Process and measurement noise are noise sources that affect the plant and the sensors respectively. They can be described by adding noise terms into the state-space equations
$$
\begin{matrix}
x_k=Ax_{k-1}+Bu_k+w_k \\
y_k=Cx_k+v_k
\end{matrix}$$
These noise values are assumed to be random but can be represented using some of the common stochastic probability distributions. For an even simpler model measurement noise can be approximated by a sine wave.
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