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ID Date Author Type Category Subject
6491   Fri Apr 6 09:57:24 2012 DenUpdateAdaptive Filteringstatic starts to work

I made static filter to work to evaluate the actuator TF.. Here is the result of static filtering:

What I did:

I did offline simulation of the MC_F Wiener filtering using 2 witness signals - GUR1X and GUR1Y. I've downsampled the data from 2048 to 128 Hz and applied the Wiener filter with 10000 for each witness channel:

Result of the filtering                                                                                     Filter coefficients for gur1x and then gur1y

Gur1x -> MC_F transfer function                                                                          Gur1y -> MC_F transfer function

Then using vectfit I approximated obtained transfer functions in the region 0.5 - 20 Hz. I used a window function and then weights to get a more precise result in this range using only 8 poles and zeros.

I obtained the zpk-model for each witness channel and entered it into the FOTON splitting it into 2 parts before that because FOTON does not like too long filters. These zpk-models are at the C1:OAF-STATIC_STATMTX_8_8 and C1:OAF-STATIC_STATMTX_8_9 filter banks.

GUR1X:

z =

7.527339430029315 +31.603999069997801i
7.527339430029230 -31.603999069997823i
27.897703898191267 + 0.000000000000071i
-6.437806394766186 + 9.893955654289517i
-6.437806394766159 - 9.893955654289510i
1.114401249545640 + 5.479278396987240i
0.176877296954015 + 0.000000000000006i
1.114401249545616 - 5.479278396987245i

p =

-0.407251778925379 + 6.263247012022007i
-0.407251778925379 - 6.263247012022007i
-0.230672968859081 + 6.846868757063707i
-0.230672968859081 - 6.846868757063707i
-2.871419857491615 +13.707864827517826i
-2.871419857491615 -13.707864827517826i
-2.134260618362721 +18.319129999777648i
-2.134260618362721 -18.319129999777648i

k =

4.113285626223658e-04

GUR1Y

z =

17.961416874092624 +13.631821252434328i
17.961416874092642 -13.631821252434353i
-8.788634771726304 + 7.653357335975781i
-8.788634771726285 - 7.653357335975777i
-0.037906973323273 + 5.133348020858679i
-0.164348392996182 + 3.588803405511463i
-0.164348392996187 - 3.588803405511474i
-0.037906973323277 - 5.133348020858679i

p =

-0.027577318242359 + 5.174655410828068i
-0.027577318242359 - 5.174655410828068i
-0.500384298611703 + 6.310552036591990i
-0.500384298611703 - 6.310552036591990i
-0.237055716999485 + 6.881204941979009i
-0.237055716999485 - 6.881204941979009i
-1.408223271160550 +14.874570175309771i
-1.408223271160550 -14.874570175309771i

k =

-2.723835471763049e-04

Then I approximated the reversed actuator TF  and placed it to the C1:OAF-SUS_MC2_OUT filter bank. The gain to the static filter output is -1.

P.S. Also the static matrix was filled with 1 for some reason. Here is the script to fix it if if will be bad again

for i in {1..8}
do
for j in {1..28}
do
element="C1:OAF-STATIC_STATMTX_"$i"_"$j"_GAIN"
ezcawrite \$element 0
done
done

I've run static and adaptive filters simultaneously. AA32 filters rotate the phase of the witness signals GUR1X and GUR1Y and now the performance of the static filter is worse. Next time I'll recalculate Wiener filter coefficients taking this into account. But still 2 filters together can deal with a stack better.

 Quote: I've run static and adaptive filters simultaneously. AA32 filters rotate the phase of the witness signals GUR1X and GUR1Y and now the performance of the static filter is worse. Next time I'll recalculate Wiener filter coefficients taking this into account. But still 2 filters together can deal with a stack better.

This is super awesome!  I'm totally excited!!

6551   Thu Apr 19 22:18:24 2012 DenUpdateAdaptive Filteringoaf algorithm: old vs new

Here are the issues that I found not quite accurate in the old oaf code:

1. There is no need to calculate the norm of the witness signal every time from zero:

Every step the witness signal vector is the same except the first and last values

This step will reduce the number of multiplications and summations from 3*M/k to 2*M/k, M - filter length, k - downsample ratio.

2. Old code filter corrects filter coefficients with a delay equal to k=downsample ratio (pretty big):

witness       o o o o o o o o o o o o o o o o o o o o o

error           o o o o o o o o o o o o o o o o o o o o o

We want the filter to work at green points and skip red points computing output and correcting coefficients at this time (downsample ratio in this example is 4). Old code

• grabs error signal
• calculates output during next k-1 red points and 1 green point
• corrects coefficients using this error during next k-1 red points and 1 green point

But LMS algorithm should correct coefficients according to the latest error. As we calculate output and correct coefficients before the latest error signal will be available, we should change the order:

• grabs error signal
• corrects coefficients using this error during next k-1 red points and 1 green point
• calculates output during next k-1 red points and 1 green point

This scheme is completely equivalent to the ordinary LMS algorithm, as now we correct coefficients according to the latest error signal and so do not add any delay.

3. So long soft start is probably not needed for the LMS filter, it makes the filter to converge longer

// modify adaptation gain for soft start

if( state.iWait < state.nFIR )

{

decayRate = 1.0;  // clear FIR coeffs after reset

}

As far as I understand this is done to prevent the filter from huge coefficients variations in the beginning when norm is not calculated yet. Instead we can just introduce some small

epsilon = 10.0;

to prevent the filter from divergence in the beginning

delta = mu * error / (wit[i].norm + epsilon);

Though some soft start might be needed by not so long - it will take several minutes before the adaptation gain will take it's specified value.

6553   Fri Apr 20 23:02:25 2012 DenUpdateAdaptive Filteringfrequency domain filter

DFT-LMS is a frequency domain adaptive filter that demonstrates faster convergence compared to the time-domain LMS filter. I've tested Discrete Fourier Transform (DFT-LMS) filter. It converts witness signal to the frequency domain using DFT and corrects the eigenvalues of the covariance matrix to make them as equal to each other as possible (does pre-whitenning of the witness signal).

Left plot compares learning curves for time domain LMS and DFT-LMS algorithms on the simulated data from seismometers and mcl (number of averages  = 30) Right plot shows the evolution of the filter coefficients norm (Euclidean norms of the coefficient vector). Though LMS algorithm works in the time domain and DFT-LMS in the frequency domain, the coefficient vectors must have the same length, because we Fourier Transform is achieved by applying a unitary operator => vector norm must not change.

Plots show that both algorithms converge to the same coefficients vector norm, but DFT-LMS does it much faster then LMS.

Online realization:

Good news: algorithm complexity is linear in filter length. Though the algorithm does Fourier transform, its complexity is still O(M), M - number of coefficients. Simulations show that DFT-LMS is ~8-9 times slower then LMS. This is not so bad, may be we can do even slightly better.

Bad news: downsample process is not simple. Due to Fourier transform, the filter needs the whole witness signal vector before calculating the output. This is sad and in contrast with LMS algorithm where we could start to calculate the new output immediately after computing the previous output. We either need to calculate the whole output immediately or introduce delay in the output or approximate Fourier transform with some previous witness signal values.

Realization in the kernel: I asked Alex about complex numbers, exponents, sin and cos functions in the kernel c and he answers that we do not have complex numbers, about exp, cos, sin he is not sure. But for DFT-LMS algorithm we are able to get round of these difficulties. Complex numbers will be presented as  2 real numbers. Then exp (a) = cos(a) + i*sin(a). All what we need for DFT-LMS are sin(2 * pi * k / M) and cos(2 * pi * k / M), k=0,1,2,...,M-1. Fortunately, M - (filter length) is big enough, typical value pi/M ~ 0.001 and we can calculate sin(2*pi/M) and cos(2*pi/M) using Taylor series. As the argument is small, 5-6 terms will be enough to get precision ~1e-20. Then we build the whole table of cos and sin according to induction cos(2*pi/M*k) = cos(2*pi/M*(k-1))cos(2*pi/M) - sin(2*pi/M*(k-1))sin(2*pi/M), sin(2*pi/M*k) = cos(2*pi/M*(k-1))sin(2*pi/M) + sin(2*pi/M*(k-1))cos(2*pi/M). We should do it only once, so the algorithm will build these values in the beginning during first several iterations, then will use them.

The main problem is downsampling. I need to think more about it.

