Noise Cancellation for
Gravitational Wave Detectors
Jenne Driggers
California Institute of Technology
December 2014
Noise Sources
and Cancellation
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Noise Sources
Seismic noise
Alignment and angular noise coupling to gravitational wave channel
Newtonian gravitational noise
Auxiliary length control coupling to gravitational wave channel
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Feedback vs. Feedforward
Residual
Feedback
Filter
S
FB
Sensor
Setpoint
Actuator
+
S
FF
Ground
Motion
-
Actuator
Plant
Feedforward
Filter
Witness
Sensor
Disturbance
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Feedback vs. Feedforward
FB
Feedback
S
Pros:
A
S
FF
Can handle small variations in
the plant
Only need rough model of plant
Cons:
Time lag
Disturbances must pass through system
Feedforward
Pros:
Cons:
Does not require disturbance to
propagate through system
Requires very accurate
model of system
Predicts incoming disturbances
Can only handle disturbances
that are externally witnessed
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Global Seismic Noise
Cancellation
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Seismic Noise Cancellation
Global seismic cancellation has been done before, but most of the
focus in recent years has been on local seismic cancellation
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Seismic Noise Cancellation
Global seismic cancellation has been done before, but most of the
focus in recent years has been on local seismic cancellation
Static noise cancellation simulations, and
adaptive implementation at the 40m:
J. C. Driggers, M. Evans, K. Pepper, R. Adhikari "Active noise cancellation
in a suspended interferometer" Rev. Sci. Instrum. 83, 024501 (2012) Go to paper
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Seismic Noise Cancellation
Global seismic cancellation has been done before, but most of the
focus in recent years has been on local seismic cancellation
Static noise cancellation simulations, and
adaptive implementation at the 40m:
J. C. Driggers, M. Evans, K. Pepper, R. Adhikari "Active noise cancellation
in a suspended interferometer" Rev. Sci. Instrum. 83, 024501 (2012) Go to paper
Static noise cancellation implementation during Enhanced LIGO:
R. DeRosa, J. C. Driggers, D. Atkinson, H. Miao, V. Frolov, M. Landry, J. A. Giame,
R. Adhikari "Global feed-forward vibration isolation in a km scale interferometer"
Go to paper
Class. Quantum Grav. 29, 215008 (2012)
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Optimal Wiener Filters
Ground
Motion
Plant /
Coupling
x(n)
Wiener
Filter w
d(n)
e(n)
Noise-suppressed signal
y(n)
R is auto-correlation matrix - how is sensor self-correlated?
p~ is cross-correlation vector - how are the sensor and the desired signal related?
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Optimal Wiener Filters
Ground
Motion
d(n)
Plant /
Coupling
x(n)
Wiener
Filter w
e(n)
Noise-suppressed signal
y(n)
R is auto-correlation matrix - how is sensor self-correlated?
p~ is cross-correlation vector - how are the sensor and the desired signal related?
Define a cost function, set derivative equal to zero:
Solve for w:
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w
~ optimum = R
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⇠ = E[e(n)2 ]
p~
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Optimal Wiener Filters
Ground
Motion
d(n)
Plant /
Coupling
x(n)
Wiener
Filter w
e(n)
Noise-suppressed signal
y(n)
R is auto-correlation matrix - how is sensor self-correlated?
p~ is cross-correlation vector - how are the sensor and the desired signal related?
Define a cost function, set derivative equal to zero:
Solve for w:
w
~ optimum = R
1
⇠ = E[e(n)2 ]
p~
Numerical precision problems arise for matrix inversion and
witness pre-filtering
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Wiener Filters
Cost function is the mean square error between the target
signal and the estimate of the target
2
1 X4
di
MSE =
2 i
N
X
j=0
x = witness sensor
32
d = target signal
w j xi j 5
w = Wiener coefficients
N = filter order
@MSE
=0
To find the extrema, we set
@wj
N
X
hj R(j
This gives us the Wiener-Hopf equations pi =
R is a symmetric Toeplitz matrix
R(j
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i)
0
B
B
=B
@
December 2014
i)
j=0
R[0]
R[1]
..
.
R[N ]
R[1]
R[0]
..
.
R[N
1]
···
···
..
.
R[N ]
R[N 1]
..
.
