J. Kissel

I'm on a slow-but-steady adventure to characterize the performance of the digital anti-aliasing system for the new 524 kHz ADC readout of the OMC DCPDs photocurrent. 
The big picture goal is too look at the 524 kHz data, and make sure that it's filtered enough before down sampling to 65 kHz and then 16 kHz, such that none of "fun and interesting" features that are real DARM signals (or analog or digital artifacts of the ADC) in the 10 - 100 kHz region appear down-converted at low frequency.

As usual, when I *look* at something for the first time, hoping for a simple "yup, works as expected," I instead get 60 new questions.

Executive summary: 
While I don't see anything obviously detrimental to the 16 kHz data, I see a confusing, frequency-independent noise floor in the filtered version of the 524 kHz OMC DCPD A channel *only* in NOMINAL LOW NOISE that I can't explain as either ADC noise, spectral leakage, or limits of numerical precision. Apparently, in NOMINAL LOW NOISE, we're not getting nearly as much digital anti-aliasing as designed.

Plots and their Explanation, Exploration, and Conclusions:

First, I compare the amplitude spectral densities of the two 524 kHz channels available for DCPD A during NOMINAL LOW NOISE vs. when DARK (the IMC OFFLINE), "as the are" 
    - H1:OMC-DCPD_A0_OUT (calibrated into milliamps [mA] on the DCPDs),                        # HAS 526-to-65 kHz and 65-to-16 kHz digital AA filters
    - H1:OMC-PI_DOWNCONV_SIG_OUT (a copy of the DCPD A channel, also calibrated into [mA]),    # DOES NOT HAVE digital AA filters

The first attachment, H1OMCDCPDs_0p1HzBW_10avgs_524kHzSignals_NLN_vs_DARK_AprtHz.png shows this comparison. 
These are the *output* of the respective filter banks, so the front-end filter banks already do *most* the work of inverting the frequency response of the TIA and Whitening filters, below 10 kHz.
That front-end calibration also already accounts for the fact that the analog voltage coming into the ADC is copied 4 times and summed, and does the "divide by four" gain of 0.25 to create an average of the voltage copies.
Importantly, the TIA and whitening's analog 11130 & 10170 and 44818.4 Hz poles are *not* inverted, so they remain a part of the frequency response of the readout -- and thus they remain uncompensated for in the plot.
The *only* calibration applied in DTT for these channels is a simple 1e-3 [A/mA].
    NOMINAL LOW NOISE, "reference" data for the two channels -- shown in CYAN and MAGENTA -- was taken a last week on 2024-06-12 20:45:46 UTC.
    DARK noise, "live" data for the two channels -- shown in RED and BLUE -- was taken this morning, with the IMC offline at 2024-06-18 15:10:28 UTC.

One immediately notices several "interesting" things:
    (1) The CYAN, OMC-PI_DOWNCONV_SIG_OUT version of the (4x copy average of the) DCPD A signal -- that doesn't have digital anti-aliasing filters applied -- shows lots of "fun and interesting" features on the DCPDs above 8 kHz in NOMINAL LOW NOISE. 
    (2) Comparing the CYAN NOMINAL LOW NOISE data with BLUE DARK data, we see that a lot of these "fun and interesting" features in the 8 kHz to 100 kHz region are real features from the detector. My guess is that this is the forest of acoustic modes of optics that appear in the DARM signal.
    (3) MAGENTA OMC-DCPD_A0_OUT version shows that most of these features *are* filtered out by 30 kHz by the digital AA filtering, BUT -- they hit some frequency-independent noise floor at "1e-12 [A/rtHz]."  This noise floor is obviously *not* a real noise floor on the DARM light coming in on the PDs, nor some sort of current noise, given how feature full the CYAN unfiltered version is. As such I will for the remainder of the aLOG put the quantitative numbers about this noise floor in quotes. What is this noise floor?
    (4) Comparing MAGENTA NLN trace and RED DARK trace, we see that when the DCPD is DARK, we see the full expected frequency response of the digital AA filters. Why are the filtered data's DARK and NLN noise floors so different?

Hoping to understand this noise floor better, I calibrated the traces into different units -- the units of input voltage in to the low-noise, 18 bit, 524 kHz ADC.
The second attachment, H1OMCDCPDs_0p1HzBW_10avgs_524kHzSignals_NLN_vs_DARK_CastAsADCInput_VprtHz.png, shows this different version of the ASD.