6642   Fri May 11 23:33:41 2012 DenUpdateAdaptive Filteringoffline vs online

I've compared offline Wiener filtering with online static + adaptive filtering for MC_F with GUR1_XYZ and GUR2_XYZ as witness signals

Note: online filter works up to 32 Hz (AI filter at 32 Hz is used). There is no subtraction up from this frequency, just MC_F was measured in different times for online and offline filtering. This difference in MC_F in frequency range 20-100 Hz showed up again as before with microphone testing.  One can see it in 1 minute. Smth is noisy.

### Reasons why online filter is worse then offline:

1. FIR -> IIR conversion produces error. Now I'm using VECTFIT approximation with 16 poles (splitting into 2 filter banks), this not enough. I tried to use 50 and split them into 5 filter banks, but this scheme is not working: zpk -> sos conversion produces error and the result filter works completely wrong.

2. Actuator TF. VECTFIT works very good here - we have only 1 resonance. However, it should be measured precisely.

3. Account for AA and AI filters that rotate the phase at 1-10 Hz by ~ 10 degrees.

7000   Sat Jul 21 18:04:02 2012 DenUpdateAdaptive Filteringfrequency domain filter

I've implemented online frequency domain filter and applied it to MC_F.

+

Magnitude of the filter output at 1 Hz is the same as MC_F. This means that it is not hard for FIR to match the resonance. The problem is with the phase. We can not match the resonance exactly. If the resonance is at f0 and we match at f0 +/- df then in the frequency range (f0, f0 +/- df) the phase is not matched for 180. I guess the filter does not diverge because df is small but also the filter can not account for this huge phase lag. We need to slightly change the simulated actuator TF and see how the filter will react.

7234   Mon Aug 20 13:02:57 2012 DenUpdateAdaptive Filtering1 Hz resonance

Static filter was adjusted to filter 1 Hz resonance in MCL and it could do it. Stack is not great in this experiment due to the phase mismatch. I'll fix it.

7252   Wed Aug 22 20:33:51 2012 DenUpdateAdaptive FilteringMC_L in ARMS

Jenne and I did adaptive filtering of MC_L and measured how X and Y ARM control signals change compared to non-filtered MC_L. We did the test during 1.5 Hz seismic noise activity and adaptive filter was able to subtract it. However, it adds noise at high frequencies, It is not seen in MC_L but it is present in the ARMs control signals.

I'll investigate this problem. May be we need to reduce adaptation gain. In this experiment it was 0.1 and adaptive filter convergence time was equal to 1-2 mins.

7589   Mon Oct 22 20:44:49 2012 AyakaUpdateAdaptive Filteringmicrophone noise

I will do some experiments on acoustic noise canceling during my stay.
Now I am planning to cancel acoustic noise from PMC and see how the acoustic noise work and how we should place microphones.

First, I measured the noise in microphones and its circuit.

-blue, green, red, solid lines; microphone signals
-blue, green, red, dashed lines; un-coherent noise in signals
-yellow, black, solid lines; circuit noise (signal input is open, not connected to the microphones)

We can see the acoustic signal above 1 Hz, and the circuit does not seem to limit its sensitivity. But I do not know why yellow and black is so different. I will check it tomorrow.

7592   Tue Oct 23 00:51:41 2012 JamieUpdateAdaptive Filteringmicrophone noise

 Quote: I will do some experiments on acoustic noise canceling during my stay. Now I am planning to cancel acoustic noise from PMC and see how the acoustic noise work and how we should place microphones.a First, I measured the noise in microphones and its circuit. -blue, green, red, solid lines; microphone signals -blue, green, red, dashed lines; un-coherent noise in signals -yellow, black, solid lines; circuit noise (signal input is open, not connected to the microphones) We can see the acoustic signal above 1 Hz, and the circuit does not seem to limit its sensitivity. But I do not know why yellow and black is so different. I will check it tomorrow.

Hi, Ayaka.  It would be good if you could give a little bit more detail about this plot:

• What exactly are the "signals"?  Are you making a sound somehow?  If so, what is producing the sound?  What is it's spectrum?
• Are the blue/green/red traces from three different microphones?
• Coherence usually implies a comparison between two signals.  Is something being compared in the dashed traces?
• Are the yellow and black traces from different amplifiers?
• What are the units of the Y axis?

7596   Tue Oct 23 10:24:42 2012 AyakaUpdateAdaptive Filteringmicrophone noise

Quote:

 Quote: I will do some experiments on acoustic noise canceling during my stay. Now I am planning to cancel acoustic noise from PMC and see how the acoustic noise work and how we should place microphones.a First, I measured the noise in microphones and its circuit. -blue, green, red, solid lines; microphone signals -blue, green, red, dashed lines; un-coherent noise in signals -yellow, black, solid lines; circuit noise (signal input is open, not connected to the microphones) We can see the acoustic signal above 1 Hz, and the circuit does not seem to limit its sensitivity. But I do not know why yellow and black is so different. I will check it tomorrow.

Hi, Ayaka.  It would be good if you could give a little bit more detail about this plot:

• What exactly are the "signals"?  Are you making a sound somehow?  If so, what is producing the sound?  What is it's spectrum?
• Are the blue/green/red traces from three different microphones?
• Coherence usually implies a comparison between two signals.  Is something being compared in the dashed traces?
• Are the yellow and black traces from different amplifiers?
• What are the units of the Y axis?

Sorry for my poor explanation.

I measured this by the same way as you measured the instrumental noise of seismometers.
I put the three microphones at the same place so that the three can hear the same sound. I did not make any sounds, just put them in the lab.
The signals from microphones are all amplified by the circuit.
And I took the correlations of each signals and two others and got the noise (dashed lines) by subtracting the correlated signal from the original signal.

So,
-The signal is the acoustic sound in the lab, amplified by the circuit.
-Three lines are from three different microphones.
-Dashed lines are subtraction of coherent signal from the original.
-Yellow and black lines are from different amplifiers in the same circuit box. The circuit has 6 channels.
-I did not calibrate the signals I got by DTT since I do not know the calibration factor now. It is just the number I got from the real time system.

7607   Wed Oct 24 14:15:34 2012 AyakaUpdateAdaptive Filteringmicrophone noise

Previous results
I am measuring the noise level of the microphones. The circuit does not seems to limit their sensitivities but the circuit's noise seems to be different from other channels.

Measurement
I measured the circuit noise of all 6 channels. (input open)
(mic_open.png)
The noise level is about 10 times different from the others.

Comparing the acoustic signal, microphone+circuit noise, and ADC noise;
(mic_noise.png)
- blue; acoustic signal
- green; microphone+circuit noise
- red; circuit (the data was not took simultaneously.)

To do
I will remake the circuit though the circuit does not limit the sensitivity. I would like to make sure that the circuit does not affect badly and to make the circuit noise level the same.
At the same time, I will get the PMC control signal and see coherence between it and acoustic sound.

Attachment 1: mic_open.png
Attachment 2: mic_noise.png
7609   Wed Oct 24 15:29:52 2012 ranaUpdateAdaptive Filteringmicrophone noise

We have to change the sample rate and AA filter for the mic channels before going too far with the circuit design.

7610   Wed Oct 24 17:02:01 2012 JenneUpdateAdaptive Filteringmicrophone noise

 Quote: We have to change the sample rate and AA filter for the mic channels before going too far with the circuit design.

To save the mic channels at higher than 2k (which we should do), we either have to move them to a different model, change the rate of the PEM model, or see if you can save data faster than the model runs (which I can't imagine is possible).

7614   Wed Oct 24 22:20:24 2012 DenUpdateAdaptive Filteringmicrophone noise

 Quote: We have to change the sample rate and AA filter for the mic channels before going too far with the circuit design.

PEM model is running at 64K now. It turned out to be tricky to increase the rate:

• BLRMS are computationally expensive and original pem model did not start at any frequency higher then 16k ( at 16k cpu meter readings were 59/60 ). Also when we go higher then 16k, front-end gives the model less resources. I guess it is assumed that this model is iop and won't need too much time. So in the end I had to delete BLRMS blocks for all channels except for GUR2Z and MIC1.
• Foton files are modified during model compilation: lines with sampling rate and declaration of filters in the beginning of the file are changed only. Sos-representation and commands are the same. I hoped that filter commands will let me change sos-representation quickly. I've opened Foton and saved the file. However, Foton modified commands in such a way that the ratio of poles and zeros to sampling rate is preserved. I guess all filters have to be replaced or this process should be done in another way.
• BLRMS block uses low-pass filters below 0.01 Hz, increasing the sampling rate by a factor of 32 might make calculations incorrect. I'll check it.

We should also increase cut off frequency of the low-pass filter in the microphone pre-amplifier from 2 kHz up to ~20-30 kHz.

