Seamless time series generator for KAGRA and GW numerical simulation

Takahiro Yamamoto, Nobuyuki Kanda, Hirotaka Yuzurihara, Kazuyuki Tanaka, KAGRA Collaboration Osaka City University 1. Introduction spectrum 1 time series 1 time series 2 We often generate time series signal which consists of noise and gravitational wave for the IFFT (f)

interferometer using FFT/IFFT. However, since FFT/IFFT method treat the data with finite time chunks, 1 it is not good to use for the study of continuous processing of signals across long duration over the h (t) (t) 1 2 h chunks. Moreover these generated will not joint smoothly to neighbor chunks. To mimic the h realistic raw data taking and calibration process, we would like to generate seamless time series data.

We try to generate seamless to interface transfer function with white gaussian noise spectrum 2 in time region. Where, transfer function is interferometer sensitivity written in complex frequencies. In

this calculation we use Laplace transform and matrix operation. About another noises (thermal noise, (f) 2 line noise etc) we can generate time series directly considering Q-value, repetition of excitation. We h t [sec] t [sec] will display these methods in detail and show result of simulation. IFFT Two time series are seamy data when we generate with IFFT

2A. Gaussian noise 2B. Quasi-stationary noise 2C. Non-stationary noise １） We get colored noise x(t) on time series by convolution of white Suspension noise (violin mode) noise w(t) and impulse response g(t). When Interferometer locked, suspension wires excited. And We modeled one spike noise following equation. t these wires excited all of the time by thermal. t t x(t)= g(t τ)w(τ)dτ − 0 − x(t)=Ae− τ cos(2πf0(t t0)) ￿−∞ One excitation depend on Q-value of wires, peak frequency w0 − and initial phase φini. A is amplitude. In fourier region colored spectrum Sx(w) written in scalar product where A is amplitude, t0 is start of spike signal, τ is damping ω0t 0 of white spectrum Sw(w) and transfer function G(w) x(t)=Ae− 2Q sin(ω0t + φini) time constant and f is typical frequency. 2 τ0 and f0 are given base on TAMA data. τ0 is similar to 0.6 Sx(ω)= G(ω) Sw(ω) Excitation at lock of interferometer is generated with this equation | | where amplitude and initial phase are given a random number. msec and f0 is similar to 1300Hz We required two conditions of transfer function, and operating At this time lock timing match in onset generating noises. transfer function to [1][2]. Interval of two spike signals allowed exponential distribution. 1) Transfer function written in idempotent of integer of 1 x p(x)dx = e− T dx complex frequencies s. T 2) When s → ±∞, G(s) → 0. T is expected value of signal interval in our simulation. m P (s) a0 + a1s + + ams G(s)= = ··· h(f) [/rHz] Q(s) b + b s + + b sn 0 1 ··· n

