Regarding http://nodus.ligo.caltech.edu:8080/40m/5867 and http://nodus.ligo.caltech.edu:8080/40m/5869 :

MC_F signal:

The measurements on p. 5867 were done using the ADC attached to the PEM computer. There was a big difference between the MC_F signal recorded directly after the server board and the signal just before the FSS board as recorded by a PEM channel.
To understand how this happens we measured the signal at different places with a spec. analyzer:

1. WIth a locked MC measuring the signal just before the PEM ADCs (meaning after a 60ft BNC cable)
2. Same position, but unlocked and seemingly dark MC
3. Locked MC, signal just before the FSS box
4. MC_F signal that is usually going into the Pentek Generic board and is recorded in C1:IOO

=> The 60ft BNC cable adds a considerable amount of noise, but doesn't fundamentally change the signal. It is weird that the signal is white from approx. 4Hz on.
Due to Jenne's measurement ( http://nodus.ligo.caltech.edu:8080/40m/5848 ) we know the TF from MCL through PD, mixer Pentek and into C1:IOO looks like this:

This is with the double HP from 15Hz on that should be in the Pentek. So one might expect a less white signal going into the FSS board...

 PEM ADCs

The dark noise in the PEM ADCs is actually a factor 10 higher than in the IOO ADCs. Still ok wrt the the seismometers.
We also tried to measure essentially the dark noise of the whole seismometer readout (seismometer box, then ADC). That seemed ok, but is of limited value since the seismometer electronics behave a bit strange when there is no seismometer connected.

We measured the coherence between the seismometer near the MC2 stack and accelerometers on the vacuum tank where MC2 is. Because accelerometers produce small signals at low frequencies, which are comparable with adc noise, we  amplified the accelerometer signal by a factor of 20. We could not do it more because though adc has 40 V range, the black box that follows the channel sockets can transmit only 2.5 V max amplitude signal. Probably, this was done because old adc accepted 2 V max amplitude.

We were able to found some coherence at 0.1-1 Hz though the accelerometer signal is rather noisy. So to consider stack as a noise source is still possible. This measurement should be better done with two seismometers, one on the floor, the other on the stack. From the figure we can also see that tilt affects the x and y seismometer signals from 0.1 Hz. Green line (z-component) is much lower that red and blue lines (x and y). Tilt affects on horisontal axes of the seismometer much more than on vertical.

What we also think about is that at low frequencies mirrors start to move approximately the same and seismometers can help us to figure out small reletive displacement of the mirrors which form the MC length. We can estimate the critical frequency by presenting the ground motion as interference of surface waves with different velocities and amplitudes. For only 1 wave we have for the relation of MC length to the seismometer read out  ~sin (2*pi*f/v*L). f - frequency of the wave, v -speed, L - length between the mirrors. We can see that below 1 Hz we have ~sin (f/2). At this point seismometer signal could lose coherence with MC length signal. We could try to subtract seismometer signals from corresponding axes, but gur1 and sts1 has different calibrations. Moreover, the noise floor of the seismometers might not allow us to measure the differential signal. We'll try to simulate this scenario and find seismometer calibration or measure it. We are basicly interested only in the ratio of calibraion fucntions of 2 seismometers.

[Jenne, Mirko]

1. We should help the OAF by compensating for the actuator TF:

The actuator TF, from adaptive filter output to MC2, through PD, mixer, Pentek and into C1:IOO looks like this:

If we assume a white-ish error signal that the adaptive code tries to compensate for its job gets extra complicated because it has to invert this TF. So we really should compensate for that. Easiest place for that is the CORR filter directly behind the adaptive code block.

Using the TF measurement from above I used the vectfit (" /cvs/cds/caltech/apps/mDV/extra/firfit_forFotonMirkoComplex.m" ) to get fit a corresponding digital filter:

If we invert swap the zeros and poles in the digital filter we get the inverted TF.
(Todo: Figure out how to invert the TF. Just switching the poles and zeros doesn't work).

2. Wiener filtering

The idea was to use the adaptive filtering only for small corrections to the wiener filtering. So we really should try to get the wiener filtering going.