Attachment 1: mic_64k.pdf
7621   Thu Oct 25 09:53:23 2012 AyakaUpdateAdaptive Filteringmicrophone noise

Quote:

 Quote: We have to change the sample rate and AA filter for the mic channels before going too far with the circuit design.

PEM model is running at 64K now. It turned out to be tricky to increase the rate:

• BLRMS are computationally expensive and original pem model did not start at any frequency higher then 16k ( at 16k cpu meter readings were 59/60 ). Also when we go higher then 16k, front-end gives the model less resources. I guess it is assumed that this model is iop and won't need too much time. So in the end I had to delete BLRMS blocks for all channels except for GUR2Z and MIC1.
• Foton files are modified during model compilation: lines with sampling rate and declaration of filters in the beginning of the file are changed only. Sos-representation and commands are the same. I hoped that filter commands will let me change sos-representation quickly. I've opened Foton and saved the file. However, Foton modified commands in such a way that the ratio of poles and zeros to sampling rate is preserved. I guess all filters have to be replaced or this process should be done in another way.
• BLRMS block uses low-pass filters below 0.01 Hz, increasing the sampling rate by a factor of 32 might make calculations incorrect. I'll check it.

We should also increase cut off frequency of the low-pass filter in the microphone pre-amplifier from 2 kHz up to ~20-30 kHz.

Thank you for changing the sample rate!
Also we have to change the Anti-Aliasing filter, as Jamie said.

Now my question is, whether S/N ratio is enough at high frequencies or not. The quality of EM172 microphone is good according to the data sheet. But as you can see in previous picture, the S/N ratio around 1kHz is not so good, though we can see some peaks, e.g. the sound that a fan will make. I have to check it later.
And, is it possible to do online adaptive noise cancellation with a high sampling rate such that computationally expensive algorithms cannot be run?

7622   Thu Oct 25 10:03:38 2012 ranaUpdateAdaptive Filteringmicrophone noise

That's no good - we need BLRMS channels for many PEM channels, not just two. And the channel names should have the same name as they had in the past so that we can look at long term BLRMS trends.

I suggest:

1. Have a separate model for Mics and Magnetometers. This model should run at 32 kHz and not have low frequency poles and zeros. Still would have acoustic frequency BLRMS.
2. Have a low frequency (f_sample = 2 kHz) model for seis an acc. Seismometers run out of poop by 100 Hz, but we want to have the ACC signal up to 800 Hz since we do have optical mount resonances up to there.
3. Never remove or rename the BLRMS channels - this makes it too hard to keep long term trends.
4. Do a simple noise analysis to make sure we are matching the noise of the preamps to the noise / range of the ADCs.
5. Immediately stop using bench supplies for the power. Use ONLY fused, power lines from the 1U rack supplies.
7623   Thu Oct 25 14:39:14 2012 DenUpdateAdaptive Filteringmicrophone noise

 Quote: That's no good - we need BLRMS channels for many PEM channels, not just two. And the channel names should have the same name as they had in the past so that we can look at long term BLRMS trends. I suggest: Have a separate model for Mics and Magnetometers. This model should run at 32 kHz and not have low frequency poles and zeros. Still would have acoustic frequency BLRMS. Have a low frequency (f_sample = 2 kHz) model for seis an acc. Seismometers run out of poop by 100 Hz, but we want to have the ACC signal up to 800 Hz since we do have optical mount resonances up to there. Never remove or rename the BLRMS channels - this makes it too hard to keep long term trends. Do a simple noise analysis to make sure we are matching the noise of the preamps to the noise / range of the ADCs. Immediately stop using bench supplies for the power. Use ONLY fused, power lines from the 1U rack supplies.

Ayaka, Den

C1PEM model is back to 2K.

We created a new C1MIC model for microphones that will run at 32K. C1SUS machine is full, we have to think about rearrangement.

For now, we created DQ channels for microphones inside iop model, so we can subtract noise offline.

We provided 0-25 kHz bandwidth noise to AA board and saw the same signal in the output of ADC in the corresponding channel. So cut-off frequency is higher then 25 kHz. There is a label on the AA board that all filters are removed. What does this mean?

We've turned off AA bench power supply, prepare to use fused from 1U.

7633   Fri Oct 26 18:25:02 2012 AyakaUpdateAdaptive FilteringMicrophone noise again

[Raji, Ayaka]

Thanks to Den, power supplies for microphone circuit are changed.
So I measured the microphone noise again by the same way as I did last time.

solid lines: acoustic noise
dashed lines: un-coherent noise
black line: circuit noise (microphone unconnected)

The circuit noise improves so much, but many line noises appeared.
Where do these lines (40, 80, 200 Hz...) come from?
These does not change if we changed the microphones...

Anyway, I have to change the circuit (because of the low-pass filter). I can check if the circuit I will remake will give some effects on these lines.

7634   Fri Oct 26 19:06:14 2012 DenUpdateAdaptive FilteringMicrophone noise again

 Quote: The circuit noise improves so much, but many line noises appeared. Where do these lines (40, 80, 200 Hz...) come from? These does not change if we changed the microphones... Anyway, I have to change the circuit (because of the low-pass filter). I can check if the circuit I will remake will give some effects on these lines.

I do not think that 1U rack power supply influenced on the preamp noise level as there is a 12 V regulator inside. Lines that you see might be just acoustic noise produced by cpu fans. Usually, they rotate at ~2500-3000 rpm => frequency is ~40-50 Hz + harmonics. Microphones should be in an isolation box to minimize noise coming from the rack. This test was already done before and described here

I think we need to build a new box for many channels (32, for example, to match adc). The question is how many microphones do we need to locate around one stack to subtract acoustic noise. Once we know this number, we group microphones, use 1 cable with many twisted pairs for a group and suspend them in an organized way.

7636   Mon Oct 29 08:41:22 2012 AyakaUpdateAdaptive FilteringMicrophone noise again

Quote:

 Quote: The circuit noise improves so much, but many line noises appeared. Where do these lines (40, 80, 200 Hz...) come from? These does not change if we changed the microphones... Anyway, I have to change the circuit (because of the low-pass filter). I can check if the circuit I will remake will give some effects on these lines.

I do not think that 1U rack power supply influenced on the preamp noise level as there is a 12 V regulator inside. Lines that you see might be just acoustic noise produced by cpu fans. Usually, they rotate at ~2500-3000 rpm => frequency is ~40-50 Hz + harmonics. Microphones should be in an isolation box to minimize noise coming from the rack. This test was already done before and described here

I think we need to build a new box for many channels (32, for example, to match adc). The question is how many microphones do we need to locate around one stack to subtract acoustic noise. Once we know this number, we group microphones, use 1 cable with many twisted pairs for a group and suspend them in an organized way.

I do not think they are acoustic sounds. If so, there should be coherence between three microphones because I placed three at the same place, tied together. However, there are no coherence at lines between them.

7708   Tue Nov 13 21:05:35 2012 DenUpdateAdaptive Filteringonline and simulation

For a last few days I've been working on oaf and simulink model to simulate it. First I did online subtraction from MC when MC_L path was enabled. Inside my code I've added a sum of squares of filter coefficients so we can monitor convergence of the filter.

To to this I've measured path from OAF output to input without AA and AI filters. Then made a vectfit using 2 poles and zeros. Foton command

zpk( [-2.491928e+03;5.650511e-02], [-4.979872e+01;-3.278776e+00], 6.011323e+00)

My simulink model consists of 3 parts:

• cavity with seismic noise at low frequencies, 1/f^2 noise at medium frequencies and white noise at high frequencies
• this cavity is locked using feedback compensation filters that we use to lock arms
• locked cavity with adaptive filter

Adaptive filter in the model uses online c-code. It is connected to simulink block through an S-function. Sampling frequency of the model is 10 kHz. It works fairly fast - 1 sec of simulation time is computed in 1 sec.

I've tested FxLMS algorithm and MFxLMS algorithm that is faster. I plan to test 2 iir adaptive algorithms that are already coded.

7764   Fri Nov 30 02:40:44 2012 DenUpdateAdaptive FilteringYARM

I've applied FIR adaptive filter to YARM control. Feedback signal of the closed loop was used as adaptive filter error signal and OAF OUT -> IN transfer function I assumed to be flat because of the loop high gain at low frequencies. At 100 Hz deviation was 5 dB so I've ignored it.

I've added a filter bank YARM_OAF to C1LSC model to account for downsampling from 16 kHz to 2 kHz and put low-pass filter inside.

