Rectification Quotient Sync Filter

[Bret Cahill]

Now we can easily guess several things:

1.  dividing 2 in sync signals is done all the time.

No, usually one samples the signals, say X and Y,
and does a least-squares fit to determine
 Y= aX + b
and on finding b approximately equal to zero, one has
a good value for the ratio, 'a'.
Division isn't a necessary step at all.   Accumulating
(in this case, integrating)  values like X, Y, X**2, X*Y,
and doing some arithmetic ON THE SUMS.

2.  the time to smooth a rectified AC to a DC output is frequently an
issue.

It's less than 'an issue', it's a failure to define the terms.   AC
measurement by making a DC conversion only works if you have
some kind of average in mind (AC is a vector, DC is a scalar; they
aren't ever 'equivalent').  

I only need the average values and I was hoping to do it all analogue.

It's not clear that the best 'm' value
in the linear equation is related to any averaging after division.

3.  someone decades ago probably tried to lower the time constant in
the situation above by using a low time constant prefilter to keep the
denominator a small amount above zero and then letting the division of
the two signals do most of the heavy lifting as far as smoothing the
quotient to DC in a short time.

Collect the data, do a least-squares fit, extract 'm' value.  There's
no particular need for any 'time constant' or prefilter, no concern
with denominators.

Bret Cahill

 

[Bret Cahill]

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 www.wolframalpha.com

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

 

[whit3rd]

On Aug 10, 12:27 pm, Bret Cahill <BretCah...@peoplepc.com> wrote:

1.  dividing 2 in sync signals is done all the time.

No, usually one samples the signals, say X and Y,
and does a least-squares fit to determine
 Y= aX + b

I only need the average values and I was hoping to do it all analogue.

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.