LIGO-G1200759
Introduction To Signal Processing & Data AnalysisGregory Mendell
LIGO Hanford Observatory
The Laser Interferometer Gravitational-Wave Observatory
http://www.ligo.caltech.edu
Supported by the United States National Science Foundation
Fourier Transforms
dextx
dtetxx
ti
ti
)(~2
1)(
)()(~
tite
fti
sincos
2
Digital Data
1,...2,1,0;;/
/1;1,...2,1,0;
2/;
)/(1
NkkTfTkf
TfNjtjt
ffNTf
tf
kk
j
sNyquists
s
t
x
j=0 j=1 j=2 j=3 j=4 j=N-1 T
…
.
.
.
.
.
. .
Sample rate, fs:
Duration T, number of samples N, Nyquist frequency:
Time index j: Frequency resolution:
Frequency index k:
Discrete Fourier Transforms(DFT)
1
0
/2
1
0
/2
~1
~
N
k
Nijkkj
N
j
Nijkjk
exN
x
exx
The DFT is of order N2. The Fast Fourier Transform (FFT) is a fast way of doing the DFT, of order Nlog2N.
DFT Aliasing
]2/,0[],0[:
~~;~~;
~~;
*
*
NkfBandUseful
xxxxmfff
xxff
Nyquist
kkNkmNks
kk
For integer m:
Power outside this band is aliased into this band. Thus need to filter data before digitizing, or when changing fs, to prevent aliasing of unwanted power into this band.
DFT Aliasing
DFT Orthogonality
jN
j
NkkiN
j
NkkijN
j
NijkNijk eeee
1
0
/)'(21
0
/)'(21
0
/2/'2
',;',0
1
1
1
1/)'(2
)'(2
/)'(2
/)'(21
0
/)'(2 kkNkke
e
e
ee
Nkki
kki
Nkki
NNkkijN
j
Nkki
'
1
0
/2/'2kk
N
j
NijkNijk Nee
r
rSrrrrSrrSrS
NN
N
j
jN
j
jjN
j
jN
j
j
1
1;1;;
1
0'
'
1'
1'1
0
11
0
Parseval’s Theorem
1
0
*1
0
1
0
21
0
2
~~1
~1
N
kkk
N
jjj
N
kk
N
jj
yxN
yx
xN
x
Correlation Theorem
*
1
0''
~~~kkk
N
jjjjj
yxc
yxc
Convolution Theorem
kkk
N
jjjjj
yxC
yxC
~~~
1
0''
Example: Calibration• Measure open loop gain G,
input unity gain to model• Extract gain of sensing
function C=G/AD from model
• Produce response function at time of the calibration, R=(1+G)/C
• Now, to extrapolate for future times, monitor calibration lines in DARM_ERR error signal, plus any changes in gains.
• Can then produce R at any later time t.
• A photon calibrator, using radiation pressure, gives results consistent with the standard calibration.
)(_)()(
)(
)(1)(
)()()()(
fERRDARMfRfx
fC
fGfR
fAfDfCfG
ext
DARM_ERR
One-sided Power Spectral Density (PSD) Estimation
T
txP
k
k
22~2
Note that the absolute square of a Fourier Transform gives what we call “power”. A one-sided PSD is defined for positive frequencies (the factor of 2 counts the power from negative frequencies). The angle brackets, < >, indicate “average value”. Without the angle brackets, the above is called a periodogram. Thus, the PSD estimate is found by averaging periodograms. The other factors normalize the PSD so that the area under the PDS curves gives the RMS2 of the time domain data.
Gaussian White Noise
1
0
21
0
222
21
0
22
~1~
1;0
N
k
N
jjkk
N
jjjj
NnnN
n
nN
nn
Parseval’s Theorem is used above.
Power Spectral Density, Gaussian White Noise
2222/
0
22/
0
22
22
2
2
12
2
22
2
RMSAreaN
N
TffP
fP
ft
T
tNP
N
k s
N
kk
s
sk
The square root of the PSD is the Amplitude Spectral Density. It has the units of sigma per root Hz.
This ratio is constant, independent of fs.
For gaussian white noise, the square root of the area under the PSD gives the RMS of time domain data.
Amplitude and Phase of a spectral line
0
00
2~
2)2cos(
22
0
ik
iiftiift
jj
eAN
x
kfT
eeAftAx
jj
If the product of f and T is an integer k, we call say this frequency is “bin centered”.
For a sinusoidal signal with a “bin centered” frequency, all the power lies in one bin.