I've used GUR 1&2 XYZ channels as witnesses. Bandpass filters 0.4-10 Hz we applied to each of them. Error signal was filters using the same bandpass filter and 16 Hz 40 dB Q=10 notch filter. As an AI filter I used 32 Hz butterworth 4 order low-pass filter. Consequently, AI, bandpass and notch filters were added to adaptive path of witness signals.

I've used an FIR filter with 4000 taps, downsampling = 16, delay = 1, tau = 0, mu = 0.01 - 0.1. Convergence time was ~3 mins.

7767   Fri Nov 30 11:49:24 2012 KojiUpdateAdaptive FilteringYARM

This is interesting. I suppose you are acting on the ETMY.
Can you construct the compensation filter with actuation on the MC length?
Also can you see how the X arm is stabilized?

This may stabilize or even unstabilize the MC length, but we don't care as the MC locking is easy.

If we can help to reduce the arm motion with the MCL feedforward trained with an arm sometime before,
this means the lock acquisition will become easier. And this may still be compatible with the ALS.

Why did you notched out the 16Hz peak? It is the dominant component for the RMS and we want to eliminate it.

7769   Fri Nov 30 22:11:50 2012 DenUpdateAdaptive FilteringARMS

 Quote: This is interesting. I suppose you are acting on the ETMY. Can you construct the compensation filter with actuation on the MC length? Also can you see how the X arm is stabilized? This may stabilize or even unstabilize the MC length, but we don't care as the MC locking is easy. If we can help to reduce the arm motion with the MCL feedforward trained with an arm sometime before, this means the lock acquisition will become easier. And this may still be compatible with the ALS. Why did you notched out the 16Hz peak? It is the dominant component for the RMS and we want to eliminate it.

I actuate on ETMY for YARM and ETMX for XARM. For now I did adaptive filtering for both arms at the same time. I used the same parameters for xarm as for yarm.

I've notched 16 Hz resonance because it has high Q and I need to think more how to subtract it using FIR filter or apply IIR.

I'll try MC stabilazation method.

Attachment 1: arms_oaf.pdf
7771   Sat Dec 1 00:13:16 2012 DenUpdateAdaptive FilteringARMS and MC

 Quote: I actuate on ETMY for YARM and ETMX for XARM. For now I did adaptive filtering for both arms at the same time. I used the same parameters for xarm as for yarm. I've notched 16 Hz resonance because it has high Q and I need to think more how to subtract it using FIR filter or apply IIR. I'll try MC stabilazation method.

Adaptive filtering was applied to MC and X,Y arms at the same time. I used a very aggressive (8 order) butterworth filter at 6 Hz as an AI filter for MC not to inject noise to ARMS as was done before

Mu for MC was 0.2, downsample = 16, delay = 1. I was able to subtract 1 Hz. Stack subraction is not that good as for arms but this is because I used only one seismometer for MC that is under the BS. I might install accelerometers under MC2.

EDIT, JCD, 18Feb2013:  Den remembers using mu for the arms in the range of 0.01 to 0.1, although using 0.1 will give extra noise.  He said he usually starts with something small, then ramps it up to 0.04, and after it has converged brings it back down to 0.01.

Attachment 1: arms_mcl_oaf.pdf
14010   Sat Jun 23 13:08:41 2018 JonUpdateAUXFirst Coherent AUX Scan of PRC Using AM Sidebands

[Jon, Keerthana, Sandrine]

Thu.-Fri. we continued with PRC scans using the AUX laser, but now the "scanned" parameter is the frequency of AM sidebands, rather than the frequency of the AUX carrier itself. The switch to AM (or PM) allows us to coherently measure the cavity transfer as a function of modulation frequency.

In order to make a sentinel measurement, I installed a broadband PDA255 at an unused pickoff behind the first AUX steering mirror on the AS table. The sentinel PD measures the AM actually imprinted on the light going into the IFO, making our measurement independent of the AOM response. This technique removes not only the (non-flat) AOM transfer function, but also any non-linearities from, e.g., overdriving the AOM. The below photo shows the new PD (center) on the AS table.

With the sentinel PD installed, we proceeded as follows.

• Locked IFO in PRMI on carrier.
• Locked AUX PLL to PSL.
• Tuned the frequency of the AUX laser (via the RF offset) to bring the carrier onto resonance with the PRC.
• Swept the AOM modulation frequency 0-60 MHz while measuring the AUX reflection and injection signals.

The below photo shows the measured transfer function [AUX Reflection / AUX Injection]. The measurement coherence is high to ~55 MHz (the AOM bandwidth is 60 MHz). We clearly resolve two FSRs, visible as Lorentzian dips at which more AUX power couples into the cavity. The SURFs have these data and will be separately posting figures for the measurements.

With the basic system working, we attempted to produce HOMs, first by partially occluding the injected AUX beam with a razor blade, then by placing a thin two-prong fork in the beam path. We also experimented with using a razor blade on the output to partially occlude the reflection beam just before the sensor. We were able to observe an apparent secondary dip indicative of an HOM a few times, as shown below, but could not repeat this deterministically. Besides not having fine control over the occlusion of the beams, there is also large few-Hz angular noise shaking the AS beam position. I suspect from moment to moment the HOM content is varying considerably due to the movement of the AS beam relative to the occluding object. I'm now thinking about more systematic ways to approach this.

14011   Sat Jun 23 20:54:35 2018 KojiUpdateAUXFirst Coherent AUX Scan of PRC Using AM Sidebands

How much was the osc freq of the marconi? And then how much was the resulting freq offset between PSL and AUX?

Are we supposed to see two dips with the separation of an FSR? Or four dips (you have two sidebands)?

And the distance between the dips (28MHz-ish?) seems too large to be the FSR (22MHz-ish).
cf https://wiki-40m.ligo.caltech.edu/IFO_Modeling/RC_lengths

14017   Tue Jun 26 10:06:39 2018 keerthanaUpdateAUXFirst Coherent AUX Scan of PRC Using AM Sidebands

(Jon, Keerthana, Sandrine)

I am attaching the plots of the Reflected and transmitted AUX beam. In the transmission graph, we are getting peak corresponding to the resonance frequencies, as at that frequency maximum power goes to the cavity. But in the Reflection graph, we are obtaining dips corresponding to the resonance frequency because maximum power goes to the cavity and the reflected beam intensity becomes very less at those points.

Attachment 1: TRANS.pdf
Attachment 2: REFL.pdf
14035   Tue Jul 3 11:59:10 2018 JonUpdateAUXAUX Carrier Scan of Y-Arm Cavity

I made the first successful AUX laser scan of a 40m cavity last night.

Attachment #1 shows the measured Y-end transmission signal w.r.t. the Agilent drive signal, which was used to sweep the AUX carrier frequency. This is a distinct approach from before, where the carrier was locked at a fixed offset from the PSL carrier and the frequency of AM sidebands was swept instead. This AUX carrier-only technique appears to be advantageous.

This 6-15 MHz scan resolves three FSR peaks (TEM00 resonances) and at least six other higher-order modes. The raw data are also enclosed (attachment #2). I'll leave it as an excercise for the SURFs to compute the Y-arm cavity Gouy phase.

Attachment 1: yarm_carrier_trans.pdf
Attachment 2: AG4395A_02-07-2018_185504.txt
# AG4395A Measurement - Timestamp: Jul 02 2018 - 18:55:04
#---------- Measurement Parameters ------------
# Start Frequency (Hz): 6000000.0, 6000000.0
# Stop Frequency (Hz): 15000000.0, 15000000.0
# Frequency Points: 801, 801
# Measurement Format: LOGM, PHAS
# Measuremed Input: AR, AR
#---------- Analyzer Settings ----------
# Number of Averages: 16
# Auto Bandwidth: Off, Off

... 807 more lines ...
14036   Wed Jul 4 19:11:49 2018 JonUpdateAUXMore Testing of AUX-Laser Mode Scanning

More progress on the AUX-laser cavity scans.

### Changes to the Setup

• For scans, the Agilent is now being used as a standalone source of the LO signal provided to the AUX PLL (instead of the Marconi), which sets the RF offset. We discovered that when the sweep is "held" in network analyzer mode, it does not turn off the RF drive signal, but rather continues outputting a constant signal at the hold frequency. This eliminates the need to use the more complicated double-deomdulation previously in use. The procedure is to start and immediately hold the sweep, then lock the PLL, then restart the sweep. The PLL is able to reliably remain locked for frequency steps of up to ~30 kHz. The SURFs are preparing schematics of both the double- and single-demodulation techniques.
• Both the Marconi and Agilent are now phase-locked to the 10 MHz time reference provided by the rabidium clock. This did noticeably shift the measured resonance frequencies.
• I raised the PI controller gain setting to 4.5, which seems to better suppress the extra noise being injected.
• I've procured a set of surgical needles for occluding the beam to produce HOMs. However, I have not needed to use them so far, as the TEM00 purity of the AUX beam appears to already be low. The below scans show only the intrinisic mode content.

