LTP Data Analysis Algorithms Development

LTP Data Analysis Algorithms

Development

Luigi Ferraioli

1Luigi Ferraioli - 7th International LISA

Symposium - 17 June 2008, Barcelona

LTP Developers team: Mauro Hueller, Nicola Alex Tateo, Martin Hewiston, Anneke Monsky, Miquel Nofrarias,

Gudrun Wanner, Ingo Diepholz, Adrien Grenagier, Walter Fichter, Josep Sanjuan, Alberto Lobo and Stefano Vitale

Outline

Lisa Pathfinder Dynamics

along x axis

Approximate Calculation of

Second Order Derivatives

Power Spectra

Estimation Tools

Mock Data Challenge II,

signal generation and

parameter estimation

Whitening Filter



LTP Dynamics

0 i n

n

D q g

g g C o o g

o S q o

0

is a vector containing the dynamical coordinates

is a force-per-unit mass vector

is the force-per-unit mass disturbance (it summarize all the possible force-noise sources)

is a force-per-un

n

q

g

g

g

it mass vector purposely applied to the system

is the available signal vactor

is an input signal vector used to apply forces trough the control loops

is the dynamical matrix

is the feedback

i

o

o

D

C

circuit matrix. It converts available signals into commanded force

converts the coordinate vector into the measured signal vectorS

System Dynamics



LTP Dynamics

Looking at the interferometer output…

11

o no D S C a o o

Is defined as the system nominal response

Is the colored noise output

11

o io D S C C o

11 1

n n no D S C g D S o

And going back…

1 11 1

o na D S C o D S C o o

Where…

1o

o

o

oo

o

and

1n

n

n

oo

o



LTP Dynamics

In imaginary angular frequency notation s…

2 2 2 2

1 1 2 2 2 2

2 2 2 2

2 1 2 2

1

2

p p p

p p p

sD

s

Where…

2

0

df lfs

lfs

H hC

h

11 1

1

S SS

S S

1 1 2 2

2 1

1 n n

n

n n

g g Gg

g g

μ i Is the ratio between TMs and S/C masses ( ~ 10-3 )

ω2pi is the electrostatic parasitic stiffness

Γ is the gravity gradient between TMs

gni is a noisy force per unit mass acting on TMs from

disturbances internal to the S/C

G is the force disturbance acting on the S/C and

generated from external sources

Sij are the terms of the calibration matrix5

Luigi Ferraioli - 7th International LISA


Derivative Estimation

We have a continuous dynamics and a discrete control circuit…

2 2 2 2

1 1 2 2 2 2

2 2 2 2

2 1 2 2

1

2

p p p

p p p

sD

s

We decided to make a discrete approximation

of the second derivative represented by s2…

2

0

df lfs

lfs

H hC

h

Parabolic fit approximation

to the second derivative

Taylor series expansion

approximation to the

second derivative



Derivative Estimation – parabolic fit

approximation

22

0 1 2 22 2i

i

t kT

d o to k m T k k mT k mT

dt

0t kT mNote:

The fit procedure

22 1 1 2

2 2

1 2 1 2 1 2

7 7 7 7 7

dz z z z z

dt T



Derivative Estimation – series expansion

approximation

2 3 4

5

0 0 0 0 0 02 6 24

I II III IVnT nT nTf x nT f x nTf x f x f x f x O T

22 1 1 2

2 2

1 1 16 30 16 1

12 12 12 12 12

dz z z z z

dt T

This method is based on the series expansion of a function representing data points…

We obtain a five point estimator if the

expansion is performed for:2, 1,0,1,2n

Putting the five expansions in a

system and solving out for the

second derivative f II[x0 ]



Derivative Estimation – General equation

22 1 1 2

2 2

1dz az bz c bz az

dt T

The two methods are part of an entire family of derivative estimators…

The parameters a, b and c

are not independent:

iz e

In Laplace notation the second

derivative estimator is:

