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Suppose a system follows this equation: $$ x(t)=A \cos(\omega t + \phi)+\eta$$

where: $\omega = 2\pi f $ and $\eta$ is a random error

using Extended Kalman Filter, how does estimated value $\hat{x}$ be?

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2 Answers 2

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I'm copying my answer to Estimate and Track the Amplitude, Frequency and Phase of a Sine Signal Using a Kalman Filter which solves a more general problem with example code:

We can build a non linear dynamic model in order to estimate the parameters of a sine signal.

Let's model the signal as $ a \sin \left( \phi \right) $ where $ \phi $ is the instantaneous phase. So the model could be also written as $ a \sin \left( \omega t + \psi \right) $.

Then the model can be:

$$ {a}_{k} \sin \left( {\omega}_{k} {t}_{k} + \psi \right) = {a}_{k} \sin \left( {\phi}_{k} \right) $$

With some math and pre processing of Kalman Filter you may derive the model with the matrices:

$$ \boldsymbol{x}_{k} = \begin{bmatrix} {a}_{k} \\ {\omega}_{k} \\ {\phi}_{k} \end{bmatrix}, F = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & \Delta t & 1 \end{bmatrix}, Q = \begin{bmatrix} \Delta t {\sigma}_{a}^{2} & 0 & 0 \\ 0 & \Delta t {\sigma}_{\omega}^{2} & \frac{ {\Delta t}^{2} {\sigma}_{\omega}^{2}}{2} \\ 0 & \frac{ {\Delta t}^{2} {\sigma}_{\omega}^{2}}{2} & \frac{ {\Delta t}^{3} {\sigma}_{\omega}^{2}}{3} \end{bmatrix} $$

Where $ {\sigma}_{a}^{2} $ is the process variance of the amplitude and $ {\sigma}_{\omega}^{2} $ is the variance of the process noise of instant angular frequency.

The measurement model is a bit more tricky. The measurement model is:

$$ {z}_{k} = h \left( \boldsymbol{x}_{k} \right) = {a}_{k} \sin \left( {\phi}_{k} \right) $$

Hence the Jacobian is given by $ \frac{\partial h \left( \boldsymbol{x}_{k} \right )}{\partial \boldsymbol{x}_{k}} = \left[ \sin \left( {\phi}_{k} \right), 0, {a}_{k} \cos \left( {\phi}_{k} \right) \right] $.

Wrapping all this into a Kalman Model will yield:

enter image description here

You may see that the model can effectively track changes in the parameters.
There are other alternatives to this dynamic model but I think this is a simple and effective one.

You may also use the Unscented Kalman Filter. I implemented it at Extended Kalman Filter (EKF) for Non Linear (Coordinate Conversion - Polar to Cartesian) Measurements and Linear Predictions.

The code is available at my StackExchange Signal Processing Q76443 GitHub Repository (Look at the SignalProcessing\Q76443 folder).

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  • $\begingroup$ Nice answer! :-) $\endgroup$
    – Peter K.
    Dec 6, 2021 at 18:46
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    $\begingroup$ @PeterK., Thanks. There are many ways to derive the model matrix for harmonic signals. $\endgroup$
    – Royi
    Dec 6, 2021 at 18:50
  • $\begingroup$ Thanks for the intuitive answer, I have a question about dT, and paramAngFreq, how do you select the appropriate dT.? do we follow Nyquist theorm, or the sampling frequency (1/dT) is at least the same with paramAngFreq.? $\endgroup$ Dec 8, 2021 at 9:53
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    $\begingroup$ The parameters dT is the rate new measurement is measured. Of course it has to be at least by Nyquist. You may use dT = 1 and then everything is in normalized frequency. $\endgroup$
    – Royi
    Dec 8, 2021 at 11:23
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    $\begingroup$ I am not sure what you mean. Assume you have which measures a sine signal. By Sampling Theorem you must sample it at sampling rate which is larger than 2 times the bandwidth. But in many cases we sample at much higher rate. In any case, dT must be the sampling rate in practice and not the Nyquist rate. $\endgroup$
    – Royi
    Dec 8, 2021 at 16:47
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This isn't quite what you're asking, because it neglects the amplitude, $A$, but it's a relatively straightforward example of application of an extended Kalman filter to the frequency tracking problem. See section 1.2 of this PDF, that I wrote some time ago.

I'd also recommend starting with B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Inc., Engle- wood Cliffs, New Jersey 07632, 1979.

EKF for frequency tracking

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  • $\begingroup$ I like the reference Peter, your PDF is a nice summary $\endgroup$ Dec 6, 2021 at 13:31
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    $\begingroup$ @DanBoschen For an unpublished article (in the journal or conference sense), that PDF has received more citations than the IEEE TSP paper for my PhD. Oh well. ;-) $\endgroup$
    – Peter K.
    Dec 6, 2021 at 15:53
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    $\begingroup$ I'm giving a callout to Dan Simon's "Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches", Wiley, 2006. I think it'll be clear to someone who's taken a senior-level statistics class and state-space control. The only downside is that after it was published someone came up with a formal way to determine the constellation for an Unscented Kalman filter, and I can't remember who wrote the paper or when (aside from "after 2006"). $\endgroup$
    – TimWescott
    Dec 6, 2021 at 20:24
  • $\begingroup$ @TimWescott thanks,Tim! I’ll see if I can get that. $\endgroup$
    – Peter K.
    Dec 6, 2021 at 20:47
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    $\begingroup$ @TimWescott, Could it be you're talking about Cubature Kalman Filter? If so, then it is a generalization of the UKF and actually it shows that in most cases the UKF is the optimal constellation. $\endgroup$
    – Royi
    Dec 8, 2021 at 11:24

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