﻿ One-dimensional Kalman Gain Derivation

# One-dimensional Kalman Gain Derivation

There are several ways to derive the one-dimensional Kalman Gain equation. I present the simplest one.

Given the measurement $$z_{n}$$ and the prior estimate $$\hat{x}_{n,n-1}$$ we are interested in finding an optimum combined estimate $$\hat{x}_{n,n}$$ based on the measurement and the prior estimate.

The optimum combined estimate is a weighted mean of the prior estimate and the measurement:

$\hat{x}_{n,n} = w_{1}z_{n} + w_{2}\hat{x}_{n,n-1}$

Where $$w_{1}$$ and $$w_{2}$$ are the weights of the measurement and the prior estimate.

$w_{1} + w_{2} = 1$

$\hat{x}_{n,n} = w_{1}z_{n} + (1 - w_{1})\hat{x}_{n,n-1}$

The relation between variances is given by:

$p_{n,n} = w_{1}^{2}r_{n} + (1 - w_{1})^{2}p_{n,n-1}$

Where:

$$p_{n,n}$$ is the variance of the optimum combined estimate $$\hat{x}_{n,n}$$

$$p_{n,n-1}$$ is the variance of the prior estimate $$\hat{x}_{n,n}$$

$$r_{n}$$ is the variance of the measurement $$z_{n}$$

Note: for any normally distributed random variable $$x$$ with variance $$\sigma^{2}$$, $$wx$$ is distributed normally with variance $$w^{2}\sigma^{2}$$

Since we are looking for an optimum estimate, we want to minimize $$p_{n,n}$$.

To find $$w_{1}$$ that minimizes $$p_{n,n}$$, we differentiate $$p_{n,n}$$ with respect to $$w_{1}$$ and set the result to zero.

$\frac{dp_{n,n}}{dw_{1}} = 2w_{1}r_{n} - 2(1 - w_{1})p_{n,n-1}$

Hence

$w_{1}r_{n} = p_{n,n-1} - w_{1}p_{n,n-1}$

$w_{1}p_{n,n-1} + w_{1}r_{n} = p_{n,n-1}$

$w_{1} = \frac{p_{n,n-1}}{p_{n,n-1} + r_{n}}$

We have derived the Kalman Gain!

Since the Kalman Gain yields the minimum variance estimate, the Kalman Filter is also called an optimal filter.