Rectification Quotient Sync Filter

Taking the quotient allows you to save 95% of the time constant in some situations:

Assume:

Signal 1 = 0.6 + 2sin6.3t

Signal 2 = 0.6 - sin6.3t

The 0.6 is DC or low frequency noise.

First each signal is integrated to get the DC noise. The integrals are then subtracted from the respective signals. After that rectification and another integration. In real life a small prefilter would be necessary just before the division.

cut and paste in

Output from signal 1:

{[2t-0.31746 cos(6.3 t)-0.01]/t} t from 0 to 35

Output from signal 2:

{(t-0.15873 cos(6.3 t))/t} t from 0 to 35

To get within 0.5% takes 30 - 35 cycles.

But taking the quotient reduces the time to 1.5 cycles:

{[2t-0.31746 cos(6.3 t)-0.01]/[t-0.15873 cos(6.3 t)]} t from 0 to 3

This saves time when SNR >1

Bret Cahill

Yes, you're right to look for a simpler way to proceed; if the signals aren't heavily noise-contaminated, you get good results by just gating out the low-denominator times. This means approximating the complicated weight function that results from full noise analysis with " 0 if abs(theta) .le. 30 degrees, 1 otherwise" or something similar.

That loses some signal, and includes some noise, but isn't unsupportable. The alternatives are more attractive, IMHO.

The full analysis of a multiplying phase-locked amplifier notes that the output includes amplitude-of-signal plus sine(2* w *t), and if you don't want the double-freqency part, EITHER you need to integrate a long time (many cycles) and let it average to zero, or you can track/hold the integrator filter, holding the value only at the reference signal zero crossings. The combination of multiply and track/hold seems just as easy as divide and gated- integrate. It doesn't run into (for instance) amplifier-saturation issues.