Best NET-CC: Extension of BCC search to GW interferometer networks - - PowerPoint PPT Presentation

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Best NET-CC: Extension of BCC search to GW interferometer networks - - PowerPoint PPT Presentation

Sep 04, 2023 143 likes •432 views

Best NET-CC: Extension of BCC search to GW interferometer networks Archana Pai Albert Einstein Institute, Potsdam with ric Chassande-Mottin AstroParticule et Cosmologie, Paris CNRS, Observatoire de la Cte de Azur, Nice (20th Dec. 2006,

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SLIDE 1

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)

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SLIDE 2

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

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SLIDE 3

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

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SLIDE 4

Signal Model : Smooth chirp

GW polarisations: h+ = A(1+cos2 ǫ)

2

cos(ϕ − ϕ0) h× = A cos ǫ sin(ϕ − ϕ0) A: amplitude, ϕ0: initial phase, ϕ: arbitrary phase

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SLIDE 5

Signal Model : Smooth chirp

GW polarisations: h+ = A(1+cos2 ǫ)

2

cos(ϕ − ϕ0) h× = A cos ǫ sin(ϕ − ϕ0) A: amplitude, ϕ0: initial phase, ϕ: arbitrary phase Network Signal, s ∈ ℜNd s = 1 2   

  • D D∗
  • D

⊗

  • Φ∗ Φ
  • ∅

           p1 p2 p3 p4        

P

≡ Π P

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SLIDE 6

Signal Model: Understand Π

What is SVD of Π = UΠΣΠV H

Π ?

SVD of ⊗ is ⊗ of SVD SVD(Π) = SVD(D)

  • ΣΠ=diag(σ1,σ2)

⊗ SVD(Φ)

  • I2

Φ and Φ∗ are orthogonal

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SLIDE 7

Signal Model: Understand Π

What is SVD of Π = UΠΣΠV H

Π ?

SVD of ⊗ is ⊗ of SVD SVD(Π) = SVD(D)

  • ΣΠ=diag(σ1,σ2)

⊗ SVD(Φ)

  • I2

Φ and Φ∗ are orthogonal Condition Number cond(Π) = σ1/σ2 σ1 > σ2 σ2 can be very small = ⇒ F+ and F× are collinear. [Klimenko et. al PRD 2005, Rakhmanov CQG 2006]

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SLIDE 8

Detector Networks: cond (Π)−1 = σ2/σ1 < 0.1

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SLIDE 9

Maximum of Network LLR

  • Maximize Network Likelihood Ratio wrt P:

Λ = −x − ΠP2 + x2 Solve Linear LSQ:- Pseudo-inverse of Π i.e. ˆ P = VΠΣ−1

Π U H Π x

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SLIDE 10

Maximum of Network LLR

  • Maximize Network Likelihood Ratio wrt P:

Λ = −x − ΠP2 + x2 Solve Linear LSQ:- Pseudo-inverse of Π i.e. ˆ P = VΠΣ−1

Π U H Π x

  • Network MLR:

Λ(ˆ P) = U H

Π x2 = (U H D ⊗ U H ∅ )x2

U H

D XTU∅

X =

  • x1 . . . xd
  • N×d

Project data on to U∅ first and then combine with weights

[Pai, Dhurandhar, Bose, PRD 2001]

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SLIDE 11

Maximum of Network LLR

  • Maximize Network Likelihood Ratio wrt P:

Λ = −x − ΠP2 + x2 Solve Linear LSQ:- Pseudo-inverse of Π i.e. ˆ P = VΠΣ−1

Π U H Π x

  • Network MLR:

Λ(ˆ P) = U H

Π x2 = (U H D ⊗ U H ∅ )x2

U H

D XT

U∅ X =

  • x1 . . . xd
  • N×d

Project data on to UD and then Matched filtering

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SLIDE 12

Synthetic Streams and Null Streams

Network MLR: Λ(ˆ P) = U H

Π x2 = (U H D ⊗ U H ∅ )x2

UD =

  • d1 d2
  • U H

D XT

U∅ U∅ =

  • Φ∗ Φ
  • (a) Project data on to UD = construct synthetic streams

(b) Matched filtering of synthetic streams

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SLIDE 13

Synthetic Streams and Null Streams

Network MLR: Λ(ˆ P) = U H

Π x2 = (U H D ⊗ U H ∅ )x2

UD =

  • d1 d2
  • U H

D XT

U∅ U∅ =

  • Φ∗ Φ
  • (a) Project data on to UD = construct synthetic streams

(b) Matched filtering of synthetic streams

  • 2 non-zero singular values ⇒ 2 synthetic streams: Y1 = Xd1 and Y2 = Xd2

ˆ Λ = Λ(ˆ P) = |ΦHY1|2 + |ΦHY2|2 N

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SLIDE 14

Synthetic Streams and Null Streams

Network MLR: Λ(ˆ P) = U H

Π x2 = (U H D ⊗ U H ∅ )x2

UD =

  • d1 d2
  • U H

D XT

U∅ U∅ =

  • Φ∗ Φ
  • (a) Project data on to UD = construct synthetic streams

(b) Matched filtering of synthetic streams

  • 2 non-zero singular values ⇒ 2 synthetic streams: Y1 = Xd1 and Y2 = Xd2

ˆ Λ = Λ(ˆ P) = |ΦHY1|2 + |ΦHY2|2 N

  • (d − 2) zero singular values ⇒ (d − 2) orthogonal null streams span null space

Null Stream Def: Xdj = 0, d ≥ j > 2 (signal only)

