Application of Non-Harmonic Analysis for Gravitational Wave Detection

Masaya NakanoUniversity of Toyama

Collaborate with

S. Hirobayashi(Univ. Toyama), H. Tagoshi(Osaka Univ.),

K. Ueno(Osaka Univ.), T. Narikawa(Osaka Univ.),

K. Miyake(Univ. Toyama), N. Kanda(Osaka city Univ.),

K. Hayama(Osaka city Univ.)

The 7th Korea-Japan workshop on KAGRA

2014/12/19-20 @ Univ. Toyama

BackgroundGravitational Waves resulting from such neutron binary stars, binary black hole and early universe, is a means of a new space observation. Also, it is possible to clarify the physical behavior of the measured signal.

Gravitational wave that is shaking and fine frequency to time-frequency analysis, the amplitude change was visualized, sorting of noise and gravitational waves, detailed analysis of noise, it want to make a detailed analysis of the physical phenomena of gravitational wave sources.

Using the NHA of high-precision analysisFrequency analysis method of the current highest accuracy

Conventional Frequency Analysis TechniqueFourier Transform

The side-lobe suppression using Hamming and Hanning window, Interpolation of frequency resolution using the zero padding－Serge Droz, et al. Physical Rev. D, 1999

Wavelet Transform

Summing by scale basis functions. Time resolution is high.－B Abbott, et al. Classical and Quantum Gravity, 23, 8, S29 2006

Instantaneous Frequency

Take the differential value of the phase, the analysis of detailed harmonic structure－Alexander Stoeer,et al. Physical Rev. D 79, 2009

– Influence on analysis window

– Non-periodic signal is also regarded as a periodic signal analysis

・Lowering of the frequency resolution

・Discrete frequency decomposition width

– There is no completely independent of the mode decomposition

・It occurs Artifact

Fourier transform

Wavelet transform

Hilbert Spectrum analysis

Non-Harmonic Analysis (NHA)

N : 窓長

1

0

ˆ1ˆˆ ˆ ˆ ˆ( , , ) ( ) cos 2N

n s

fF A f x n A n

N f

2

2

DFT(zero-padding)

1 Original spectrum

NHA

NHA estimates the Fourier coefficient by solving a non-linear

equation. (least square method)

NHA shows near original spectrum

without undesired side-lobes.

Advantages of NHAWe compared the accuracy of frequency analysis achieved by two approaches.

Method Accuracy

DFT 1 order of magnitude

NHA10 or more orders of

magnitude

Better axail resolution can be expected when NHA is

used.

The accuracy of DFT analysis is

relatively low when the objective

signal is not a multiple of the

fundamental frequency.

Gravitational Wave Detection Using Non-Harmonic Analysis

At normalized frequencies below 1 Hz, NHA is demonstrated to have

greater analysis accuracy than DFT. Accurate estimation at frequencies

below 1 Hz implies that object signals with periods longer than the

window length can be analyzed accurately.

The square error of each estimated parameter.

¥

(c) STFT

(f) NHA

(d) HSA

0

freq

uen

cy [

Hz]

time [s]

200

100

0

(b) source

(a) waveform1

0

-1

amp

litu

de

freq

uen

cy [

Hz]

0 0.5 1time [s]

Comprison of simulation signal

0 0.5 1time [s]

200

100

0

200

100

0

200

100

0

freq

uen

cy [

Hz]

time [s]

(d) NHA

(c) HSA

(b) STFT

0 1 2

(a) waveform1

0

-1

amp

litu

de

200

100

0

freq

uen

cy [

Hz]

Comparison of BBH AnalysisMass:𝑚1 = 𝑚2 = 10𝑀⊙ 𝐼𝑆𝐶𝑂 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦: 𝑓𝑖𝑠𝑐𝑜 = 220[𝐻𝑧]

200

100

0

200

100

0

200

100

0

Problems under Noisy Conditions

Source spectrum and noise spectrum may overlap in harmonic

analysis based on DFT, in which the frequency resolution is

generally low.

Source spectrum + Noise spectrum = Noise environment spectrum

Source spectrum + Noise spectrum = Noise environment spectrum

Since NHA is affected very little by the frame length, the noise

and source spectra are less likely to overlap than with DFT.

Also, NHA can potentially preserve amplitude and initial phase.

NHA

DFT

(a) waveform

Experiment Condition

¥time [s]

0 1 2

1

0

-1ampli

tude

・To cut of wasted bandwidth, use the

Low-pass filter :𝑓𝑐 = 300[𝐻𝑧]

Before Processing :𝑓𝑠 = 16384 𝐻𝑧After Processing:𝑓𝑠 = 512 𝐻𝑧

・Matched filter S/N

𝜌2 = 4 0

∞ 𝑓 ℎ2

𝑆𝑛(𝑓)

𝑑𝑓

𝑓= 4

0

𝑖𝑠𝑐𝑜 𝑓 ℎ2

𝑆𝑛(𝑓)

𝑑𝑓

𝑓

Frequency Characterization

Under the noisy conditioni n the case of SN=100

Line noise: Violin mode of KAGRA

time [s]

freq

uen

cy [

Hz]

NHA

SN = 100

SN = 10

Noisy Condition

200

100

0

200

100

0

200

100

0

0 1 2

ConclusionsGravitational wave have large frequency variation in a short time,

the reproducibility of the waveform is an important problem in gravitational wave observation.

In this report, for the binary black hole waveform, evaluated in time-frequency domain under the noisy condition.

Under the noisy condition of SN = 10 and 100, it was visualized the frequency trajectory due to merger waveform.

As challenges for the future, to conduct and review of GPU acceleration process, the analysis of detector data calculation can be expensive NHA. In addition, to perform accurate comparison with other frequency analysis methods.

Feature Works

That’s all.

Thank you for your attention!

Concept of NHA

1. Set the initial value

of the spectrum

parameters.

2. Adjust the frequency

and the initial phase by

expanding or contracting

or translating.

3. Adjust the amplitude.

initial phase

The spectrum parameter of NHA is obtained by the signal shape fitting.

High Coherence Source Probe Arm

lc/⊿L

t ➔ f

FFT

OCT

signaldz

Δλ

⊿L1 ⊿L2

l Scan

Depth

Reference Arm

Fixed mirror

Coupler

Sampl

e

OCT Image Based on NHA

OCT cross-sectional images of finger skin.

何か一言コメント OCTのNHAの応用性から重力波にも応用させる