ H Y 2 | 2 Λ = Λ(ˆ ˆ N • ( d − 2) zero singular values ⇒ ( d − 2) orthogonal null streams span null space Null Stream Def: Xd j = 0 , d ≥ j > 2 (signal only)
Synthetic Streams and Null Streams � � Π x � 2 = � ( U H Λ(ˆ ∅ ) x � 2 Network MLR: P ) = � U H D ⊗ U H U D = d 1 d 2 � � Φ ∗ Φ U H D X T � �� � U ∅ U ∅ = (a) Project data on to U D = construct synthetic streams (b) Matched filtering of synthetic streams • 2 non-zero singular values ⇒ 2 synthetic streams: Y 1 = Xd 1 and Y 2 = Xd 2 P ) = | Φ H Y 1 | 2 + | Φ H Y 2 | 2 Λ = Λ(ˆ ˆ N • ( d − 2) zero singular values ⇒ ( d − 2) orthogonal null streams span null space Null Stream Def: Xd j = 0 , d ≥ j > 2 (signal only) 2 detectors: No null stream 3 detectors: 1 null stream d 3 = d 1 × d 2 = { ǫ ijk d 1 j d 2 k } [ Wen, Schutz, CQG, 2005]
Synthetic Streams and Null Streams � � Π x � 2 = � ( U H Λ(ˆ ∅ ) x � 2 Network MLR: P ) = � U H D ⊗ U H U D = d 1 d 2 � � Φ ∗ Φ U H D X T � �� � U ∅ U ∅ = (a) Project data on to U D = construct synthetic streams (b) Matched filtering of synthetic streams • 2 non-zero singular values ⇒ 2 synthetic streams: Y 1 = Xd 1 and Y 2 = Xd 2 P ) = | Φ H Y 1 | 2 + | Φ H Y 2 | 2 Λ = Λ(ˆ ˆ N • ( d − 2) zero singular values ⇒ ( d − 2) orthogonal null streams span null space Null Stream Def: Xd j = 0 , d ≥ j > 2 (signal only) 2 detectors: No null stream 3 detectors: 1 null stream d 3 = d 1 × d 2 = { ǫ ijk d 1 j d 2 k } [ Wen, Schutz, CQG, 2005] In general, for d detectors, ( d − 2) null streams: d l i = ǫ ijk...n d 1 j d 2 k . . . d ( l − 1 ) n for d ≥ l > 2
Treating Rank Defficiency Π might be ill-conditioned i.e. cond(Π) >> 1 ( σ 2 ∼ 0 ⇒ � − 1 diverges : Y 2 insensitive to GW) Ill posed LSQ ⇒ Needs treatment, regularisation
Treating Rank Defficiency Π might be ill-conditioned i.e. cond(Π) >> 1 ( σ 2 ∼ 0 ⇒ � − 1 diverges : Y 2 insensitive to GW) Ill posed LSQ ⇒ Needs treatment, regularisation • Truncated SVD approach: Y 2 adds noise to the statistics, discard it. Λ t = | Φ H Y 1 | 2 ˆ N Truncation criterion cond(Π) > SNR
Treating Rank Defficiency Π might be ill-conditioned i.e. cond(Π) >> 1 ( σ 2 ∼ 0 ⇒ � − 1 diverges : Y 2 insensitive to GW) Ill posed LSQ ⇒ Needs treatment, regularisation • Truncated SVD approach: Y 2 adds noise to the statistics, discard it. Λ t = | Φ H Y 1 | 2 ˆ N Truncation criterion cond(Π) > SNR • Tikhonov Regularisation: similar to [Rakhmanov CQG 2006] Regularise Λ == Add a quadratic regulator to Λ � � | Φ H Y 1 | 2 + | Φ H Y 2 | 2 Λ r = 1 ˆ N cond(Π) Larger the cond(Π) smaller is the contribution from Y 2
Best Net-CC : Extension of BCC Signal phase Φ is unknown : maximising ˆ Λ over possible phases
Best Net-CC : Extension of BCC Signal phase Φ is unknown : maximising ˆ Λ over possible phases Time-frequency (TF) mapping of ˆ Λ – TF map – Wigner-Ville transform [Chassande-mottin, Pai, IEEE SPL 2005] + Simplified template of smooth chirp – 1-dim ridge in WV i.e. δ ( t, f ( t )) == TF path integral on WV map [Chassande-mottin, Pai, PRD 2006]
Best Net-CC : Extension of BCC Signal phase Φ is unknown : maximising ˆ Λ over possible phases Time-frequency (TF) mapping of ˆ Λ – TF map – Wigner-Ville transform [Chassande-mottin, Pai, IEEE SPL 2005] + Simplified template of smooth chirp – 1-dim ridge in WV i.e. δ ( t, f ( t )) == TF path integral on WV map [Chassande-mottin, Pai, PRD 2006] Λ = | Φ H Y 1 | 2 + | Φ H Y 2 | 2 = 2 � ˆ w Y ( t n , f ( t n )) w Y = w Y 1 + w Y 2 N 2 N n Maximise ˆ Λ over Φ = longest path problem in TF map of w Y
Best Net-CC : Extension of BCC Signal phase Φ is unknown : maximising ˆ Λ over possible phases Time-frequency (TF) mapping of ˆ Λ – TF map – Wigner-Ville transform [Chassande-mottin, Pai, IEEE SPL 2003] + Simplified template of smooth chirp – 1-dim ridge in WV i.e. δ ( t, f ( t )) == TF path integral on WV map [Chassande-mottin, Pai, PRD 2006] Λ = | Φ H Y 1 | 2 + | Φ H Y 2 | 2 = 2 � ˆ w Y ( t n , f ( t n )) w Y = w Y 1 + w Y 2 N 2 N n Maximise ˆ Λ over Φ = longest path problem in TF map of w Y
Best Net-CC : Extension of BCC Signal phase Φ is unknown : maximising ˆ Λ over possible phases Time-frequency (TF) mapping of ˆ Λ – TF map – Wigner-Ville transform [Chassande-mottin, Pai, IEEE SPL 2003] + Simplified template of smooth chirp – 1-dim ridge in WV i.e. δ ( t, f ( t )) == TF path integral on WV map [Chassande-mottin, Pai, PRD 2006] Λ = | Φ H Y 1 | 2 + | Φ H Y 2 | 2 = 2 � ˆ w Y ( t n , f ( t n )) w Y = w Y 1 + w Y 2 N 2 N n Maximise ˆ Λ over Φ = longest path problem in TF map of w Y
Simulations: Linear chirp in Gaussian white noise
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