To calibrate into ADC input voltage units, I calibrated all the DTT traces by the following calibration (with detailed poles and zeros pulled from the calibration group's DARM loop model parameter file, pydarm_H1.ini from the 20240601T183705Z report): 
    zpk([2.613;2.195;6.556],[5.766+i*22.222;5.766-i*22.222;32.77;11130;10170],219.65,"n") # The TIA response [V/A], normalized to have a gain of 100e3 [V/A] at 1 kHz
  * zpk([0.996556],[9.90213;44818.4],1,"n")                                               # The Whitening Response [V/V]
  * gain(0.001)                                                                           # The inverse of the [mA]
where notably, I continue to exclude the two 11130, 10170 Hz poles and the 44818 Hz pole from the TIA and whitening filter's analog response. 

To compare against our model of the noise of the low-noise, 18-bit ADC sampled at 524 kHz (derived from the data in LHO:61913, shown to be comparable in G2201909), I took that model and divided by sqrt(4) = 2.0 to account for the 4 copies of the ADC noise that appear during the digitization of the 4 copies of the analog voltage. 

I also show the RMS voltage (color coded to match the corresponding ASD) in case that's important for discussions of noise floors.

This exposes more interesting things, and rules out one thing that this 1e-12 [A/rtHz] noise floor might be -- it's *not* the ADC noise floor.
    (5) Both version of the DARK noise traces, BLUE and RED show that the data agree with the noise model below 5 Hz, which gives me confidence that the scale of the model is correct across all frequencies.
    (6) The MAGENTA trace shows that, when recast into ADC input voltage, the filtered nominal low noise data's "1e-12 [A/rtHz]" noise floor is "1e-6 [V/rtHz]," which is a factor of ~4 above modeled ADC noise floor.
    (7) The BLUE trace shows the dark noise -- without the digital anti-aliasing filters -- is *also* above the ADC noise, is frequency-dependent, and *below* the MAGENTA filtered NOMINAL LOW NOISE data in the 20 to 200 kHz region. 
(5), (6), and (7) all give me confidence that this frequency independent noise is *not* ADC noise
    (8) The ADC input voltage 
        (a) during NOMINAL LOW NOISE data spans 5 orders of magnitude (0.2 [V_RMS] total in the CYAN NLN unfiltered trace, w.r.t. this 1e-6 [V/rtHz] high frequency noise floor in the filtered MAGENTA trace), and
        (b) the DARK data spans 6 orders of magnitude (5.4e-4 [V_RMS] of the BLUE unfiltered trace, w.r.t the 1e-10 [V/rtHz] at the lowest 65 kHz notch in the RED trace). But BOTH are still above the dynamic range of where floating point precision would be causing problems -- dynamic range of 8 orders of magnitude. So, I don't think it's a numerical precision issue, unless I'm misunderstanding how that works.
    (9) Also, given that the NOMINAL LOW NOISE data spans less orders of magnitude that the DARK noise data, and yet the filtered RED data does *not* show any such frequency-independent noise floor, I also don't think it's an issue what window I've chosen on the ASDs (it's the default Hanning window, for the record).

Just to add another view in further different units, to explore (8) a little better, I attach a third version of the plot, but cast into the 18 bit ADC's counts.
This is the third attachment, H1OMCDCPDs_0p1HzBW_10avgs_524kHzSignals_NLN_vs_DARK_CastAsADCInput_ctprtHz.png.
The further calibration is a simple multiplication of the ADC Input Voltage by a further 2^18/40 [ct/V]. 
(The inverse of this ADC voltage calibration has already been done in the "raw" channels in [mA], and the factor of 4 for the number of copies being summed has already been accounted, so I *can* treat these channels like a single ADC channel).
    (10) Not much more interesting here, the MAGENTA trace showing "1e-12 [A/rtHz]" or "1e-6 [V/rtHz]" high frequency noise floor of the filtered NOMINAL LOW NOISE data, now in ADC counts, lands at an uninteresting "7e-3 [ct/rtHz]", compared to an RMS of ~1000 [ct_RMS]. So, it's not like we're bottoming out on ADC counts or ADC precision.

I'd really like to understand this noise, since it's present during NOMINAL LOW NOISE, and it's indicating that there's some flaw in our digital anti aliasing.