Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises, 4th Edition

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Discrete Kalman Filter Basics

Modern filter theory began with N. Wiener's work in the 1940s (1). His work was based on minimizing the mean-square error, so this branch of filter theory is sometimes referred to as least-squares filtering. This is an oversimplification though, because a more exact description would be “linear time-domain minimum mean-square error filtering.” This is a bit wordy though, so the shortened version will suffice for now. Regardless of what it is called, the central problem is simply a matter of separation of the signal from an additive combination of signal and noise. In hindsight, the Wiener solution turned out to be one of those subjects that was much discussed in textbooks, but little used in practice. Perhaps Wiener's main contribution was the way in which he posed the the problem in terms of minimizing the mean-square error in the time domain. This is in contrast to the frequency-separation methods that were in use at the time. However, in fairness to Wiener's work, the weighting function approach (which is central in the Wiener theory) still has some merit. More is said of this in Section 6.8.

In 1960 R.E. Kalman considered the same problem that Wiener had dealt with earlier, but in his 1960 paper he considered the noisy measurement to be a discrete sequence in time in contrast to a continuous-time signal (2) He also posed the problem in a state-space setting that accommodated the time-variable multiple-input/multiple-output scenario nicely. Engineers, ...