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SLIDE 15

Synthetic Streams and Null Streams

Network MLR: Λ(ˆ P) = U H

Π x2 = (U H D ⊗ U H ∅ )x2

UD =

  • d1 d2
  • U H

D XT

U∅ U∅ =

  • Φ∗ Φ
  • (a) Project data on to UD = construct synthetic streams

(b) Matched filtering of synthetic streams

  • 2 non-zero singular values ⇒ 2 synthetic streams: Y1 = Xd1 and Y2 = Xd2

ˆ Λ = Λ(ˆ P) = |ΦHY1|2 + |ΦHY2|2 N

  • (d − 2) zero singular values ⇒ (d − 2) orthogonal null streams span null space

Null Stream Def: Xdj = 0, d ≥ j > 2 (signal only) 2 detectors: No null stream 3 detectors: 1 null stream d3 = d1 × d2 = {ǫijkd1jd2k} [Wen, Schutz, CQG, 2005]

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SLIDE 16

Synthetic Streams and Null Streams

Network MLR: Λ(ˆ P) = U H

Π x2 = (U H D ⊗ U H ∅ )x2

UD =

  • d1 d2
  • U H

D XT

U∅ U∅ =

  • Φ∗ Φ
  • (a) Project data on to UD = construct synthetic streams

(b) Matched filtering of synthetic streams

  • 2 non-zero singular values ⇒ 2 synthetic streams: Y1 = Xd1 and Y2 = Xd2

ˆ Λ = Λ(ˆ P) = |ΦHY1|2 + |ΦHY2|2 N

  • (d − 2) zero singular values ⇒ (d − 2) orthogonal null streams span null space

Null Stream Def: Xdj = 0, d ≥ j > 2 (signal only) 2 detectors: No null stream 3 detectors: 1 null stream d3 = d1 × d2 = {ǫijkd1jd2k} [Wen, Schutz, CQG, 2005] In general, for d detectors, (d − 2) null streams: dli = ǫijk...nd1jd2k . . . d(l−1)n for d ≥ l > 2

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SLIDE 17

Treating Rank Defficiency

Π might be ill-conditioned i.e. cond(Π) >> 1 (σ2 ∼ 0 ⇒ −1 diverges : Y2 insensitive to GW) Ill posed LSQ ⇒ Needs treatment, regularisation

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SLIDE 18

Treating Rank Defficiency

Π might be ill-conditioned i.e. cond(Π) >> 1 (σ2 ∼ 0 ⇒ −1 diverges : Y2 insensitive to GW) Ill posed LSQ ⇒ Needs treatment, regularisation

  • Truncated SVD approach: Y2 adds noise to the statistics, discard it.

ˆ Λt = |ΦHY1|2 N Truncation criterion cond(Π) > SNR

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SLIDE 19

Treating Rank Defficiency

Π might be ill-conditioned i.e. cond(Π) >> 1 (σ2 ∼ 0 ⇒ −1 diverges : Y2 insensitive to GW) Ill posed LSQ ⇒ Needs treatment, regularisation

  • Truncated SVD approach: Y2 adds noise to the statistics, discard it.

ˆ Λt = |ΦHY1|2 N Truncation criterion cond(Π) > SNR

  • Tikhonov Regularisation: similar to [Rakhmanov CQG 2006]

Regularise Λ == Add a quadratic regulator to Λ ˆ Λr = 1 N

  • |ΦHY1|2 + |ΦHY2|2

cond(Π)

  • Larger the cond(Π) smaller is the contribution from Y2
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SLIDE 20

Best Net-CC : Extension of BCC

Signal phase Φ is unknown : maximising ˆ Λ over possible phases

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SLIDE 21

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]

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SLIDE 22

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] ˆ Λ = |ΦHY1|2 + |ΦHY2|2 N = 2 N 2

  • n

wY(tn, f(tn)) wY = wY1 + wY2 Maximise ˆ Λ over Φ = longest path problem in TF map of wY

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SLIDE 23

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] ˆ Λ = |ΦHY1|2 + |ΦHY2|2 N = 2 N 2

  • n

wY(tn, f(tn)) wY = wY1 + wY2 Maximise ˆ Λ over Φ = longest path problem in TF map of wY

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SLIDE 24

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] ˆ Λ = |ΦHY1|2 + |ΦHY2|2 N = 2 N 2

  • n

wY(tn, f(tn)) wY = wY1 + wY2 Maximise ˆ Λ over Φ = longest path problem in TF map of wY

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SLIDE 25

Simulations: Linear chirp in Gaussian white noise

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SLIDE 26

Concluding Remarks

Chirp detection problem with a detector network in a “new formalism” (LSQ)

  • Evidence of degeneracy in the signal model.

Parameter estimation may be unreliable. Need for proper regularisation. Possible implication for inspiral search.

  • Coherent Network detection == Process 2 synthetic streams

Straightforward extension of Best CC search == Best Net-CC Best Net-CC —- a feasible full sky search of GW chirps Fraction of a sec duration ∼ few 100 GFlops

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SLIDE 27

Treating Rank Defficiency

Π might be ill-conditioned i.e. cond(Π) >> 1 (σ2 ∼ 0 ⇒ −1 diverges : Y2 insensitive to GW) Ill posed LSQ ⇒ Needs treatment, regularisation

  • Truncated SVD approach: Y2 adds noise to the statistics, discard it.

ˆ Λt = |ΦHY1|2 N parameter estimation??!!

  • Tikhonov Regularisation: [Rakhmanov CQG 2006]

Regularise Λ == Add a quadratic regulator PHΩP to Λ ˆ Λr = 1 N

  • |ΦHY1|2 + |ΦHY2|2

cond(Π)

  • LSQ estimator ⇒ ˆ

Pr = (ΠHΠ + Ω)−1ΠHx Larger the cond(Π) smaller is the contribution from Y2