···
R[0]
1
9
C
C
C
A
Static Seismic Noise Cancellation
Sensor locations relative to
core optics
Seis
Horizontal axes only,
no vertical information used
Seis
Laser
Faraday
Seis
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Static Seismic Noise Cancellation
LLO Power Recycling Cavity Residual Length
−5-5
10
-6
10
-7

m
p
Control Signal
Hz
[m/√Hz]
1010
On
Off
Ground Motion
Local damping
a
10
-8
10
-9
−10
-10
trol Signal
1010
−5
10
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-1
10
0
Frequency [Hz]
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MICH Both Paths
MICH First Path
PRC Both Paths
PRC First Path
1
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Static Seismic Noise Cancellation
LLO Power Recycling Cavity Residual Length
−5-5
10
-6
10
-7

m
p
Control Signal
Hz
[m/√Hz]
1010
On
Off
Ground Motion
Local damping
Online
feedforward
a
10
-8
10
-9
−10
-10
trol Signal
1010
−5
10
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-1
10
0
Frequency [Hz]
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MICH Both Paths
MICH First Path
PRC Both Paths
PRC First Path
1
11
Static Seismic Noise Cancellation
LLO Power Recycling Cavity Residual Length
−5-5
10
-6
10
-7

m
p
Control Signal
Hz
[m/√Hz]
1010
On
Off
Ground Motion
Online
feedforward
a
10
10
-8
Outcomes:
-9
Lower glitch rate
−10
-10
trol Signal
1010
−5
10
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Local damping
10
-1
10
0
Frequency [Hz]
December 2014
Facilitated 1S6 high power
10
MICH Both Paths
MICH First Path
PRC Both Paths
PRC First Path
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Offloading control signals to actuators located in the extern
systems mitigates several of the problems mentioned in Section 1.
subtract the filtered witness signals, the transfer function from ou
to the cavity control signal must be measured and divided out.
the relevant pieces of the mechanical structure can be seen in Figu
mechanical components separating the mirrors from the ground c
transfer function with many resonant features.
Static Seismic Noise Cancellation
Challenges:
Measure transfer function between
actuator and desired signal to 1%
Fit measured transfer function, for
pre-filtering use, before Wiener
filter was calculated
Then fit Wiener filter to implement
in digital real-time system
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Isolation
Stack
Sensor &
Actuator
Suspension
Optic
Seismometer
Figure 4. As described in Section 1, the interferometer opti
isolation stack. The signals from the ground seismometer are
stack via piezo (LHO) or hydraulic (LLO) actuators as indic
circles on the optic represent the four electromagnetic actua
December
mirror.
The2014
dashed line represents the vacuum chamber.12
systems mitigates several of the problems mentioned in Section 1. In order to
subtract the filtered witness signals, the transfer function from our point of
to the cavity control signal must be measured and divided out. A diagram
the relevant pieces of the mechanical structure can be seen in Figure 4. The n
mechanical components separating the mirrors from the ground creates a co
transfer function with many resonant features.
Static Seismic Noise Cancellation
Challenges:
Isolation
Stack
Sensor &
Actuator
Measure transfer function between
actuator and desired signal to 1%
Suspension
Optic
Seismometer
Fit measured transfer function, for
pre-filtering use, before Wiener
filter was calculated
Figure
40 4. As described in Section 1, the interferometer optic is suspend
isolation stack.
The signals from the ground seismometer are applied jus
Measurement
stack via piezo (LHO) or hydraulic (LLO) actuators as indicated in the
20 on Fit
circles
the optic represent the four electromagnetic actuators on the
10x
Residual
mirror. The
dashed
line represents the vacuum chamber.
0
!20
!40
Phase [deg]
Then fit Wiener filter to implement
in digital real-time system
Magnitude [dB]
Global feed-forward vibration isolation in a km scale interferometer
180
135 Plant transfer function
90
45
0
!45
!90
!135
!180
!1
0
10
10
Frequency [Hz]
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Figure 6. Example plant transfer function, fit using Vectfit. The averag
December 2014 error of points shown was ∼ 2.5% in magnitude and ∼ 1.5◦ in phase.
12
systems mitigates several of the problems mentioned in Section 1. In order to
subtract the filtered witness signals, the transfer function from our point of
to the cavity control signal must be measured and divided out. A diagram
the relevant pieces of the mechanical structure can be seen in Figure 4. The n
mechanical components separating the mirrors from the ground creates a co
transfer function with many resonant features.
Static Seismic Noise Cancellation
Challenges:
Isolation
Stack
Sensor &
Actuator
Measure transfer function between
actuator and desired signal to 1%
Suspension
Optic
Seismometer
Fit measured transfer function, for
pre-filtering use, before Wiener
filter was calculated
Figure 4. As described in Section 1, the interferometer optic is suspend
isolation
stack. The signals from the ground seismometer are applied jus
0
stack via piezo (LHO) or hydraulic (LLO) actuators as indicated in the
circles on the optic represent the four electromagnetic actuators on the
mirror. The dashed line represents the vacuum chamber.
!20
Measurement
Fit
10x Residual
!40
!60
Phase [deg]
Then fit Wiener filter to implement
in digital real-time system
Magnitude [dB]
Global feed-forward vibration isolation in a km scale interferometer
180
135
90
45
0
!45
!90
!135
!180
Wiener filter
!1
10
0
10
Frequency [Hz]
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December 2014 Figure 7. Example Wiener filter, corrected for the mechanical plant
12respon
using Vectfit.
