P. P. Vaidyanathan wrote a chapter in the book "Handbook of Digital Signal Processing: Engineering Applications" called "Low-Noise and Low-Sensitivity Digital Filters" (Chapter 5 pg. 359). I took a quick look at it and wanted to give some thoughts in case they are useful. The experts in the field would be Leland B. Jackson, P. P. Vaidyanathan, Bernard Widrow, and István Kollár. Widrow and Kollar wrote the book "Quantization Noise Roundoff Error in Digital Computation, Signal Processing, Control, and Communications" (a copy of which is at the 40m). it is good that P. P. Vaidyanathan is at Caltech.
Vaidyanathan's chapter is serves as a good introduction to the topic of quantization noise. He starts off with the basic theory similar to my own document on the topic. From there, there are two main relevant topics to our goals.
The first interesting thing is using Error-Spectrum Shaping (pg. 387). I have never investigated this idea but the general gist is as poles and zeros move closer to the unit circle the SNR deteriorates so this is a way of implementing error feedback that should alleviate this problem. See Fig. 5.20 for a full realization of a second-order section with error feedback.
The second starts on page 402 and is an overview of state space filters and gives an example of a state space realization (Fig. 5.26). I also tested this exact realization a while ago and found that it was better than the direct form II filter but not as good as the current low-noise implementation that LIGO uses. This realization is very close to the current realization except uses one less addition block.
Overall I think it is a useful chapter. I like the idea of using some sort of error correction and I'm sure his other work will talk more about this stuff. It would be useful to look into.
One thought that I had recently is that if the quantization noise is uncorrelated between the two different realizations then connecting them in parallel then averaging their results (as shown in Attachment 1) may actually yield lower quantization noise. It would require double the computation power for filtering but it may work. For example, using the current LIGO realization and the realization given in this book it might yield a lower quantization noise. This would only work with two similarly low noise realizations. Since it would be randomly sampling two uniform distributions and we would be going from one sample to two samples the variance would be cut in half, and the ASD would show a 1/√2 reduction if using realizations with the same level of quantization noise. This is only beneficial if the realization with the higher quantization noise only has less than about 1.7 times the one with the lower noise. I included a simple simulation to show this in the zip file in attachment 2 for my own reference.
Another thought that I had is that the transpose of this low-noise state-space filter (Fig. 5.26) or even of LIGO's current filter realization would yield even lower quantization noise because both of their transposes require one less calculation. |