Howto:

1. Get data for STS1X and GUR1X and MC_F in matlab. E.g. via ligodv
2. Check the MC was in lock the entire time.
3, Filter MC_F with the actuator TF, so the wiener filter knows about that and compensates for it
4. Calculate the wiener filter " h1winolevLigoDV.m "
5. Export the data to the workspace. It is also saved to the disc as "h1filtcoeffTS.mat". Make sure there are first the witnesses, then MC_F
6. Execute " /cvs/cds/caltech/apps/mDV/extras/LHO/firfit_for_FotonMirko.m" while one directory higher. 
7. Copy the digital filter in SOS form that is printed into the matlab command line and put it into the corresponding filter in the OAF model via foton.

With data from 11-11-15 04:00 to 05:45. Sampling freq. 256Hz. 8000 Taps => length = 30.2s. Prefiltered to notch the 60Hz line in MC_F, but not compensation the actuator TF. This results in the following wiener filter and corresponding SOS filter to be copied into foton.
STS1X:

GUR1X:

[Micro, Den]

Analyzing coherence of seismic noise and mode cleaner length we've figured out that at some days the coherence below 1 Hz is still present. For example, at Nov 13 we can see some coherence compared to most other dates when we are not able to see coherence as shown on the figure. On the top plot - psd of MC_L and GUR1_X at Nov 13 (red and blue) and Nov 15 (black and cyan). On the bottom plot is presented coherence between MC_L and GUR1_X on Nov 13 (red) and Nov 15 (black)

We can divide the psd plot for 2 parts - below 1 Hz and above 1 Hz. Above 1 Hz seismic noise on Nov 15 (cyan) was higher then on Nov 13 (blue) and correspondently MC_L at that region was higher on Nov 15. Below 1 Hz seismic noise was higher on Nov 13 but MC_L is still lower that on Nov 15. That is surprising. From the coherence plot we can say that once we have some more seismic noise than usually, we immediately see coherence.

Because of this we wanted to find out the level of the X noise that makes seismic noise invisible. We injected seismic noise by doing smooth physical exercises near MC_2 (1.5 m and 3 m apart). The MC_2 was in lock during the experiment.

In the coherence plot we can see that coherence between GUR1_X and MC_L increased with noise injection. The highest coherenced we obtained sittind down and standing up smoothly near MC_2 at distance 1.5 m. We did not want to come clother and break the lock. This measurement tells us that the X noise is approximately 3-4 times higher than seismic noise in the range 0.1 - 1 Hz. That means that it is approximately 1e-6 - 1e-8 m/sqrt(Hz) in this region. This noise goes down at frequencies from 2 Hz and not seen because of seismic noise. Actually, seismic noise can be filtered out with the Wiener filter and then we'll see the spectrum of X noise.

We now try to figure out the method to estimate the contribution of OSEM noise to the X noise.

Analyzing coherence between MC length and signals on MC1, MC2 and MC3 coils we have noticed that MC1 COIL signal is not coherent to MC length at all at interesting frequencies 0.1 - 1 Hz.

We try to explain this phenomena.

I've looked through the coherence between the MC length and seismometers after the if-statement problem was fixed. Coherence improved for all seismometers but is still not 1. It is possible that contribution from X, Y, Z directions split the coherence between them but at ~0.2-03 Hz we do not see much coherence for all these directions.

I looked at the coherence between MC2 OSEM signal and MC_F when the AUTO LOCKER is ON and OFF. I thought that we'll ses the same coherence for both regimes as laser is locker to the MC length. However, I figured out the coherence is worse when AUTO LOCKER is ON at frequencies 0.2-0.3 Hz.

The first idea that comes to mind is that when feedback to the laser is provided, the pressure to the mirrors from the laser beam is changed.

I've added filter banks for correction path in the C1OAF model to use AA filters.  I compiled and installed the new version. I runs but does not sync. Probably, I've made a mistake in the some names of epics channels. Leave it for now, figure out tomorrow. If someone needs an old version, it is in the /opt/rtcds/caltech/c1/userapps/trunk/isc/c1/models/c1oaf_BACKUP20111204.mdl file. Corresponding medm screen is in the /opt/rtcds/caltech/c1/userapps/trunk/isc/common/medm/OAF_OVERVIEW.adl file.

 The general rule in the 40m is that if it's not an 'emergency', i.e. something is wrong with the computers and preventing the main locking work, no model recompiling-type activities at nighttime. 

Also, if you do things and recompile, you need to do an svn check-in.  That's where backups are kept.  We don't want to clutter folders with backups anymore.

