Estimate and Track the Amplitude, Frequency and Phase of a Sine Signal Using a Kalman Filter

Question

There is sinusoidally controlled signal, which other than being noisy, can change values for amplitude, frequency, phase and offset. At every new sample a new sine is fitted for the last N samples. These fitted signals might be different due to noise or due to the signal changing value. To filter it, I would like to use a Kalman filter to estimate the actual sine and to smoothen the transition of the above mentioned parameters.

I tried to get familiar with Kalman filters, but most of the examples deal with only estimating one parameter, and in my case the parameters are not independent.

Could somebody provide some hints on how to get started, or knows how to do it?

Answer 1

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:

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).