Adaptive Noise Cancellation
Residual
Similar to static
technique, but can
follow changes in
transfer function
Adjust the
feedforward filter in
realtime for optimal
cancellation
Setpoint
+
Adaptation
Algorithm
Actuator
Plant
Feedforward
Filter
Plant
Estimate
Witness
Sensor
Disturbance
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Adaptive Noise Cancellation
Similar to static technique, but can follow changes in transfer function
Should converge to the static Wiener solution
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Adaptive Noise Cancellation
Similar to static technique, but can follow changes in transfer function
Should converge to the static Wiener solution
We use a leaky normalized filtered-x least mean squared (LMS) algorithm
Combination of 3 modifications to the simple LMS algorithm
w(n + 1) = w(n) + µ(1
⌧ )x(n)e(n)
w
~ is the vector of filter weights
µ is the step size
~x is the witness signal
~e is the residual error signal
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Adaptive Noise Cancellation
Similar to static technique, but can follow changes in transfer function
Should converge to the static Wiener solution
We use a leaky normalized filtered-x least mean squared (LMS) algorithm
Combination of 3 modifications to the simple LMS algorithm
w(n + 1) = w(n) + µ(1
⌧ )x(n)e(n)
w
~ is the vector of filter weights
µ is the step size
~x is the witness signal
~e is the residual error signal
Leaky: allow response to decay using (1
⌧)
µ
Normalized: µ is a function of time µ(n) = T
~x (n)~x(n)
Filtered-x: pre-filter ~x(n) with an estimate of the plant transfer function
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Adaptive Noise Cancellation
Power spectrum
Power spectrum
C1:LSC-XARM_OUT
Implemented at the 40m
C1:LSC-XARM_OUT(REF0)
102
C1:IOO-MC_F
C1:IOO-MC_F(REF2)
10
1010101
1
-1
10
11
1
10
-1-1
11
1
*T0=01/12/2012 07:59:20
*T0=01/12/2012 07:59:20
*Avg=84
Frequency [Hz]
Frequency (Hz)
Frequency (Hz)
BW=0.1875
*Avg=84
*Avg=84
22
Magnitude
Magnitude Magnitude
2
C1:LSC-YARM_OUT
C1:LSC-YARM_OUT
C1:LSC-YARM_OUT(REF1)
C1:LSC-YARM_OUT(REF1)
10
1
10
-1-1
10101
*T0=01/12/2012
07:59:20
10-1
10
10
Frequency (Hz)
Frequency
[Hz]
*Avg=84
1
1
*T0=01/12/2012 07:59:20
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BW=0.1875
10
Frequency (Hz)
*Avg=84 (Hz)
Frequency
*Avg=84
BW=0.18
10
1
1
*T0=01/12/2012 07:59:20
*T0=01/12/2012 07:59:20
10
10
Frequency (Hz)
Frequency (Hz)
Frequency
[Hz]
*Avg=84
*Avg=84
BW=0.1875
BW=0.1875
1
102
1
10
Frequency (Hz)
*T0=01/12/2012 07:59:20
No seismic noise cancellation
C1:IOO-MC_F / C1:LSC-XARM_OUT
0.9
C1:LSC-YARM_OUT(REF1)
1
1
101010
C1:LSC-YARM_OUT
10 1010
-1
10-1
Coherence
Power spectrum
Power spectrum
1
111
1
BW=0.1875
BW=0.1875
10 10
1
C1:IOO-MC_F
C1:IOO-MC_F
C1:IOO-MC_F(REF2)
C1:IOO-MC_F(REF2)
10
10
10
Y arm
Power spectrum
1
-1-1-1
Frequency (Hz)
10
*T0=01/12/2012
07:59:20
10-1
Magnitude
10
Mode Cleaner
Power
Powerspectrum
spectrum
Magnitude
Magnitude
Magnitude
C1:LSC-XARM_OUT(REF0)
C1:LSC-XARM_OUT(REF0)
10
0.8
Coherence
Coherence
10.7
1
0.9 0.6
0.9
0.8 0.5
0.8
0.7 0.4
0.7
0.6 0.3
0.6
0.5 0.2
0.5
0.4 0.1
0.4
0.3
0
0.3
0.2
Coherence
Coherence
Coherence
Magnitude
10
10
Magnitude
Magnitude
Magnitude
2
C1:LSC-XARM_OUT
C1:LSC-XARM_OUT
2 2
10
Magnitude
X arm
Power
spectrum
Power
spectrum
C1:IOO-MC_F / C1:LSC-YARM_OUT
Adaptive noise cancellation ON
C1:IOO-MC_F / C1:LSC-XARM_OUT
C1:IOO-MC_F
/ C1:LSC-XARM_OUT
C1:IOO-MC_F
/ C1:LSC-YARM_OUT
C1:IOO-MC_F / C1:LSC-YARM_OUT
µ(Mode Cleaner) = 0.2
µ(Arms) ~ 0.04
1
0.2
0.1
T0=01/12/2012 07:59:20
0.1
0
1
0
1
T0=01/12/2012 07:59:20
December
2014
BW=0.1875
10
Frequency (Hz)
Avg=84
BW=0.18
10
Frequency (Hz)
Avg=84
Frequency (Hz)
10
15
BW=0.1875
Future Extensions
Offline analysis of S5 H1/H2 to potentially improve stochastic searches
Other external sensors
Laser power monitors
Microphones
Magnetometers
Offline analysis on One Arm Test data to see aLIGO potential
Remove auxiliary length degrees of freedom from gravitational wave channel
Implement online (work already begun at LLO by others)
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