 I did offline adaptive filtering with yesterday's 3 hours of MC-F and GUR1X data. It turns out that normalized-lms can strongly outperform static Wiener filtering!

This is interesting. It might be something inside MC_F that Wiener static does not see. I think the problem is either with seismometer noise or tilt.

I tried to figure out why red NO SYNC label became present in the C1OAF_GDS_TP screen after I added AA filters to the C1OAF model.

C1OAF model contains 8 libraries C1OAF_ADAPT for 8 DOF. I changed C1OAF_ADAPT library to C1OAF_ADAPT_AA library where I added 28 AA filters for 28 witness channels. It turns out that if I use this library for all 8 DOF then I see NO SYNC label, if only for one DOF (MCL) then I see green IOP label. This means that using AA filters for each DOF too much channels of filters are created for online system to operate. I think there is some number inside the code that one can not exceed. Analyzing compilation output after "make c1oaf" I figured out that without using AA filters we have 632 filters and using AA we have 856 filters.

For now I'll use AA filters for MCL only.

 I have a feeling we are not fitting into pre-allocated memory space in the shared memory between the front-end process and the epics process. Filter module data is overwriting some other data and that's why we are not getting a sync light. I suggest we upgrade to 2.4 code first and then we will figure out a way to expand memory areas to fit 856 filters.

[Jenne, Den]

AA filters for witness channels are added to the oaf model. It is working now and the number of memory used is not critical.  NO SYNC is not present any more.

 Why did you place Matt's code inside the simulink library and use the same library for all DOFs? I think this won't work out. Inside the .c code there are static variables. If all DOF use the same ADAPT_XFCODE() function, it means that they all mess there signals and coefficients with each other! Or the RCD during the compilation creates a copy of the function with the name of a library name in front? For example, ADAPT_MCL_ADAPT_XFCODE(). But then in the RCG manual it is claimed to name the .c file the same.

This problem can be fixed by creating .c files with proper names for each DOF. But here a memory question may arise. For 1 DOF we now have 28 witness channel. If we have a several minute filter, we use 28 * 10⁴(filter length) * 3 (FIR coefficients, adapt input, corr input) * 8 (number of bytes in 1 double)  = 6.7 Mb / DoF. For 8 DOF we'll allocate ~55 Mb of memory in the kernel. The c1lsc cache size is 6 Mb per cpu. So we are definitely out of cache and it will take some time for a processor to communicate with ram. I wonder if it is OKEY for us to allocate this amount of memory as static arrays inside the kernel.

Now we use 6.7 Mb of memory because it seems to be a mistake with placing the same function for all DOF and we actually allocate for 1.

Online filter diverges. I did offline simulations with current c-code. Offline filter also diverges, even in the simplest case 

witness = randn(1e6, 1); target = witness + 0.01*randn(1e6, 1);

I tried to create a new implementation of FXLMS algorithm as a c code. Then with this c code I did offline filtering with MCL and GUR signals and compared the error signals depending on the length of the filter.

One can see the code at the svn

adaptOnline - start here and choose algorithm

adaptive_filtering - Matlab implementation of AF

current_version.c - current version of the Filter (Matt's)

fxlms_filter.c - new version of the FXLMS filter

oaf.c - agent between Matlab and C (edited Matt's file)

Data samples can be found at nodus /users/den/wiener_filtering/data

We can account for delays in the oaf system by compensating it in the adaptive path of the filter. But using only this procedure is not enough. Parameters mu and tau should be chosen accurately:

w = (1 - tau) * w;

w += mu * dw / norm;

NLMS algorithm without considering delays works well for mode cleaner length and gur1 seismometer signals, significantly reducing MC_F  with parameters mu=1, tau=0. These parameters are considered because nlms algorithm should converge with the highest speed when mu=1. However, if the system has a delay so at time moment n:

error_signal [n] = desired_signal [n] - filter_output [n-delay];

then the adaptive filter diverges for the same parameters mu=1 and tau=0 even for delay=1. For that reason we make the same calculations with tau = 1e-4 and tau = 1e-2 without reducing mu conserving the adaptation rate and get the same result as nlms algorithm without delays. Next figure shows MC_F signal, error after applying e-nlms filter with tau=1e-4 and tau=1e-2. "e-" is added to show that  a small number (epsilon) is added to the norm of the signal in order to prevent the filter from diverging in the beginning of the process when the norm is not well-determined yet.

