State Space conversion of Sinusoidal model

i would like to learn how to convert sinusoidal model into state space form which has following equation

our model consist of sum of periodic components with additive of white noise, given by following form

$y(t) = A_1\sin(\omega_1 t) + A_2\sin(\omega_2 t+) + ... + A_p\sin(\omega_p t) + z(t)$ where $z(t)$ is additive white noise and

$\omega_i=2*\pi*F_i$

i was reading special document Tracking a Sine Wave

i somehow understand step when we have single sinusoidal components, but what about when we have several ones? please help me

• It seems you are confusing signals and systems. State space model applies to "systems", i..e things that have an input and output. Your sine+noise model is just a signal – Hilmar Jun 29 '18 at 19:28
• so can't apply Kalman filter to this model? – dato datuashvili Jun 29 '18 at 20:27
• hi: based on what you explained, you only have a measurement equation. There is no system equation. so, if you let $F$ be the $p \times 1$ vector of where each element is $A_i$ and you let $\theta$ be the $p \times 1$ vector where each element is $sin(w_i \times t)$, then $y_t = F^\prime \theta_t + \epsilon_t$ and that's it. Nothing is being estimated so there are no updates, atleast based on the way you explained it. – mark leeds Jun 30 '18 at 1:46
• P.S: I just read Hilmar's answer again and I'm probably saying the same thing as him so my bad for duplication. – mark leeds Jun 30 '18 at 1:51