Understanding the Kalman filter with a simple radar example

1. understand weighted least squares and how you can update an initial estimate (prior mean and variance) with a new measurement and its uncertainty (i.e. inverse variance weighted least squares)

2. this works because the true mean hasn't changed between measurements. What if it did?

3. KF uses a model of how the mean changes to predict what it should be now based on the past, including an inflation factor on the uncertainty since predictions aren't perfect

4. after the prediction, it becomes the same problem as (1) except you use the predicted values as the initial estimate

There are some details about the measurement matrix (when your measurement is a linear combination of the true value -- the state) and the Kalman gain, but these all come from the least squares formulation.

Least squares is the key and you can prove it's optimal under certain assumptions (e.g. Bayesian MMSE).

Basically, a Kalman filter is part of a larger class of "estimators", which take the input data, and run additional processing on top of it to figure out the true measurement.

The very basic estimator a low pass filter is also an "estimator" - it rejects high frequency noise, and gives you essentially a moving average. But is a static filter that assumes that your process has noise of a certain frequency, and anything below that is actual changes in the measured variable.

You can make the estimator better. Say you have some idea of how the process variable should behave.For a very simple case, say you are measuring temperature, and you have a current measurement, and you know that change in temperature is related to current being put through a winding. You can capture that relationship in a model of the process, which runs along side the measurement of the actual temperature. Now you have the noisy temperature reading, the predicted reading (which acts like a mean), and you can compute the covariance of the noise, which then you can use to tune the parameter of low pass filter. So if your noise changes in frequency for some reason, the filter will adjust and take care of it.

The Kalman filter is an enhanced version of above, with the added feature of capturing correlation between process variables and using the measurement to update variables that are not directly measurement. For example, if position and velocity are correlated, a refined measurement on the position from gps, will also correct a refined measurement on velocity even if you are not measuring velocity (since you are computing velocity based of an internal model)

The reason it can be kind of confusing is because it basically operates in the matrix linear space, by design to work with other tools that let you do further analysis. So with restriction to linear algebra, you have to assume gaussian noise profile, and estimate process dependence as a covariance measure.

But Kalman filter isnt the end/all be all for noise rejection. You can do any sort of estimation in non linear ways. For example, I designed an automated braking system for an aircraft that tracks a certain brake force command, by commanding a servo to basically press on a brake pedal. Instead of a Kalman filter, I basically ran tests on the system and got a 4d map of (position, pressure, servo_velocity)-> new_pressure, which then I inverted to get the required velocity for target new pressure. So the process estimation was basically commanding the servo to move at a certain speed, getting the pressure, then using position, existing pressure, and pressure error to compute a new velocity, and so on.