The test was done offline with the sampling frequency 2048 Hz, without downsampling and any filters. We can see that tau=1e-4 is still not enough, tau=1e-3 or tau=1e-2 is as good as nlms without delays, tau=1e-1 and high are also bad.

Correctly choosing tau we have some freedom for delay compensation in the adaptation path. This is important as we do not know exactly what is the delay in the real system. We can measure it approximately. In order to figure out the range of reasonable delay errors we make a test with delay = 1, but to the adaptation path we give delays from 0 to 10. It turns out that adaptation path delays greater then 5 make the filter diverge, delays in the range 0-3 produce a reasonable error. In the figure below errors with adaptation path delays = 1 (correct) and 3 are presented.

Here for the downsampling process we use a low-pass Bessel digital filter of order 6, normalized cut-off frequency = 0.1. In the plot presented below we compare the results with downsampling ratio = 1, 2, 4.

We can see that increasing the downsampling ratio, we increase the error of the filter. Moreover, the error at some particular frequency f seems to depend on the ratio f/Fs, where Fs - sampling frequency (2048 Hz) devided by downsampling ratio. Error is the same for all curves below 1 Hz but then begins to increase as we increase the sownsampling ratio. In order to figure out what the problem is - mistake in the filter code, inaccurate upsample algorithm or this is NLMS particularity, I've changed sampling frequency in the chans/daq/C1PEM.ini and C1IOO.ini files from 2048 Hz to 512 Hz for corresponding channels. Now, we compare the error from the filter working with 2048 Hz frequency, downsampling ratio = 4, low-pass filter = Bessel of order 6, normalized cut-off = 0.1 and filter working with 512 Hz sampling frequency, without downsampling and with corresponding Bessel low-pass filter with normalized frequency 0.4.

MC_F measurement at 2048 Hz was done during the day, for that reason red curved is slightly higher then green in the resonance frequencies. But still we can see that these two cases are very much alike. For this reason, it seems that NLMS filter works better with higher sampling frequencies.

 In order to figure out what downsampling ratio we can take, we need to determine the running time of the fxlms_filter() function. If the filter length is equal to 5000, downsampling ratio is equal to 1, number of witness channels is 1 then with ordinary compilation without speed optimization one call runs for 0.054 ms (milli seconds). The test was done on the 3 GHz Intel processor. With speed optimization flags the situation is better

-O1 0.014 ms, -O3 0.012 ms

However, Alex said that speed optimization is not supported at RCG because it produce unstable execution time for some reason. However, by default the kernel should optimize for size -Os. With this flag the running time is also 0.012 ms. We should check if the front-end machine compilers indeed use -Os flag during the compilation and also play with speed optimization flags. Flags -O3 and -Os together might also give some speed improvement.

But for now we have time value = 0.012 ms as running time for 5000 coefficient filter, 1 witness channel and downsample ratio = 1. Now, we need to check how this time is scaled if we change the parameters.

5000 cofficients - 0.012 ms

10000 coefficients - 0.024 ms

15000 coefficients - 0.036 ms

20000 coefficients - 0.048 ms

We can see that filter length scaling is linear. Now we check downsampling ratio

ratio=1 - 0.048 ms

ratio=2 - 0.024 ms

ratio=4 - 0.012 ms

Running time on the dependance of downsample ratio is also linear as well as on the dependence of the number of witness channels and degrees of freedom. 

If we want to filter 8 DOF with approximately 10 witness channels for each DOF, then 5000 length filter will make 1 cycle for ~1 ms, that is good enough to make the downsample ratio equal to 4.

Things get a little bit complicated when the code is called for the first time. Some time is taken to initialize variables and check input data. As a result the running time of the first cycle is ~0.1 ms for 1 DOF that is ~10 times more then running time of an ordinary cycle. This effect takes place at this moment when one presses reset button in the c1oaf model - the filter becomes suspended for a while. To solve this problem the initialization should be divided by several (~10) parts.

In order to prevent different DOF from redetermining static variables in the adaptive code, I've created a separate code for each DOF with the name ADAPT_XFCODE_{$DOF}.c

I've provided the links for these files in the c1oaf.mdl, compiled and run it. Now there are no conflicts between DOFs.

I tried to filter MC_F from seismic noise measured by GUR1 seismometer. I've used 8000 tap filter, downsample ratio=8, delay=1. In the Figure the output of the filter is presented with MC_F signal.

