# A State space model for discrete Sine wave Using kalma filter

I'm looking to apply an optimal LQR filter to a discrete signal of the form $$x[n]=A\sin(\omega_0n+\phi)+v[n]$$ The amplitude $A$ and the phase $\phi$ are unknown variables I want to estimate using the filter, and $v[n]$ is an uncorrelated noise signal of variance $\sigma_v^2$.

I don't know how to build a state model to generate this sine wave, and proceed from there.

• I did something similar in the past except we wanted to track the frequency. From what I recall we used 4 states states : Amplitude, derivate of x[n], frequency, frequency derivate – Ben May 27 '18 at 17:55
• Is $\omega_0$ known? – Peter K. May 27 '18 at 21:18
• Yes $\omega_0$ is known – Steve.G Ayeni May 28 '18 at 15:57
• @Ben Please could you explain how you did that better? Any materials to help on this ? I am still confused. – Steve.G Ayeni May 28 '18 at 22:33

## 1 Answer

You could use a nonlinear Kalman filter, such as the extended Kalman filter (EKF), and track the phase and frequency as your state variables.

In this case, your Kalman filter is essentially acting like a phase-locked loop (PLL).

Example reference