study of correlated noise
z
UF
correlated noise from power lines
¾ goal: Look if it have “bad” components (~T) with very large time
scale.
9 method: Look at the coherence of power monitors.
¾ goal: How strong is the power correlated noise compare to
uncorrelated noise?
9 method: Look at correlation of H & L IFO output using the Sign
Correlation Test.
z
other possible sources of correlated noise
¾ goal: Look if there is another significant broadband correlated
noise in addition to power lines.
9 method: Use the SCT on data with power lines removed.
S.Klimenko
Coherence (T)
z
γ (T ) ~
γ(T)=const, (small T<1min);
~
1
T
UF
, (large T)
1
T
z
Conclusion: no terms ~T are observed on 17 days data.
z
E7 power correlated noise should not be a problem for optimal correlation
method in Fourier domain, however removal of power lines is desired.
S.Klimenko
Cross-Correlation in Wavelet Domain
z
z
UF
x-correlation in wavelet domain
¾ τ – time lag
S = ∑ N k wk (τ )rk (τ )
¾ n – wavelet layer number
n ,τ
¾ Nk – number of samples in layer k
¾ rk(τ) – correlation coefficients as a function of lag time t
wk(τ) – optimal filter
∞
wk (τ ) = ∫ df ψ k ( f ) f
−∞
2
−3
ΩGW ( f ) ⋅ γ ( f , Ω L , Ω H )
exp(− j 2πfτ )
L
H
RL ( f ) PL ( f ) / σ k ⋅ RH ( f ) PH ( f ) / σ k
9Ψk – Fourier image of mother wavelet for layer k
9 γ – overlap reduction function
L
H
9 σ n ,σ n - noise rms in wavelet domain for detector L (H)
9R(f) – detector responses
S.Klimenko
X-correlation in wavelet domain
E7 data
band 32-64Hz
H2xL1, LSC-AS_Q
power lines
lines removed
H1xL1, LSC-AS_Q
Injected SB signal
S.Klimenko
UF
Coherence of Power Monitors
UF
more details in J.Castiglione’s talk on March LSC meeting
z
Coherence of sL(t) (L0:PEM-LVEA_V1) and sH(t) (H0:PEM-LVEA_V1).
s(t ) = s L (t ) + s H (t ) = A ⋅ sin(ωt + θ )
z
w150sec
∆φ(t)
Ε4
Average square amplitude
A 2 = aL2 + aH2 + 2aL aH cos(φL − φH ).
¾ φL,φH – measured with LineMonitor
z
Ε7
Coherence
γ=
z
1
N
∑
N
k =1
exp(i∆φk )
Coherence at long time scale?
1
γ (T ) ~
T
S.Klimenko
∆φ
Autocorrelation Function
z
z
sign statistics s(t)={uxuy}
a(t) - autocorrelation function of s(t)
¾ a measure of correlated noise.
wE7 data
wX-correlation of
wL1:AS_Q & H2:AS_Q
win wavelet domain:
w32-64 Hz band
S.Klimenko
UF
Variance of Correlation Coefficient
z
uncorrelated noise
¾ autocorrelation function:
¾ variance:
z
varu (γ ) = σ 2 = 1 / n
correlated noise with time scale <Ts
¾ autocorrelation function:
¾ variance:
z
a(0) = 1, a(τ ≥ ∆t ) = 0
a(τ < Ts ) = an (τ ), a(τ > Ts ) = 0
1
varc (γ ) = R
n
variance ratio – measure of © noise (depends on an(t) only)
R = 1+
Ts / ∆t
∑ (n − m)a (m∆t )
n
m =1
S.Klimenko
UF
Power correlated noise measured with SCT
32-64Hz
Sign Correlation Test
is a tool to look at
broadband correlated noise
γ
σ
UF
dots – before line removal
solid - after line removal
Ts
± 3σ
32-64Hz
S.Klimenko
σγ – correlation coefficient
σσ – rms for uncorrelated noise
Variance Ratio (Ts)
z
UF
11 data segments 4096 sec each (total 12.5 h of E7 data)
16-32Hz
32-64Hz
Ts
Ts
128-256Hz
64-128Hz
S.Klimenko
Ts
Ts
Sign Correlation Test
z
Sign transform:
¾
x̂
ui = sign( xi − xˆ )
- median of x
si = sign( xi − xˆ ) ⋅ sign( yi − yˆ )
z
Sign statistics:
z
Correlation coefficient γ:
z
γ distribution (n – number of samples):
¾ Gaussian (large n):
z
very robust:
¾ error from
S.Klimenko
UF
x̂ and ŷ
γ = mean( si )
 nγ 2 
n

P ( n, γ ) ≈
⋅ exp −

2π
2


~2/n2, much less then var(γ )=1/n for large n
Sign Cross-Correlation
z
UF
S s = ∑ N kω k (τ ) ρ k (τ )
Sign x-correlation
n ,τ
¾ ρk(τ) – sign correlation coefficients
¾ ωk(τ) – optimal filter
z
optimal filter
ω k (τ ) =
ε k ⋅ wk (τ )
vk (τ )
¾ εk – sign correlation efficiency
¾ vk(τ) – contribution from correlated noise
¾ ε k ⋅ wk (τ ) should not vary with time
z
Variance of ρ
S.Klimenko
var( ρ k (τ )) =
1
Nk
vk (τ )
Data Analysis Status & Plans
z
Software development is complete
z
Use datacondAPI to calculate
¾
¾
¾
¾
¾
¾
z
ρκ(τ)
vκ(τ)
σ Lk
σ Hk
rκ(τ,Ω)
ρκ(τ,Ω)
- sign correlation coefficient for layer k and lag τ
- variance of the sign correlation coefficients
- rms of wavelet coefficients for IFO L
- rms of wavelet coefficients for IFO H
- linear correlation coefficients for (xh+hh)*(xl+hl)
- sign correlation coefficients (xh+hh)*(xl+ hl)
use stochastic DSO to calculate
¾ Wκ(τ)
z
-------------------------------
UF
------ - optimal filter in wavelet domain
run analysis on playground data (xh) and simulation hh(Ω)
S.Klimenko
Upper Limit
z
Cross-correlation & variance:
Vs = ∑ N kω k2 (τ )vk (τ )
S s = ∑ N kω k (τ ) ρ k (τ )
n ,τ
n ,τ
z
x-correlation expectation value:
z
signal to noise ratio:
z
z
UF
confidence level:
upper limit:
S.Klimenko
SNR = Ω
µ = Ω∑ N kω k2 (τ )vk (τ ) = ΩVs
n ,τ
2
N
ω
∑ k k (τ )vk (τ ) = Ω Vs
n ,τ
 Ss 
1
 → S~s (95CL)
CL = erf 
 V 
2
 s
~
~ S s (95CL)
Ω=
Vs