Particle Filter and Extended Kalman filter

Hello, I am going to talk about math, suppose that you want to have de signal x(t) (for example volatility signals for asset allocation, Electrocardiogram Signal) but you perform observation y(t) but this observation has a measuring error, i.e, y(t)=x(t)+"error".

The filtering problem is: what is the best estimate of x(t) based on the observations y(t). Intuitively, the problem is to "filter" the noise away from the observation in an optimal way.

I present two picture of two filters that filter the noise away from the observation using Particle Filter and Extended Kalman filter

The red line represents the observation y(t) and the black line x(t) (Real state that it is obtained with the filter)

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