J. Kissel, E. Goetz

Executive Summary: after thinking about things a bit more, and considering all the conflicting metrics, we need to change frequency of the PCALX 410.2 Hz calibration line, and update/revert *all* the CALCS DEMOD filters and parameters, in order to achieve a compromise between Line-to-Line Bias, IFO parameter change time scales, and accuracy in uncertainty estimate. We'll do this on Monday 2023-05-01.



The other day, when Evan and I increased the time-constant of the I & Q low pass filters in all of the calibration line DEMODS (LHO:69117), our metrics under consideration were:
    (1) We need to narrow up the band-pass for the PCALY 410.3 Hz line because we found that the adjacent PCALX 410.2 Hz line was polluting the TDCF estimates of the relative optical gain and cavity pole (LHO:68479)
    (2) As a general principle for any DEMOD process, one must make the impulse response of the I and Q low pass filters longer than the impulse response of the SIG band-pass filters, or else the DC component of the I and Q signals (the output of the low pass) is going to be entirely confused by the impulses of the filter rather than actual changes in the line amplitude and phase.

So, based on those criteria, we figured "let's just make all the low-pass filters long, i.e. with a 40 sec time constant, or 0.025 Hz corner frequency" so that regardless of whether the band pass on the PCAL lines is narrow (+/- 0.03 Hz) or wide (+/- 0.1 Hz), we can just consistently "average for longer and get a less noisy answer of the (I and Q) cast into (Real and Imag) cast into (magnitude and phase)." That's LHO:69117. Today as a part of the thinking that went into this aLOG, we confirm that Metric (2) is satisfied by comparing the step responses of the two filter pairings -- see 2023-04-28_DEMOD_Filter_ImpulseResponses.png. Indeed in both cases (red and blue), the impulse response of the SIG band-pass is shorter than the I & Q low-pass. 



But, then we remembered that we also want to compute the uncertainty in the answers we get from the TDCFs and systematic error monitoring. The uncertainty calculation depends on the coherence, C, of the transfer function as well as the "number of averages", Navg. In the frequency domain, the thing that you're averaging is consecutive FFTs. The relationship between coherence and uncertainty we've been using for decades is the ol' Bendat and Piersol formula,
    Unc [rad] = sqrt( (1 - C) / (2*Navg*C) )

Here, in the case of a live calculation, we treat the FFT length as equivalent to the time constant of I & Q low-pass filters. The critical thing to remember there is that increasing the number of averages does NOT improve the uncertainty, it improves the accuracy of the estimate of coherence or uncertainty. 

So, this adds two more metrics to consider: if we want to keep the time over which we're creating this live rolling average constant -- at ~2 minutes, or ~120 seconds -- the time scale that we think is about the right time scale to capture the speed at which the IFO thermalizes and/or changes alignment (a number we collectively settled on after an old, crude, study by Sudarshan in LHO:22753) then we should favor more averages of shorter FFTs. 
    (3) We want FFT stride * Navg to be ~120 seconds, and
    (4) in order to achieve a more accurate estimate of the coherence and thus uncertainty we want higher corner frequencies on our I and Q low-pass filters, and more averages

In the CALCS front-end code as it stands now, the FFT length and Number of averages for all DEMOD's uncertainty estimate are determined by one pair of parameters, Navg = H1:CAL-CS_TDEP_COH_BUFFER_SIZE and FFT length = I & Q Low Pass Time Constant = H1:CAL-CS_TDEP_COH_STRIDE. This means
    - whatever we choose for the I & Q filters' low-pass time constant, you MUST update the FFT Length = COH_STRIDE to match it, and
    - Once one update the FFT length = COH_STRIDE, you have to update the Navg = BUFFER_SIZE to keep the overall time period sampled constant (and in our case we want this at 2 minutes).

So Metric (4) -- wanting high corner frequencies, low time-constants on our I & Q low passes in order to get the best estimate, is in direct opposition to Metric (1) -- wanting narrow SIG band-passes to isolate the line frequencies from other lines, because of Metric (2) -- a narrow band pass has a long impulse response, and we'd need low corner frequencies, high time-constants  on out I & Q low passes.




So how do we decide on the compromise?
At the moment, we're leaning towards "it's right before the engineering run, so the newest thing that's causing problems must change."
That means:
    - Move the PCALX 410.2 Hz PCALXY comparison line further away from the pre-existing PCALY 410.3 Hz TDCF line. Joe, for entirely different reasons, recommends 0.5 Hz separation instead of 0.1 Hz.
    - Update the "DARM Model transfer function values at calibration line frequencies" EPICs records for the PCALX 410.2 Hz line,
    - Revert all DEMOD the band-passes to have a pass band that's +/- 0.1 Hz wide (what we had in O3)
    - Revert all DEMOD I & Q low passes to a 10 second time constant, or 0.1 Hz corner frequency
    - Change COH_STRIDE back to 10 seconds to match the low pass, and Change the BUFFER_SIZE back to 13.0 in order to preserve the rolling average of 2 minutes.

This is too much to do on a Friday night at 6pm local time, so we'll make all these changes on Monday morning. That'll also give some time for others to weigh in if they're leaning a different way on the compromise, or they've got other metrics we haven't yet considered.