### New Results

• YARM scan at 70 uW injection power (Attachment #1). The previously reported YARM scan was measured with 9 mW of injected AUX power, 100x larger than the power available from the SQZ laser at the sites. This scan repeats the measurement with the AUX power attenuated to uW. It still resolves the FSR and at least three HOMs.
• PRC scan (Attachment #2) at 9 mW injection power. It appears to resolve the FSR and at least three HOMs. Angular injection noise was found to cause large fluctuations in the measured signal power. This dominates the error bars shown below, but affects only the overall signal amplitude (not the peak frequency locations). The SQZ angular alignment loops should mitigate this issue at the sites.

Both data sets are attached.

Attachment 1: yarm_trans_70uW.pdf
Attachment 2: prc_trans_9mW.pdf
Attachment 3: yarm_carrier_trans_70uW.tar.gz
Attachment 4: prc_carrier_trans_9mW.tar.gz
14044   Sun Jul 8 12:20:12 2018 JonSummaryAUXGouy Phase Measurements from AUX-Laser Scans

This note reports analysis of cavity scans made by directly sweeping the AUX laser carrier frequency (no sidebands). The measurement is made by sweeping the RF offset of the AUX-PSL phase-locked loop and demodulating the cavity reflection/transmission signal at the offset frequency.

# Y-Arm Scan

Due to the simplicity of its expected response, the Y-arm cavity was scanned first as a test of the AUX hardware and the sensitivity of the technique. Attachment 1 shows the measured cavity transmission with respect to RF drive signal.

The AUX laser launch setup is capable of injecting up to 9.3 mW into the AS port. This high-power measurement is shown by the black trace. The same measurement is repeated for a realistic SQZ injection power, 70 uW, indicated by the red curve. At low power, the technique still clearly resolves the FSR and six HOM resonances. From the identified mode resonance frequencies the following cavity parameters are directly extracted.

YARM Gautam's Finesse Model Actual
FSR 3.966 MHz 3.967 MHz
Gouy phase 54.2 deg 52.0 deg

# PRC Scan

An analogous scan was performed for the PRC, with the IFO locked on PSL carrier in PRMI. Attachment 2 shows the measurement of PRC transmission with respect to drive signal.

The scan resolves HOM resonances to at least ~13th order, whose frequencies yield the following cavity parameters.

PRC Gautam's Finesse Model Actual
FSR 22.30 MHz 22.20 MHz
Gouy phase 13.4 deg 15.4 deg

# SRC Scan

Ideally (and at the sites) the SRC mode resonances will be measured in SRMI configuration. Because every other cavity is misaligned, this configuration provides an easily-interpretable spectrum whose resonances can all be attributed to the SRC.

Due to time constraints at the 40m, the IFO could not be restored to lockability in SRMI. It has been more than two years since this configuration was last run. For this reason the scan was made instead with the IFO locked in DRMI, as shown in Attachment 3. The quantity measured is the AUX reflection with respect to drive signal.

This result requires far more interpretation because resonances of both the SRC and PRC are superposed. However, the resonances of the PRC are known a priori from the independent PRMI scan. The SRC mode resonances identified below do not conincide with any of the first five PRC mode resonances.

Based on the identified mode resonance frequencies, the SRC parameters are measured as follows.

SRC Gautam's Finesse Model Actual
FSR 27.65 MHz 27.97 MHz
Gouy phase 10.9 deg 8.8 deg

# Lessons Learned

From experience with the 40m, the main challenges to repeating this measurement at the sites will be the following.

• Pointing jitter of the input AUX beam. This causes the PSL-AUX beam overlap to vary at transmission (or reflection), causing variation in the amplitude of the AUX-PSL beat note. As far as we can tell, the frequency of the resonances (the only object of this measurement) is not changing in time, only the relative amplitudes of the diferent mode peaks. I believe the SQZ alignment loops will mitigate this problem at the sites.
• Stabilization of the network analyzer time base. We found the intrinsic frequency stability of the network analyzer (Agilent 4395A) to be unacceptably large. We solved this problem by phase-locking the Agilent to an external reference, a 10-MHz signal provided by an atomic clock.
Attachment 1: yarm_aux_carrier_trans.pdf
Attachment 2: prmi_aux_carrier_trans.pdf
Attachment 3: drmi_aux_carrier_trans.pdf
Draft   Wed Jul 11 18:13:19 2018 keerthanaSummaryAUXGouy Phase Measurements from AUX-Laser Scans

From the Measurement Jon made, FSR is 3.967 MHz and the Gouy phase is 52 degrees. From this, the length of the Y-arm cavity seems to be 37.78 m and the radius of curvature of the mirror seems to be 60.85 m.

$Guoy Phase = \cos^{-1} \sqrt{g1.g2}$

$\\ g = 1- \frac{L}{R}$

$L = \frac {c} {2*FSR}$

FSR = Free spectral Range

L = Lenth of the arm

R = Radius of curvature of the mirror (R1 =$\infty$  , R2= unknown)

Quote:

This note reports analysis of cavity scans made by directly sweeping the AUX laser carrier frequency (no sidebands). The measurement is made by sweeping the RF offset of the AUX-PSL phase-locked loop and demodulating the cavity reflection/transmission signal at the offset frequency.

# Y-Arm Scan

Due to the simplicity of its expected response, the Y-arm cavity was scanned first as a test of the AUX hardware and the sensitivity of the technique. Attachment 1 shows the measured cavity transmission with respect to RF drive signal.

The AUX laser launch setup is capable of injecting up to 9.3 mW into the AS port. This high-power measurement is shown by the black trace. The same measurement is repeated for a realistic SQZ injection power, 70 uW, indicated by the red curve. At low power, the technique still clearly resolves the FSR and six HOM resonances. From the identified mode resonance frequencies the following cavity parameters are directly extracted.

YARM Gautam V. Finesse Model Actual
FSR 3.966 MHz 3.967 MHz
Gouy phase 54.2 deg 52.0 deg

14062   Fri Jul 13 00:15:13 2018 Annalisa, TerraConfigurationAUXY arm cavity scan

[Annalisa, Terra, Koji, Gautam]

Summary: We find a configuration for arm scans which significantly reduces phase noise. We run several arm scans and we were able to resolve several HOM peaks; analysis to come.

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

As first, we made a measurement with the already established setup and, as Jon already pointed out, we found lots of phase noise. We hypothesized that it could either come from the PLL or from the motion of the optics between the AUX injection point (AS port) and the Y arm.

• We first characterized the PLL loop phase noise by comparing the beat signal against the Agilent reference signal, and we found that the beat had lots of phase noise with respect to the reference. Decreasing the PLL gain, we got rid of the phase noise in the beat signal.
• Next, for the optical path length induced phase noise, we took the transfer function between TransMon and REFL signal rather than TransMon and Agilent reference signal. This takes advatage of the fact that the TransMon and REFL both see optical path length phase noise, which therefore gets canceled out in the transfer function.

In this configuration, we were able to do arm scans where the phase variation at each peak was pretty clear and well defined. We took several 10MHz scan, we also zoomed around some specific HOM peak, and we were able to resolve some frequency split.

We add some pictures of the setup and of the scan.

The data are saved in users/OLD/annalisa/Yscans. More analysis and plots will follow tomorrow.

Attachment 1: IMG_6492.JPG
Attachment 2: IMG_6494.JPG
14091   Fri Jul 20 18:30:47 2018 JonConfigurationAUXRecommend to install AUX PZT driver

I recently realized that the PLL is only using about 20% of the available actuation range of the AUX PZT. The +/-10 V control signal from the LB1005 is being directly inputted into the fast AUX PZT channel, which has an input range of +/-50 V.

I recommend to install a PZT driver (amplifier) between the controller and laser to use the full available actuator range. For cavity scans, this will increase the available sweep range from +/-50 MHz to +/-250MHz. This has a unique advantage even if slow temperature feedback is also implemented. To sample faster than the timescale of most of the angular noise,  scans generally need to be made with a total sweep time <1 sec. This is faster than the PLL offset can be offloaded via the slow temperature control, so the only way to scan more than 100 MHz in one measurement is with a larger dynamic range.