We can see that output is close to the MC_F, but the phase for some reason is not zero. It should not be at 1 Hz - 10 Hz due to the actuator. But below these frequencies I do not see any reasons for the output phase to differ from MC_F phase. But it is possible, the phase of the actuator is evaluated very rough and the adaptive filter can't match it.

 Just to be clear what I said at the meeting, I write all this down here. Adaptive filtering of real signals (MC_F and GUR1_X) with all noises inside is 

This is offline filtering but with real signals and with the C-code that is compiled at the 40m now. We can reduce the MC_F signal by ~100 below 10 Hz, but  the problem is that reducing the adaptation gain, the error increases. As a result when we move towards FxLMS algorithm with AA, AI and downsampling, we have to take the gain equal to ~1e-2 and we do not reduce any noise. 

The second demonstration of this problem is static Wiener filtering. This is the result

We can see that adaptive filtering outperforms the "optimal" filtering. This is because an adaptive filter can follow the changes of coefficients immediately while the Wiener filter averages them. This is the mathematical formulation:

mcl_real = coeff _real* seismic_noise_real + other_noise

mcl_real - the real length of the MC,

coeff_real - real coefficients, that represent the transfer function between seismic noise and MC length,

other_noise - noise uncorrelated to the seismic noise seismic_noise_real

But in the world of our measured signals we have the equation

mcl_measured = coeff * seismic_noise_measured + other_noise

mcl_measured = TF_mcl * mcl_real

seismic_noise_measured = TF_seis * seismic_noise_real

where TF_mcl and TF_seis - transfer functions from the real world to measurements.

It seems to me that TF_mcl or TF_seis are not constants and for that reason the TF between measured seismic noise and mcl is not constant. But it is exactly what an adaptive or Wiener filter tries to define:

coeff(time) = average(coeff(time)) + delta(coeff(time))

The result of applying average(coeff) is the green line in the Figure 2 - error after applying the Wiener filtering.

delta(coeff) - the changing part of the transfer function is caught by the adaptive filtering. The lower the gain, the lower is the capability of adaptive filter to catch these changes. Theoretically. the error after applying adaptive filter can be presented like this:

E(error*error) = E(other_noises*other_noises) + 1/(2-mu)*mu*E(other_noises*other_noises)  + 1/ {mu*(2-mu)} * Tr(Q) * A

where mu = adaptation gain

Q - covariance matrix of delta(coeff)

A - norm of the seismic signal

The first term in this equation is the dispersion of other noises, the second term is the error of the adaptive filter due to non-zero gain, the third term is due to the changes in the transfer function - we can see that it is proportional to 1/mu. This term explains why the error increases while mu decreases.

Now I'm looking for the part in the path of the signals where the transfer function can change. As I mentioned above, this is not a change in the real world, it is the change in the measured signlals. My first guess is the quantization error - we do not have not enough counts. If this is not the case, I'll move to other things of the signal path.

Today I tried to make the lms filter to work online. I played around with the signals (GUR1_X and MC_F) to pre-whiten them and in the end the following configuration worked out:

1. mu = 0.03, tau = 1e-5, downsample=8, nCoeff = 4000, delay = 5 (sample-and-hold delay is not included in the new code, it should be added here!)

2. witness pass: AA32 = cheby1("LowPass", 4, 1, 32) AND 0.1:0

3. witness adaptation path: AA32 AND AI32 = cheby1("LowPass", 4, 1, 32) AND 0.1:0

4. error path: AA32 AND 0.1:0 AND anti_1Hz. Before I added anti_1Hz filter oaf did nothing. This filter tries to approximate the actuator transfer function. Note, it is not in the witness adaptation path. This is some sort of whitening.

5. correction path: AI32, gain = -1

Convergence time ~ 5 mins. The performance of the filter is far not perfect compared to the offline implementation. But it deals with a stack though.

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.

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

 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: 

norm += (*pBufAdapt) * (*pBufAdapt); // add to the norm 

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

wit[i].norm += histAdpt[nCoeff]*histAdpt[nCoeff] - histAdpt[0]*histAdpt[0];

 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 )

    {

      adaptGain = 0.0;

      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. 

 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.

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.

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.

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.

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.

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.

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.

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.)
- sky blue; ADC noise

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.

  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).

 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?

  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.

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.

[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.

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.

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.

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.

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.

 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.

[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.

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