Wow, that's great info Robert, thanks. Actually, you reminded me that
it's not as simple as I first described, because I will be including a
temperature sensor on the VCXO. But you've given me a great start, and
I'll get that working before I try to get it to compensate for
temperature as well.

Neil

On 10/20/05, Robert A LaBudde <ral@lcfltd.com> wrote:
> At 05:41 PM 10/20/2005, Neil wrote:
> >Can anyone point me to some info about implementing a simple Kalman filter
> >in a PIC or other microcontroller?
> >
> >I'm building a disciplined oscillator controlled by a PIC. So, it will
> >receive intermittent frequency samples from WWV or LORAN, and will generate
> >a voltage to tune a VCXO. 1 input, 1 output.
> >
> >I've seen lots of stuff about Kalman filters on the web, but it's all way
> >too complex for me. Can anyone simplify it?
>
> A Kalman filter is most useful when you have a vector set of differential
> equations that govern the plant model. In this case, this would include the
> drift and scale errors, effects of temperature fluctuations, etc., of your
> controlled oscillator. This complex electromechanical model typically
> allows an increase of accuracy of about 10x when it's included.
>
> It sounds more that what you are looking for is a simple scalar Kalman
> filter tracker, similar to a digital phase-locked-loop.
>
> First, a good starting point is the asymptotic (in time) result of a scalar
> Kalman filter, called an Exponentially-weighted Moving Average filter
> ("EWMA"). In this case, the formula is particularly simple:
>
>          Prediction (n+1) = Prediction (n) + factor * [Data(n) - Prediction
> (n)]
>
> The "factor" is typically a value > 0 and less than 0.2. It's optimal
> choice is related to the first-order autocorrelation of the data, or, in
> Kalman filter terms, the "ideal" factor is
>
>          Factor = (P + Q ) / (1 + P + Q)
>
> where P is the propagation matrix element and Q is the error factor. As
> time gets large, this asymptotically becomes
>
>          Factor ~ Q / (1 + Q)
>
> which leads to the EWMA.
>
> Because you don't know Q to start with (unless you have done a prior
> capability study), usually a guess works just as well. Typical values for
> "factor" are 0.05 to 0.2 (low number for longer averaging, shorter numbers
> for quicker response).
>
> It takes about 1/factor recursions to get to where EWMA settles down to the
> asymptotic response. A simple algorithm is to use a straight average for
> the first 1/factor terms, and then switch to the EWMA thereafter. This
> gives very close to the Kalman filter optimal response for the equivalent
> value of Q. The straight average can be implemented with only a cumulative
> sum and a count, so it is low memory as well.
>
> Another advantage of the EWMA is that the single multiply is constant and
> can be hard-wired in binary, so a fully arithmetic package is not needed.
>
> The simplest implementation:
>
>          Prediction (n+1) = Prediction (n) + [Data(n) - Prediction(n)]/K
>
> where K = 4 to 16 (done by shifting). Just realize the first K to 2*K
> predictions will be heavily biased until the tracker locks on.
>
> ================================================================
> Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral@lcfltd.com
> Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
> 824 Timberlake Drive                     Tel: 757-467-0954
> Virginia Beach, VA 23464-3239            Fax: 757-467-2947
>
> "Vere scire est per causas scire"
> ================================================================
>
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