Matt, Rolf, Dave, Alex (remote)
We got the Biquad filter module working in the new CDS system and made a few simple tests. For these test I created a few somewhat complex filters:
- Pass 1Hz - a resonant filter which passes 1Hz and the cuts off steeply
- Not 1Hz - a 1Hz notch filter with 120dB of attenuation, also has a pair nearly canceling poles and zeros at 10mHz
- HP 1k - a high-pass filter, flat above about 1kHz
The first test was to check the transfer function. The biquad filter matched the the standard filter (direct form 2) with various combinations of the above filters. No problems were observed.
The noise test was designed to match the test performed at the 40m lab in March 2008. With all of the above filters engaged, a 1 Hz sine wave with amplitude 1 was injected into the input of the biquad filter and a standard filter module. In the frequency region where the noise is amplified by the HP filter, the biquad filter produced a numerical noise floor 8 orders of magnitude lower than that of the standard filter. Furthermore, the harmonic distortion seen in the standard filter was absent in the biquad.
A little more information... the trouble with DF2 is that it essentially applies the poles first, and then the zeros. This is a problem only when there are low-frequency poles and zeros, since the input is attenuated by the poles (into the numerical noise generated by the filter), and then amplified by the zeros (along with the noise). The "Not 1Hz" filter is designed to demonstrate this by placing poles and zeros at the same frequency, both for the notch at 1Hz, and the pole/zero pair at 10mHz which accetuates the effect. The input 1Hz sine wave is also attenuated by this filter, so that the noise generated in the filters is easier to measure. The noise is then amplified by the "HP 1k" filter which follows the "Not 1Hz" filter to prevent masking by numerical noise added later in the data recording and processing. (The "Pass 1Hz" input filter is present only to reduce the on and off glitches produced when starting and stopping the excitation.) See also G0900928.