2 cos(2 ) 2 cos( )a b c

2 2s

We expect our estimator going

to zero at φ = 0 so:2 2 0a b c

Then we expect the estimator

tending to φ2 for small φ so:4 1a b

The final result is: 2 cos(2 ) 2 1 4 cos( ) 2 1 3a a a9




Parabolic fit:2

7a Has a root at 2.42 0.39

s

frad

f

Series expansion:1

12a Does not have roots between 0 and π

Root at π :1

4a Has a root at π

0 0.5 1 1.5 2 2.5 3 3.5-10

-8

-6

-4

-2

0

2

[rad]

Am

pli

tud

e

s2

Parabolic Fit

Series Expansion

Zero at

10-6

10-4

10-2

100

10-15

10-10

10-5

100

105

f / fs

Am

pli

tud

e

s2

Parabolic Fit

Series Expansion

Zero at



10-1

100

10-2

10-1

100

101

f / fs

Am

pli

tud

e

s2

Parabolic Fit

Series Expansion

Zero at


Difference is roughly one

order of magnitude.

We know this can generate

problems in estimating the

PSD of the acceleration

noise at low frequencies



Power Spectral Density Estimation

We have to different tools in the toolbox

pwelch

Generalizes the standard Matlab pwelch

allowing for several detrending options

• linear frequency spacing

• FFT algorithm

lpsd*

• logarithmic frequency spacing

• DFT algorithm

* M. Tröbs and G. Heinzel, Measurement 39 (2006) 120-129 12Luigi Ferraioli - 7th International LISA



10-5

10-4

10-3

10-2

10-1

100

101

10-14

10-13

10-12

10-11

10-10

10-9

10-8

Am

plitu

de

[m

s-2

Hz-1

/2]

Frequency [Hz]

a12 - parfit - lpsd - 100 ave - linfit detrend

a12 - parfit - pwelch - 10 ave - linfit detrend

a12 - series - lpsd - 100 ave - linfit detrend

a12 - series - pwelch - 10 ave - linfit detrend




10-5

10-4

10-3

10-2

10-1

100

101

10-14

10-13

10-12

10-11

10-10

10-9

10-8

Am

plitu

de

[m

s-2

Hz-1

/2]

Frequency [Hz]





No difference between parfit

and series with lpsd




10-5

10-4

10-3

10-2

10-1

100

101

10-14

10-13

10-12

10-11

10-10

10-9

10-8

Am

plitu

de

[m

s-2

Hz-1

/2]

Frequency [Hz]





Different noise levels

between parfit and

series with pwelch



Signal generation

We will adopt an analytical approach, where signal and colored noise are generated

