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Summed Parallel Infinite Impulse Response (SPIIR) Filters For Low-LatencyGravitational Wave Detection
Shaun Hooper1, Linqing Wen1, David Blair1, Jing Luan2, Shin Kee Chung1 and Yanbei Chen2
1The University of Western Australia, 2California Institute of Technology
Abstract
We present a new time-domain low-latency detection method forgenerating the signal to noise ratio (SNR) of inspiral gravitationalwaveforms (GWs) in detector data. Fast GW triggering will enableelectromagnetic observations of the prompt optical emission related togamma-ray burst (GRB) events.
Real-Time Low-Latency GW Triggering isEssential for Prompt Optical Follow-Up
Neutron star binary mergers are widely thought to be the progenitors ofshort hard gamma-ray bursts (GRBs). Shown below is the current idea ofGRB - GW connection;
Coalescence(GWs produced)
-1000 - 0 seconds
Merger
Gamma-ray burst(beamed)
0
Prompt OpticalEmission
10's-100's seconds
? ?
?
?
?
?
?
Prompt Radio Emission
OpticalAfterglow
100's seconds - days
hours - days
Generate GW
trigger &
point telescopes
Figure 1: Basic time-line of GRB event. Note that the times are model dependent.
Time-Domain Low-Latency GW DetectionMethod
GWs are detected by performing a cross-correlation of the waveform andthe detector data, weighted 1/Sn(f ) (matched filter).
I Frequency domain correlation introduces latency (data segmenting)
I Time-domain correlation computationally expensive
I Infinite Impulse Response (IIR) filters provide a computationally cheaperalternative
Consider the simplest IIR filter,
yk = a1yk−1 + b0xk, (1)
which has the solution,
yk =
k∑j=−∞
xjb0ak−j1 (2)
+
×
×
Figure 2: A flow chart showing thefiltering of xk to yk
This is the cross-correlation of the input data xk a complex sinusoid wherea1 = e−(γl−iω)∆t and b0 is a complex constant.
Approximation of Inspiral Waveform as aSummation of Sinusoids
The inspiral waveform A(t)eiφ(t) can approximated by a linear summationof exponentially increasing constant frequency sinusoid’s with cutoff timestl,
A(t)eiφ(t) '∑l
Ale(γl+iωl)(t−tl)Θ(tl − t) (3)
(e)
(d)
...
(c)
...
(b)
(a)
Figure 3: The top three sinusoids (a-c) have different γ, ω and cutoff time factors. Thefourth panel (d) shows the linear addition of these, plus more scaled sinusoids. The fifthpanel (e) shows the exact inspiral-like waveform (Note that this figure is only for illustrativepurposes).
We create a bank of IIR filters, each with coefficients a1, b0 correspondingto a sinusoid with parameters ωl, γl, Al, and cutoff time tl.
99% Overlap Easily Achieved
By varying the number of sinusoids needed to approximate the inspiralwaveform, we can recover a very good overlap.
200 400 600 800 1000 1200 1400
0.976
0.978
0.98
0.982
0.984
0.986
0.988
0.99
0.992
0.994
Number of sinusoids per waveform
Ov
erla
p
1.4+1.4 M⊙1.0+1.0 M⊙1.0+3.0 M⊙2.0+2.0 M⊙2.0+3.0 M⊙3.0+3.0 M⊙
Figure 4: We show the overlap (the cross-correlation between the exact inspiral waveformand the approximate inspiral waveform) as a function of number of sinusoids.
Real Time Filter Output same as MatchedFilter
The summation of parallel infiniteimpulse response filters (SPIIR)run on mock detector data, cancalculate an SNR almost the sameas the matched filter, but calcu-lated in real-time without segment-ing of data.
Figure 5: SNR output of both the SPIIRmethod and a traditional matched filtermethod as function of time
SNR
t − τc (ms)−15 −10 −5 0 5 10 15
0
1
2
3
4
5
6
7
8
9
10
IIR filter output
Matched filter output
−1 0 17.5
8
8.5
Detection Efficiency Compared withMatched Filter
The SPIIR method recovers almost all of the detections made by thematched filter method at an equivalent false alarm rate.
False Alarm Rate
Det
ecti
on R
ate
SNR ~8 (250 Mpc)
SNR ~6.6 (300 M
pc)
SNR ~
5.7(
350
Mpc
)
SNR
~5
(400
Mpc
)
10−6
10−5
10−4
10−3
10−2
10−1
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SPIIR Method
Matched Filter
Figure 6: The detection rate vs false alarm rate of both the SPIIR method and the matchedfilter method. Shown is the detection efficiency for four different injections scaled ateffective distances of 250, 300, 350 and 400 Mpc (SNR ∼ 8, 6.6, 5.7 and 5 respectively).
Summary
I SPIIR filter method enables real-time detection of inspiral waveforms
I Good signal recovery compared to matched filter
I High computational efficiency compared to full time-domain correlation
Iχ2 tests are easily included
I Expandable to search for waveforms from spinning black holes
I Parallelizable algorithm, already tested using graphics processing units
Shaun Hooper et. al SPIIR Filters For Low-Latency Gravitational Wave Detection (LIGO - G1100030) Mail: [email protected]
