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