A Kalman-Filter gets applied to a state-space model in order to obtain estimates of the state-vector.
Example: Assume a car moving on a straight line in x-direction (this is a 1-dimensional problem). The State-space model could be described by:
$$ \begin{bmatrix} x_k \\ \dot{x}_k \\ \ddot{x}_k \end{bmatrix} = \begin{bmatrix} 1 & T & \frac12\,T^2 \\ 0 & 1 & T \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_{k-1} \\ \dot{x}_{k-1} \\ \ddot{x}_{k-1} \end{bmatrix} + w_{k-1} $$
$$ \begin{bmatrix} y_k \\ \dot{y}_k \\ \ddot{y}_k \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_k \\ \dot{x}_k \\ \ddot{x}_k \end{bmatrix} + v_k $$
The y-vector is our measurement-sequence. And here comes my question: For the Kalman-Filter to work, is it necessary to measure each of the entries of the y-vector?
Coming back to the car: I can measure the number of rotations of the driving-shaft using a wheel-encoder. From this I can infer the distance travelled, which is my x-coordinate assuming a movement on a straight line in x-direction.
I cannot measure the velocity directly. But I can infer the velocity from the number of edges detected within a certain time-gap, using the wheel encoder. And I can derivate velocity in order to obtain acceleration.
So, what I am saying is: In order to obtain each entry of the y-vector, I am using but one sensor: The wheel encoder. Is this valid when applying the Kalman-Filter equations?