DFT Bin Centered Frequency
DFT Leakage
22
2222
22
0
)(sin
4~
2
)2cos(1
2
)2sin(
2~
;2
)2cos(
0
00
NAx
ieAN
x
kfT
eeAftAx
k
ik
iiftiift
jj
jj
For a signal (or spectral disturbance) with a non- bin-centered frequency, power leaks out into the neighboring bins.
For kappa not an integer get sinc function response:
DFT Leakage
Windowing reduces leakage
1;442
~
1
2cos1
2
1:
1,1,0,
NNforNNN
w
N
jwWindowHann
kkkk
j
11
0'''
~4
1~4
1~2
1~
~~1~
kkkw
k
N
kkkk
wkjj
wj
xxxx
wxN
xxwx
Corrections to PSD
ff
S
TA
S
TA
S
TASNR
nnCorrectionEnergy
xxCorrectionAmplitude
kkk
wkk
wkk
2
3
)3/2(
2
1
)8/3(2
2/)2/(2
2
1
2
1
~3
8~:
~2~:
2
2
2222
22
Example PSD Estimation
For iLIGO and aLIGO
PSD Statistics
dedrdrdr
drredrrdrdredrdr
rdrddxdyxyyxr
dxdyeedxdyyx
yxyxifyxn
iyxn
rr
yx
2/2
2/2/
122
2/2/
22222
2
1)(;
2
1;
)(;2
1),(
);/(tan;
2
1
2
1),(
1;0;~
~
22
22
Rayleigh Distribution:
Chi-squared for 2 degrees of freedom:
Histograms, Real and Imaginary Part of DFT
For gaussian noise, the above are also gaussian distributions.
Rayleigh Distribution
Histogram of the square root of the DFT power.
Histogram DFT Power
Result is a chi-squared distribution for 2 degrees of freedom.
Chi-Squared Statistics
ded
rdrdrddrdrdxdydzge
zyxzyxr
zyxifzyx
2/2/1
22
2222222
222
)(
2;;sin.,.
1......;
0......;
ded 2/12
/2 )2/(2
1)(
A chi-squared variable with ν degrees of freedom is the sum of the squares of ν gaussian distributed variables with zero mean and unit variance.
0,0,0cos
,0
),,cos,(.,.
2
12
)(
...2
1
2
1
2
1)|(
2
0
22
0
2
00
222
22
2
)(
3
2
)(
2
2
)(
1
23
233
22
222
21
211
h
hfhge
hhhxhx
eeehxP
j
j j
jj
j j
jj
j j
jj
hxhxhx
Likelihood of getting data x for model h for Gaussian Noise:
Chi-squared
Minimize Chi-squared = Maximize the Likelihood
Maximum Likelihood
k k
kk
k k
kk
k k
kk
k k
kk
j j
jj
j j
jj
S
hh
S
hxSNR
hh
hx
hh
hxA
hhAhxA
hhA
hxAAhhge
hhhx
S
hh
S
hxhhhx
~~/
~~;
),(
),(
2
1ln;
),(
),(
0),(),(/ln
),(2
),(ln;.,.
),(2
1),(ln
~~
2
1~~
2
1ln
**2
max
2
**
22
Matched Filtering
This is called the log likelihood; x is the data, h is a model of the signal.
Maximizing over a single amplitude gives:
This can be generalized to maximize over more parameters, giving the formulas for matched-filtering in terms of dimensionless templates.
LIGO-G050492-00-W
Frequentist Confidence Region
• Estimate parameters by maximizing likelihood or minimizing chi-squared
• Injected signal with estimated parameters into many synthetic sets of noise.
• Re-estimate the parameters from each injection
ftBftAts 2sin2cos)( Matlab simulation for signal:
Figure courtesy Anah Mourant, Caltech SURF student, 2003.
Ch
hxhx
dxxhPC
hxPxP
hPxhP
eehxP
baPbPabPaP
0
2
)(
2
2
)(
1
)|(
)|()(
)()|(
...2
1
2
1)|(
)|()()|()(
22
222
21
211
Bayes’ Theorem:
Likelihood:
Confidence Interval:
Posterior Probability:
Prior
Bayesian Analysis
LIGO-G050492-00-W
Bayesian Confidence Region
x
Matlab simulation for signal: ftBftAts 2sin2cos)( Uniform Prior: )2/exp();,( 2xBAP
Figures courtesy Anah Mourant, Caltech SURF student, 2003.
The End
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