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Entry  Mon Sep 20 15:42:44 2021, Ian MacMillan, Summary, Computers, Quantization Code Summary 472px-Typical_State_Space_model.svg.pngBiquad_filter_DF-IIx.svg.pngSS-IIR-TF.pdf
    Reply  Wed Sep 22 14:22:35 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary IIR64-bitnoisespectrum.pdf
       Reply  Mon Sep 27 12:12:15 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary 
          Reply  Mon Sep 27 16:03:15 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary DeltaNoiseSpectrum.pdf
             Reply  Mon Sep 27 17:04:43 2021, rana, Summary, Computers, Quantization Noise Calculation Summary 
                Reply  Thu Sep 30 11:46:33 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary SS-IIR-Bode.pdfPSD_32bit.pdfPSD_64bit.pdf
                   Reply  Wed Nov 24 11:02:23 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary PSD_IIR_all.pdf
                      Reply  Wed Nov 24 13:44:19 2021, rana, Summary, Computers, Quantization Noise Calculation Summary 
Message ID: 16360     Entry time: Mon Sep 27 12:12:15 2021     In reply to: 16355     Reply to this: 16361
Author: Ian MacMillan 
Type: Summary 
Category: Computers 
Subject: Quantization Noise Calculation Summary 

I have not been able to figure out a way to make the system that Aaron and I talked about. I'm not even sure it is possible to pull the information out of the information I have in this way. Even the book uses a comparison to a high precision filter as a way to calculate the quantization noise:

"Quantization noise in digital filters can be studied in simulation by comparing the behavior of the actual quantized digital filter with that of a refrence digital filter having the same structure but whose numerical calculations are done extremely accurately."
-Quantization Noise by Bernard Widrow and Istvan Kollar (pg. 416)

Thus I will use a technique closer to that used in Den Martynov's thesis (see appendix B starting on page 171). A summary of my understanding of his method is given here:

A filter is given raw unfiltered gaussian data f(t) then it is filtered and the result is the filtered data x(t) thus we get the result: f(t)\rightarrow x(t)=x_N(t)+x_q(t)  where x_N(t) is the raw noise filtered through an ideal filter and x_q(t) is the difference which in this case is the quantization noise. Thus I will input about 100-1000 seconds of the same white noise into a 32-bit and a 64-bit filter. (hopefully, I can increase the more precise one to 128 bit in the future) then I record their outputs and subtract the from each other. this should give us the Quantization error e(t):
e(t)=x_{32}(t)-x_{64}(t)=x_{N_{32}}(t)+x_{q_{32}}(t) - x_{N_{64}}(t)-x_{q_{64}}(t)
and since x_{N_{32}}(t)=x_{N_{64}}(t) because they are both running through ideal filters:
e(t)=x_{N}(t)+x_{q_{32}}(t) - x_{N}(t)-x_{q_{64}}(t)
e(t)=x_{q_{32}}(t) -x_{q_{64}}(t)
and since in this case, we are assuming that the higher bit-rate process is essentially noiseless we get the Quantization noise x_{q_{32}}(t).

If we make some assumptions, then we can actually calculate a more precise version of the quantization noise:

"Since aLIGO CDS system uses double precision format, quantization noise is extrapolated assuming that it scales with mantissa length"
-Denis Martynov's Thesis (pg. 173)

From this assumption, we can say that the noise difference between the 32-bit and 64-bit filter outputs:  x_{q_{32}}(t)-x_{q_{64}}(t)  is proportional to the difference between their mantissa length. by averaging over many different bit lengths, we can estimate a better quantization noise number.

I am building the code to do this in this file

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