apart and then added each other…

t s t n t

Colored noise generation with a defined spectrum…

n is assumed Gaussian and stationary

* †

n niS H S H H is the system transfer function

Sni is the input noise signal

Generating signals from white noise…

†*

nS h I h I is the cross spectral matrix of an unitary variance

white noise process

This is equivalent to…

1/2 1/2 1

nS V I V V is the eigenvector matrix and Λ is

the eigenvalue matrix of Sn 16Luigi Ferraioli - 7th International LISA


Signal generation

From the equivalence…

The transfer function for noise generation can be found…

* 1/2h V We have written a LTPDA Toolbox function psd2tf

to perform the calculation of the innovation transfer

function h

And at the same time the whitening filter transfer function…

†*

nS h I h

A set of functions is now disposable in the LTPDA

Toolbox allowing for the whitening of the data starting

from a model of the frequency response of the noise

PSD and CSD

1/2 1/2 1

nS V I V

1w h



Whitening Filter

Coming back to the system dynamics…

2 2 2 2

1 1 2 2 2 2

2 2 2 2

2 1 2 2

1

2

p p p

p p p

sD

s

We see the parameters to

be estimated…

2

0

df lfs

lfs

H hC

h

11 1

1

S SS

S S

1 1 2 2

2 1

1 n n

n

n n

g g Gg

g g

Hdf and Hlfs are the gains of the control circuit loops

ω2pi is the electrostatic parasitic stiffness

Γ is the gravity gradient between TMs

gni is a noisy force per unit mass acting on TMs from

disturbances internal to the S/C

G is the force disturbance acting on the S/C and

generated from external sources

Sij are the terms of the calibration matrix18



Whitening Filter

Parameters estimation can be afforded by maximum likelihood

estimation. That is equivalent to minimize the function…

2,

, , ,

, 1 , 1

, , , , , ,dataN

k j k i k j i j

k j

o t o D S C t o t o D S C t

Where…

, ,

1,

, ,n nk j o o sampR k j T R is the cross correlation between

the two channel signals noise

If the noise is white…

, ,, , ,n no oR k j

The minimization process

is highly simplified



Whitening Filter

Model PSD

generation

mPSD o1

mPSD o12

mCSD

psd2wf

white filter

generation

wf11

wf12

wf21

wf22

Wf2_freq

white filter in

frequency domain

Input data (colored noise)

o1 o12

wo1

wo12

Output data (white noise)



Whitening Filter

10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-24

10-23

10-22

10-21

10-20

10-19

10-18

10-17

10-16

PS

D [

m2 H

z-1

]

Frequency [Hz]

moled So

So ave

+1/2 std

-1/2 std

PSD of interferometer output noise. Average on 11 simulated datasets

Method: pwelch, linear fit detrend, 10 average, BlackmannHarris window, 69% overlap

Spacecraft

Diff. Channel



Whitening Filter

PSD Average on 11 simulated datasets

Method: pwelch, linear fit detrend, 10 average, BlackmannHarris window, 69% overlap

Whitened Spacecraft Noise Whitened Diff. Channel Noise

10-4

10-3

10-2

10-1

100

101

10-2

10-1

100

101

PS

D [

m2 H

z-1

]

Frequency [Hz]

Swo1 ave

+1/2 std

-1/2 std

10-4

10-3

10-2

10-1

100

101

10-2

10-1

100

101

PS

D [

m2 H

z-1

]

Frequency [Hz]

Swo2 ave

+1/2 std

-1/2 std



Whitening Filter

Colored noise and model Whitened data

10-5

10-4

10-3

10-2

10-1

100

101

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Cro

ss C

oh

ere

nce

Frequency [ Hz ]

re - data

im - data

re - model

im - model

10-4

10-3

10-2

10-1

100

101

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Cro

ss C

ohere

nce

Frequency [ Hz ]

re - whiten data

im - whiten data

Cross Coherence is defined as1 12

CSD

PSD PSD



Whitening Filter - Signals

Looking at the

interferometer

output…

o no o o

System response to a commanded

input in the control circuit

Is the colored noise output

11

o io D S C C o

11 1

n n no D S C g D S o

1 01 11

0

sin 2

sin 2

ii

i

i i

A f too

o A f t

6

1

7

3

1

3

1

1 10 [ ]

8 10 [ ]

3.45 10 [ ]

4.7 10 [ ]

0

i

i

i

i

i i

A x m

A x m

f x Hz

A x Hz

10-4

10-3

10-2

10-1

100

101

10-30

10-25

10-20

10-15

10-10

10-5

PS

D [

m2H

z-1]

Frequency [Hz]

oi1

oi

oo1

oo




10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-25

10-20

10-15

10-10

10-5

PS

D [

m2H

z-1

]

Frequency [Hz]

oo1

+ on1

oo

+ on

model on1

model on

10-4

10-3

10-2

10-1

100

101

10-5

100

105

1010

1015

PS

D [

m2H

z-1

]

Frequency [Hz]

w(oo1

+ on1

)

w(oo

+ on

)

Output noise + signal Whitened (output noise + signal)




Real and Imaginary part of colored and whitened output noise + signal

10-4

10-3

10-2

10-1

100

101

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Cro

ss

Co

he

ren

ce

Frequency [Hz]

Re[oo1

+ on1

]

Im[oo1

+ on1

]

Re[w(oo1

+ on1

)]

Im[w(oo1

+ on1

)]



Thank You



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LTP Data Analysis Algorithms Development

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