14501   Fri Mar 29 15:47:58 2019 gautamUpdateAUXAUX laser fiber moved from AS table to PSL table

[anjali, gautam]

To facilitate the 1um MZ frequency stabilization project, I decided that the AUX laser was a better candidate than any of the other 3 active NPROs in the lab as (i) it is already coupled into a ~60m long fiber, (ii) the PSL table has the most room available to set up the readout optics for the delayed/non-delayed beams and (iii) this way I can keep working on the IR ALS system in parallel. So we moved the end of the fiber from the AS table to the SE corner of the PSL table. None of the optics mode-matching the AUX beam to the interferometer were touched, and we do not anticipate disturbing the input coupling into the fiber either, so it should be possible to recover the AUX beam injection into the IFO relatively easily.

Anjali is going to post detailed photos, beam layout, and her proposed layout/MM solutions later today. The plan is to use free space components for everything except the fiber delay line, as we have these available readily. It is not necessarily the most low-noise option, but for a first pass, maybe this is sufficient and we can start building up a noise budget and identify possible improvements.

The AUX laser remians in STANDBY mode for now. HEPA was turned up while working at the PSL table, and remains on high while Anjali works on the layout.

14504   Sun Mar 31 18:39:45 2019 AnjaliUpdateAUXAUX laser fiber moved from AS table to PSL table
• Attachment #1 shows the schematic of the experimental setup for the frequency noise measurement of 1 um laser source.

• AUX laser will be used as the seed source and it is already coupled to a 60 m fiber (PM980). The other end of the fiber was at the AS table and we have now removed it and placed in the PSL table.

• Attachment # 2 shows the photograph of the experimental setup. The orange line shows the beam that is coupled to the delayed arm of MZI and the red dotted line shows the undelayed path.

• As mentioned, AUX is already coupled to the 60 m fiber and the other end of the fiber is now moved to the PSL table. This end needs to be collimated. We are planning to take the same collimator from AS table where it was coupled into before. The position where the collimator to be installed is shown in attachment #2. Also, we need to rotate the mirror (as indicated in attachment #2) to get the delayed beam along with the undelayed beam and then to combine them. As indicated in attachment #2, we can install one more photo diode to perform  balanced detection.

• We need to decide on which photodetector to be used. It could be NF1801 or PDA255.

• We also performed the power measurement at different locations in the beam path. The different locations at which power measurement is done is shown attachment #3

• There is an AOM in the beam path that coupled to the delayed arm of MZI. The output beam after AOM was coupled to the zero-order port during this measurement. That is the input voltage to the AOM was at 0 V, which essentially says that the beam after the AOM is not deflected and it is coupled to the zero-order port. The power levels measured at different locations in this condition are as follows. A)282 mW B)276 mW C)274 mW D)274 mW E)273 mW F)278 mW G)278 mW H)261 mW I)263 mW J)260 mW K)131 mW L)128 mW M)127 mW N)130 mW

• It can be seen that the power is halved from J to K. This because of a neutral density filter in the path of the beam

• In this case, we measured a power of 55 mW at the output of the delayed fiber. We then adjusted the input voltage to the AOM driver to 1 V such that the output of AOM is coupled to the first order port. This reduced the power level in the zero-order port of AOM that is coupled to the delayed arm of the MZI. In this case we measured a power of 0.8 mW at the output of delayed fiber.

•  We must be careful about the power level that is reaching the photodetector such that it should not exceed the damage threshold of the detector.

• The power measured at the output of undelayed path is 0.8 mW.

• We also must place the QWP and HWP in the beam path to align the polarisation.

 Quote: [anjali, gautam] To facilitate the 1um MZ frequency stabilization project, I decided that the AUX laser was a better candidate than any of the other 3 active NPROs in the lab as (i) it is already coupled into a ~60m long fiber, (ii) the PSL table has the most room available to set up the readout optics for the delayed/non-delayed beams and (iii) this way I can keep working on the IR ALS system in parallel. So we moved the end of the fiber from the AS table to the SE corner of the PSL table. None of the optics mode-matching the AUX beam to the interferometer were touched, and we do not anticipate disturbing the input coupling into the fiber either, so it should be possible to recover the AUX beam injection into the IFO relatively easily. Anjali is going to post detailed photos, beam layout, and her proposed layout/MM solutions later today. The plan is to use free space components for everything except the fiber delay line, as we have these available readily. It is not necessarily the most low-noise option, but for a first pass, maybe this is sufficient and we can start building up a noise budget and identify possible improvements. The AUX laser remians in STANDBY mode for now. HEPA was turned up while working at the PSL table, and remains on high while Anjali works on the layout.

Attachment 1: Schematic_of_experimental_setup_for_frequency_stabilisation_of_1_micron_source.png
Attachment 2: 1_micron_setup_for_frequency_noise_measurement.JPG
Attachment 3: 1_micron_setup_for_frequency_noise_measurement_power_levels.png
16194   Wed Jun 9 11:46:01 2021 Anchal, PacoSummaryAUXXend Green Laser PDH OLTF measurement

We measured the Xend green laser PDH Open loop transfer function by following method:

• We first measured the feedback transfer function 'K' directly.
• See attachment 2 for this measurement. We measured Out2/exc here.
• Then, we closed the loop as shown in attachment 1with SR560 as a summing juntion at error point.
• We injected excitation through B channel in SR560 and measured transfer function Out1/Out2.
• This measurement should give us $G_{OL} / K$ by loop alegbra.
• Then we multiplied the two transfer function measurements to get open loop transfer function.

## Result:

• Our measurement gives the same UGF of 10kHz and phase margin of 53.5 degrees as reported in 13238.
• The shape of measurement also follows 1/f above 10 Hz atleast.
• Our measurement might not be correct below 10 Hz but we did not see any saturation or loss of lock in 1Hz to 10 Hz measurement.
• This OLTF is different from the modelled OLTF here even though the UGF matches.
• The feedback gain is supposed to roll-off faster than 1/f in 30Hz to 1kHz region but it does not seem to in our measurement.
• This suggests that the actual uPDH box is shaping the loop different from what schematic suggests. This might mean that the gain is much lower in the low frequency region than we would like it to be.
• We will investigate the reason of difference between model and measurement unless someone has a better explaination for the descripancy.
Attachment 1: image-6f2923a3-01ce-4d04-bc53-d8db0238e195.jpg
Attachment 3: X_Green_ARM_PDH_OLTF.pdf
16197   Thu Jun 10 14:01:36 2021 AnchalSummaryAUXXend Green Laser PDH OLTF measurement loop algebra

Attachment 1 shows the closed loop of Xend Green laser Arm PDH lock loop. Free running laser noise gets injected at laser head after the PZT actuation as $\eta$. The PDH error signal at output of miser is fed to a gain 1 SR560 used as summing junction here. Used in 'A-B mode', the B port is used for sending in excitation $\nu_e e^{st}$ where $s = i\omega$.

We have access to three ports for measurement, marked $\alpha$ at output of mixer, $\beta$ at output of SR560, and $\gamma$ at PZT out monitor port in uPDH box. From loop algebra, we get following:

$\large \left[ (\alpha - \nu_e) K(s)A(s) + \eta \right ]C(s)D(s) = \alpha$

$\large \Rightarrow (\alpha - \nu_e) G_{OL}(s) + \eta C(s)D(s) = \alpha$, where $\large G_{OL}(s) = C(s) D(s) K(s) A(s)$ is the open loop transfer function of the loop.

