Kalman filter, defining the measurement model

I would like to implement a Kalman filter to estimate the velocity and position of an object. I have an accelerometer, therefore the acceleration is known. The approach is same as:

Kalman filter for position and velocity: introducing speed estimates

however, I don't understand well, how I should define the measurement model:

$\hat{X}(k+1) = \hat{X}(k) + K * ( y(k) - H*\hat{X}(k) );$

K is the kalman gain and is calculated in every step, but What is my "y" here, i know it is called the observations, but how i have my observations ? should i calculate the:

$v(k+1) = a(k)*t + v(k);$

$x(k+1) = 0.5*a(k)*t^2 + v(k)*t +x(k);$

Thank you!

Answer

Your observation are the $y_k$ your observation model is defined by $Y_k=HX_k+\eta_k$, where $\eta_k$ if a sample from a normal distribution.

If you want your kalman filter to estimate position, velocity and acceleration your state vector is : $X_k=[x_k,\dot{x}_k,\ddot{x}_k]^T$ and therefore your observation matrix is : $H=[0,0,1]$.

As stated by Rhei, your observations are what you get from the system at each time step, i.e. the measured acceleration.