s = iω, n > m an,bn = const. Spike

Time Series x(t) can be written with differential operator. f [Hz] P (D) d Left figure is suspension noise and right is spectrum. Spectrum of just locked(red) x(t)= w(t) D = decayed to one of after 10 minutes(green). Noise include from 1st to 7th violin mode. dt Q(D) ￿ ￿ If We generate Excitation by thermal in same method, calculation Q(D)φ(t)=w(t) ・・・① cost becomes very expensive. So We add in Brownian motion. x(t)=P (D)φ(t) ・・・② Thermal noise x(t) change complex signals X(t), Y(t), Z(t). ⇔ t+δt X(t)= x(t￿) cos(ω0t￿)dt￿ The first we make up vector z(t), vector f(t) and matrix A to t Left figure is TAMA spike noise of which f0 similar to 1300Hz. ￿ t+δt compute φ(t) [3]. Right is generated spike consulting parameter of noise of TAMA. Y (t)= x(t￿) sin(ω0t￿)dt￿ (n) t ① ⇔ b φ + b φ￿ + + b φ = w(t) ￿ 0 1 ··· n Z(t)=X(t)+iY (t) 2D. Line noise dz(t) ⇔ = Az(t)+f(t) Z(t) moves δZ(t) during sampling interval, where δZ(t) is rayleigh Line noise is sine curve fluctuating frequency. At KAMIOKA center dt distribution. frequency is 60 Hz. This noises are given assuming that frequency 010 0 ω0δt (n 1) T ··· Z(t )=Z(t )e− 2Q + δZ(t ) is constant during sampling interval δt. z =[φ(t) φ￿(t) φ − (t)] 001 0 i+1 i i+1 ···  ···  2 A = ...... δZ δZ . . . . . 2 x(t)=A sin(φ(t)) T   P (δZ)= e− 2σ f(t) = [0 0 w(t)]  000 1  rayleigh 2 ···  ···  σ  b0 b1 b2 bn 1  where A is amplitude and φ(t) is phase of signal.  − − − ··· − −  where σ^2 is variance of gaussian distribution (Re[δZ], Im[δZ]).   Advance of phase per δt is calculated with following formula. 2 2 2 <δZ >=<δZ> +σδZ < x > is expected value of x To compute this differential equation, we can calculate φ(t). 2 2 φ(ti+1)=φ(ti)+ω(ti+1)δt = µrayleigh + σrayleigh Next using φ(t) solved from differential equation, We compute ω(t)=2πf(t) π 2 4 π 2 = σ + − σ f(t) is random number around 60Hz noise signal x(t). 2 2 (m) <δZ2 > We suppose frequencies of Harmonics signals depend on freque- ② ⇔ x(t)=a0φ + a1φ￿ + + amφ σ2 = ··· 2 ncies of fundamental wave. Expectation of Brownian motion is written in sampling interval δt. e.g) 2 time wave x2time(t)=B sin(φ2time(t)) 2 kBT δt 2 <δZ > δt + (1 e− ) φ (t )=φ (t )+2ω(t )δt ∼ Q 2 − 2time i+1 2time i i+1 ￿ ￿ where kB is Boltzmann constant and T is absolute temperature. h(t) h(f) [/rHz]

Left figure is generated time series and right is 100 times average of spectrum. We see generated spectrum(red) is along (blue) and radiation t [sec] f [Hz] pressure(green) curve. Left figure is time series of line noise and right is spectrum We checked gaussianity of generated noise with Kolomogrov-Smirnov test and These figures are time series of thermal noise(left) with Quantum noise curve(green). Anderson-Darling test [4]. and 100 times average of spectrum of one(right). Today this noise include fundamental, 2 time and 3time wave.

3. Generated noise 4. Summary and Future work

Interferometerʼs noise is linear sum of shown noises. All type of noises generated without IFFT, so we can use Summary noise datas without regard to seam. Also to memorize initial parameter of generating routine and data length of ・We showed seamless simulation noise allowed KAGRA spectrum. generated, we can generate continuation of previous time. ・These noise includes gaussian-stationary, quasi-stationary and non-stationary noises. Following figures are generated total noises include quantum, suspension, thermal, spike, line noise. The left is ・ time series and center is spectrum of just lock interferometer(red), 10 minutes after from lock and 100 minutes We modeled quasi-stationary and non-stationary noises with TAMA signals. after from lock. Right figure is 100 times average of spectrum of generated noise. Future work These noises allow KAGRA spectrum and mimic to decay excitation of suspension wire. ・To generate Interferometerʼs row data from KAGRA spectrum noise. ・To read out generated noises in frame format to use analysis simulation.

5. References

[1] Heinzel, G. (2007) S2-AEI-TN-3034. [2] Franklin, J. N. (1965) SIAM Review, 7, 68-80. [3] Franklin, J. N. (1963) The Annals of Mathematical Statistics, 34, 1259-1264 [4] T. W. Anderson (1954) Journal of the American Statistical Association, 49, 765-769

2012年6月4日月曜日 TAMA line

TAMA spectrum around 50Hz v(f) [v/rHz]

±0.1Hz

f [Hz]

2012年6月4日月曜日