$\large \Rightarrow \alpha = \eta \frac{C(s) D(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{G_{OL}(s)}{1 - G_{OL}(s)}$

$\large \Rightarrow \beta = \eta \frac{C(s) D(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{1}{1 - G_{OL}(s)}$

$\large \Rightarrow \gamma = \eta \frac{1}{K(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{K(s)}{1 - G_{OL}(s)}$

So measurement of $\large G_{OL}(s)$ can be done in following two ways (not a complete set):

1. $\large G_{OL}(s) \approx \frac{\alpha}{\beta} = \frac{G_{OL}(s) - \frac{\eta C(s)D(s)}{\nu_e}}{1 - \frac{\eta C(s)D(s)}{\nu_e}}$, if excitation amplitude is large enough such that $\large \frac{\eta C(s)D(s)}{\nu_e} \ll 1$over all frequencies.
• In this method however, note that SR785 would be taking ratio of unsuppresed excitation at $\large \alpha$ with suppressed excitation at $\large \beta.$
• If the closed loop gain (suppression) $\large 1/(1 - G_{OL}(s))$is too much, the excitation signal might drop below noise floor of SR785 while measuring $\large \beta$.
• This would then appear as a flat response in the transfer function.
• This happened with us when we tried to measure this transfer function using this method. Below few hundered Hz, the measurement will become flat at around 40 dB.
• Increasing the excitation amplitude where suppression is large should ideally work. We even tried to use Auto level reference option in SR785.
• But the PDH loop gets unlocked as soon as we put exciation above 35 mV at this point in this loop.
2. $\large \frac{G_{OL}(s)}{K(s)} \approx \frac{\alpha}{\gamma} = \frac{G_{OL}(s) - \frac{\eta C(s)D(s)}{\nu_e}}{K(s)\left(1 - \frac{\eta C(s)D(s)}{\nu_e}\right )}$, if excitation amplitude is large enough such that $\large \frac{\eta C(s)D(s)}{\nu_e} \ll 1$over all frequencies.
• In this method, channel 1 (denominator) on SR785 would remain high in amplitude throughout the measurement avoiding the above issue of suppression below noise floor.
• We can easily measure the feedback transfer funciton $\large K(s)$ with the loop open. Then multiplying the two measurements should give us estimate of open loop transfer function.
• This is waht we did in 16194. But we still could not increase the excitation amplitude beyond 35 mV at injection point and got a noisy measurement.
• We checked yesterday coherence of excitation signal with the three measurment points $\large \alpha, \beta, \gamma$ and it was 1 throughout the frequency region of measurement for excitation amplitudes above 20 mV.
• So as of now, we are not sure why our signal to noise was so poor in lower frequency measurement.
Attachment 1: AUX_PDH_LOOP.pdf
16200   Mon Jun 14 18:57:49 2021 AnchalUpdateAUXXend is unbearably hot. Green laser is loosing lock in 10's of seconds

Working in Xend with mask on has become unbearable. It is very hot there and I would really like if we fix this issue.

Today, the Xend Green laser was just unable to hold lock for longer than 10's of seconds. The longest I could see it hold lock was for about 2 minutes. I couldn't find anything obviously wrong with it. Attached are noise spectrums of error and control points. The control point spectrum shows good matching with typical free running laser noise.

Are the few peaks above 10 kHz in error point spectrum worrysome? I need to think more about it in a cooler place to make sure.

I wanted to take a high frequency spectrum of error point to make sure that higher harmonics of 250 kHz modulation frequency are not leaking into the PDH box after demodulation. However, the lock could not be maintained long enough to take this final measurement. I'll try again tomorrow morning. It is generally cooler in the mornings.

This post is just an update on what's happening. I need to work more to get some meaningful inferences about this loop.

Attachment 1: XAUX_PDH_Err_In_ASD.pdf
Attachment 2: XAUX_PZT_Out_Mon_ASD.pdf
16202   Tue Jun 15 15:26:43 2021 Anchal, PacoSummaryAUXXend Green Laser PDH OLTF measurement loop algebra, excitation at control point

Attachment 1 shows the case when excitation is sent at control point i.e. the PZT output. As before, free running laser noise $\eta$ in units of Hz/rtHz is added after the actuator and I've also shown shot noise being added just before the detector.

Again, we have a access to three output points for measurement. $\alpha$ right at the output of mixer (the PDH error signal), $\beta$ the feedback signal to be applied by uPDH box (PZT Mon) and $\gamma$ the output of the summing box SR560.

Doing loop algebra as before, we get:

$\large \alpha = \frac{\eta}{K(s) A(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} + \frac{\chi}{C(s) K(s) A(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} - \frac{\nu_e}{K(s) } \frac{G_{OL}(s)}{1 - G_{OL}(s)}$

$\large \beta = \frac{\eta}{A(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} + \frac{\chi}{C(s) A(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} - \nu_e \frac{G_{OL}(s)}{1 - G_{OL}(s)}$

$\large \gamma= \frac{\eta}{A(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} + \frac{\chi}{C(s) A(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} - \nu_e \frac{1}{1 - G_{OL}(s)}$

So measurement of $\large G_{OL}(s)$ can be done by

$\large G_{OL}(s) \approx \frac{\beta}{\gamma}$

• For frequencies, where $\large G_{OL}(s)$ is large enough, to have an SNR of 100, we need that ratio of $\large \nu_e$ to integrated noise is 100.
• Assuming you are averaging for 'm' number of cycles in your swept sine measurement, time of integration for the noise signal would be $\large \frac{m}{f}$where f is the frequency point of the seeping sine wave.
• This means, the amplitude of integrated laser frequency noise at either $\large \beta$ or $\large \gamma$ would be $\large \sqrt{\left(\frac{\eta(f)}{A(f)}\right)^2\frac{f}{m}} = \frac{\eta(f) \sqrt{f}}{A(f)\sqrt{m}}$
• Therefore, signal to laser free running noise ratio at f would be $\large S = \frac{\nu_eA(f)\sqrt{m}}{\eta(f) \sqrt{f}}$.
• This means to keep a constant SNR of S, we need to shape the excitation amplitude as $\large \nu_e \sim S \frac{\eta(f) \sqrt{f}}{A(f)\sqrt{m}}$
• Putting in numbers for X end Green PDH loop, laser free-running frequency noise ASD is 1e4/f Hz/rtHz, laser PZT actuation is 1MHz/V, then for 10 integration cycles and SNR of 100, we get: $\large \nu_e \sim 100 \times \frac{10^4 \sqrt{f}}{f \times10^6 \sqrt{10}} = \frac{30\, mV}{\sqrt{f}}$
• Assuming you are averaging for a constant time $\large \tau$ in swept sine measurement, then the amplitude of integrated laser free noise would be $\large \sqrt{\left(\frac{\eta(f)}{A(f)}\right)^2 \frac{1}{\tau}} = \frac{\eta(f) }{A(f)\sqrt{\tau}}$
• In this case, signal to laser free-running noise ratio at f would be $\large S = \frac{\nu_eA(f)\sqrt{\tau}}{\eta(f)}$
• This means to keep a constant SNR of S, we need to shape the excitation amplitude as $\large \nu_e \sim S\frac{\eta(f)}{A(f)\sqrt{\tau}}$
• Again putting in numbers as above and integration time of 1s, we need an excitation amplitude shape $\large \nu_e \sim 100 \times \frac{10^4 }{f \times10^6 \sqrt{1}} = \frac{1\, V}{f}$

This means at 100 Hz, with 10 integration cycles, we should have needed only 3 mV of excitation signal to get an SNR of 100. However, we have been unable to get good measurements with even 25 mV of excitation. We tried increasing the cycles, that did not work either.

This post is to summarize this analysis. We need more tests to get any conclusions.

Attachment 1: AuxPDHloop.pdf
16213   Fri Jun 18 10:07:23 2021 Anchal, PacoSummaryAUXXend Green Laser PDH OLTF with coherence

We did the measurement of OLTF for Xend green laser PDH loop with excitation added at control point using a SR560 as shown in attachment 1 of 16202. We also measured coherence in our measurement, see attachment 1.

## Measurement details:

• We took the $\beta/\gamma$ measurement as per 16202.
• We did measurement in two pieces. First in High frequency region, from 1 kHz to 100 kHz.
• In this setup, the excitation amplitude was kept constant to 5 mV.
• In this region, the OLTF is small enough that signal to noise ratio is maintained in $\gamma$ (SR560 sum output, measured on CH1). The coherence can be seen to be constant 1 throughout for CH1 in this region.
• But for $\beta$ (PZT Mon, measured on CH2), the low OLTF actually starts damping both signal and noise and to elevate it above SR785 noise floor, we had a high pass (z:0Hz, p:100kHz, k:1000) SR560 amplifying $\beta$ before measurement (see attachment 2). This amplification has been corrected in Attachment 1. This allowed us to improve the coherence on CH2 to above 0.5 mostly.
• Second region is from 3 Hz to 1 kHz.
• In this setup, the excitation was shaped with a low pass (p: 1Hz, k:5) SR560 filter with SR785 source amplitude as 1V.
• We took 40 averaging cycles in this measurement to improve the coherence further.
• In this freqeuency region, $\beta$ is mostly coherent as we shaped the excitation as $1/f$ and due to constant cycle number averaging, the integrated noise goes as $1/\sqrt{f}$(see 16202 for math).
• We still lost coherence in $\gamma$ (CH1) for frequencyes below 100 Hz. the reason is that the excitation is suppressed by OLTF while the noise is not for this channel. So the $1/f$ shaping of excitation only helps fight against the suppression of OLTF somewhat and not against the noise.
$\gamma = \left( \frac{\eta}{A(s)} - \frac{\nu_e}{G_{OL}(s)} + \frac{\chi}{A(s) C(s)} \right)\frac{G_{OL}(s)}{1-G_{OL}(s)}$
• We need $1/f^2$ shaping for this purpose but we were loosing lock with that shaping so we shifted back to $1/f$ shaping and captured whatever we could.
• It is clear that the noise takes over below 100 Hz and coherence in CH1 is lost there.

