One-dimensional Kalman Filter

It is useful to consider, as an example, the simplified case of a one-dimensional Kalman filter whose state coincides with the observable. The transition and observation equations are

$$\begin{array}{l} x_{i} = x_{i-1} + u_{i} + w_{i} \\ z_{i} = x_{i} + v_{i} \end{array}$$ (3.24)

where $w_i$ is the process noise, whose variance $q_i$ represents the estimated probability of a change in the signal itself (low if the signal changes little over time, high if it changes substantially), whereas $v_i$ is the observation noise, with variance $r_i$, associated with observing the state.

The prediction cycle is very simple and becomes:

$$\begin{array}{l} x^{-}_{i} = x_{i-1} + u_{i}\\ p^{-}_{i} = p_{i-1} + q_i \end{array}$$ (3.25)

The Kalman gain $k$ becomes

$$k_i = \frac{p^{-}_{i}}{p^{-}_{i} + r_i}$$ (3.26)

and finally the observation phase becomes

$$\begin{array}{l} x_{i} = x^{-}_{i} + k_i (z_i - x^{-}_i) =... ...(1 - k_i) x^{-}_i \\ p_{i} = (1 - k_i) p^{-}_{i} \end{array}$$ (3.27)

It is generally possible to estimate $r$ a priori, whereas $q$ must be set experimentally.

As shown in the first of equations (3.27), the factor $k$ is effectively a blending factor between the state observation and the previous state estimate.

In the one-dimensional case, it is easy to see that the gain $k$ and the variance $p$ are processes independent of the state, the observations, and even the error. If $r$ and $q$ do not vary over time, $k$ and $p$ are numerical sequences that converge to a constant determined solely by the noise characterization, independently of their initial values. Compare this result with that obtained from equation (2.68).