# Alpha Beta Filter¶

- Also known as the g-h filter

- g is the scaling used for measurement (also known as \alpha)
- h is the scaling for the change in measurement over time (also known as \beta)

- Basis for a huge number of filters including the Kalman filter
- Each type of filter has a different way of assigning values to g and h
- Kalman filter varies g and h dynamically at each time step

## Formal Terminology¶

*System/plant*: the object we want to estimate*State*: actual**hidden**value of system*Measurement*: measured,**observable**value of system*(State) estimate*: filter's estimate of state*Process/system model*: mathematical model of the system*Process/system error*: error in mathematical model of the system*System propogation/evolution*: uses process model to form new state estimate; imperfect because of process error

## Algorithm¶

**Initialization**

```
1. Initialize the state of the filter
2. Initialize our belief in the state
```

**Predict**

```
1. Use system behavior to predict state at next time step
2. adjust belief to account for uncertainty in prediction
```

**Update**

```
1. Get measurement and associated belief about its accuracy
2. Compute residual between estimated state and measurement
3. New estimate is somewhere on residual line
```

## Notation¶

- z: measurement (some literature uses y)
- k: time step
- z_{k} is the measurement for time step k

- In general, we have n sensors and n measurements
**x**denotes the state; represents both initial weight and initial weight gain rate- \dot{x} is the derivative of x with respect to time $$ \mathbf{x} = \begin{bmatrix} x \ \dot{x} \end{bmatrix} $$
- e.g weight of 62 kg with a gain of 0.3 kg/day $$ \mathbf{x} = \begin{bmatrix} 62\ 0.3 \end{bmatrix} $$
- The state is initialized with x_{0}, the initial estimate.
- We then enter a loop, predicting the state for time or step k from the values from time or step k-1.
- We get the measurement z_{k} and choose intermediate point between measurements and prediction, creating the estimate x_{k}