## Inferences:

• Yes, the OLTF does not look how it should look but:
• The green region in attachment 1 shows the data points where coherence on both CH1 and CH2 was higher than 0.75.  So the saturation measured below 1 kHz, particularly in 100 Hz to 500 Hz (where coherence on both channels is almost 1) is real.
• This brings the question, what is saturating. As has been suggested before, our excitation signal is probably saturating some internal stage in the uPDH box. We need to investigate this next.
• It is however very non-intuitive to why this saturation is so non-uniform (zig-zaggy) in both magnitude and phase.
• In past experiences, whenever I saw somehting saturating, it would cause a flat top response in transfer function.
• Another interesting thing to note is the reduced UGF in this measurement.
• UGF is about 40-45 kHz. This we believe is due to reduced mode matching of the green light to the XARM when temperature of the end increases too much. We took the measurement at 6 pm and Koji posted the Xend's temperature to be 30 C at 7 pm in 16206. It certainly becomes harder to lock at hot temperatures, probably due to reduced phase margin and loop gain.
Attachment 1: XEND_PDH_OLTF_with_Coherence.pdf
Attachment 2: Beta_Amp.pdf
387   Thu Mar 20 17:45:36 2008 ranaSummaryASSAdaptive Filtering in the ASS system
Over the past couple weeks we (Matt, Alex, Rob, me) have worked on getting an adaptive filter
system working. We wanted to load this system into c1ass to use this processor. The dither alignment
system hasn't been employed here for awhile and so we have just used this box.

The signals are acquired in the PEM ADCU. Alex modified the code there to send the signals over to
the new system. We also get the SUS-LSC_OUT signals from each of the suspensions so that we can
try to minimize them.

The outputs of the adaptive filter go into the unused SUS-MCL inputs of all the suspensions (except
for MC2). In the current setup, we have 6 accelerometers and 1 seismometer around the MC to be used
to demonstrate the principle of the whole thing.

Much more documentation and description of everything is necessary. We'll try to get Matt, Rob, and Alex
to use the elog.
428   Fri Apr 18 19:46:08 2008 ranaUpdateASScheck adaptive
I restarted the adaptive code today using 'startass' and 'upass'.
I moved them into the scripts/ASS/ subdirectory.

Things seem OK. With a MU=0.03 and a TAU=0.00001, there is a still
a good factor of 10 reduction of the 3 Hz stack peak from the MC2
drive by doing FF into MC1.

I edited the ASS-TOP screen so that we could see such small numbers. I
also re-aligned the MC SUS to match the input beam (mainly MC3). The
cavity was locking on a TEM10 mode mostly -- we should look in the SUS
OSEM trends to see if MC3 has moved a lot in the last month or so.

Caryn Palatchi (a Caltech undergrad who just started working with us)
illustrated to me today that using even 1000 FIR taps is not very effective
for low frequency noise cancellation if you have a 2048 Hz sample rate. More
precisely, the asymptotic Wiener filter which our 'LMS' algorithm converges
to, can often amplify the noise at frequencies below f_sample/N_taps.

A less obvious thing that she also noticed is that there is almost no cancellation
of the 16.25 Hz bounce mode when using such a short filter. That's because that
mode is fairly high Q: the transfer function from the Z-ACC to the cavity signal
goes through the high-Q vertical suspension resonance; the FF signal we send back
goes through the low-Q horizontal pendulum response only. Therefore the filter
needs to be able to simulate ~100 cycles at 16.25 Hz in order to cancel that peak.

Duh.

The message here is: we need to find a computationally efficient way to do FIR filtering
or its not going to ever be cool enough to help us find the Crab.
Attachment 1: 0052_xray_thm45.jpg
432   Mon Apr 21 12:58:42 2008 robUpdateASScheck adaptive

 Quote: Caryn Palatchi (a Caltech undergrad who just started working with us) illustrated to me today that using even 1000 FIR taps is not very effective for low frequency noise cancellation if you have a 2048 Hz sample rate. More precisely, the asymptotic Wiener filter which our 'LMS' algorithm converges to, can often amplify the noise at frequencies below f_sample/N_taps. A less obvious thing that she also noticed is that there is almost no cancellation of the 16.25 Hz bounce mode when using such a short filter. That's because that mode is fairly high Q: the transfer function from the Z-ACC to the cavity signal goes through the high-Q vertical suspension resonance; the FF signal we send back goes through the low-Q horizontal pendulum response only. Therefore the filter needs to be able to simulate ~100 cycles at 16.25 Hz in order to cancel that peak. Duh. The message here is: we need to find a computationally efficient way to do FIR filtering or its not going to ever be cool enough to help us find the Crab.

This is the reason for "RDNSAMP" parameter in the ASS code. The FIR filtration is applied at the downsampled rate, not the machine rate. So, if RDNSAMP=32, the effective sampling rate of the FIR filter is 64Hz, and thus noise cancellation should be good down to 64Hz/1000, or 64mHz, and the filter has an impulse response time that extends to 15 secs. I'm not convinced the filter length is what's limiting the performance at the bounce mode, but I agree that a faster FIR implementation would be good.
579   Thu Jun 26 21:07:11 2008 ranaConfigurationASSdust & MC1
I realized today, that yesterday while we were trying to debug the adaptive noise canceler, I turned
off the analog dewhitening on MC1. I did this by changing settings on the Xycom screen but I
forgot to elog this -- this may have caused problems with locking via increased frequency noise.
I have now returned it to its nominal, dewhitening on, configuration.

I also used mDV to look at the last year of dust trend. I have plotted here the cumulative
histogram in percentile units of the 0.5 micron dust level. The x-axis is in units of particles per cu. ft.
and the y-axis is percentage. For example, the plot tells us that over the last year, the counts were
below 6000, 90% of the time. I have set the yellow and red alarm levels to alarm at the 95-th and 99-th
percentile levels, respectively.
Attachment 1: Screenshot-2.png
981   Mon Sep 22 21:54:05 2008 ranaUpdateASSNew Wiener result with x10 gain in ACC
The 2 attached PDF files show the performance of the Wiener filter code on 2 hours of data
with a 4000 tap filter on 64 Hz data. All 6 accelerometers around the MC and the Ranger seismometer
were used.

I attribute the improved performance in the 3-10 Hz band to the better SNR of the ACC channels. To
do better below 1 Hz we need the Guralps.
Attachment 1: f.pdf
1105   Sun Nov 2 20:44:58 2008 ranaUpdateASSWiener Filter performance over 5 hours
I took one 2 hour stretch of data to calculate a MISO Wiener filter to subtract the Ranger seismometer
and the 6 Wilcoxon accelerometers from the IOO-MC_L channel. I then used that static filter to calculate
the residual of the subtraction in 10 minute increments for 5 hours. The filter was calculated based upon
the first 2 hours of the stretch.

The MC lock stretch is from Oct 31 03:00 UTC (I think that we are -8 hours from UTC, but the DST confounds me).
So its from this past Thursday night.

I wrote a script (/users/rana/mat/wiener/mcl_comp.m) which takes the static filter and does a bunch of loops
of subtraction to get a residual power spectrum for each 10 minute interval.

In the attached PNG, you can see the result. The legend is in units of minutes from the initial t0 = 03:00 UTC.

BLACK-DASHED -- MCL spectrum before subtraction

I have also used dashed lines for some of the other traces where there is an excess above the unsubtracted data.
Other than those few times, the rest are all basically the same; this indicates that we can do fine with a very
slow adaptation time for the feed-forward filters
-- a few hours of a time constant is not so bad.

After making the plot I noticed that the Ranger signal was totally railed and junky during this time.
This probably explains the terrible performance below 1 Hz (where are those Guralps?)

The second attached image is the same but in spectrogram form.
Attachment 1: f.png
Attachment 2: f1.png
1111   Mon Nov 3 22:35:40 2008 ranaUpdateASSWiener Filter performance over 5 hours
To speed up the Wiener filter work I defined a 256 Hz version of the original 16kHz IOO-MC_L signal. The
attached plots show that the FE decimation code works correctly in handling the anti-aliasing and
downsampling as expected.
Attachment 1: DAQ.pdf
ELOG V3.1.3-