Filtering

If we think of our signal as a discrete time random process, then like a normal deterministic signal, we can try filtering our random process.

Filtering can either be accomplished with an LTI system or some other non-linear/non-time-invariant system just like with deterministic signals.

LTI Filtering on WSS Processes

If we use an LTI filter on a WSS process, then we can easily compute how the filter impacts the spectrum of the signal.

Theorem 13

Wiener Filter

Non-Causal Case

Starting with noncausal case, we can apply the orthogonality principle,

When we cascade these filters,

Definition 21

Causal Case

Taking the unilateral Z-transform of both sides,

Definition 22

Theorem 14

Vector Case

Definition 23

In matrix form, this means

Theorem 15

Applying the LDL decomposition, we see that

Definition 24

Hidden Markov Model State Estimation

Causal Distribution Estimation

Non-Causal Distribution Estimation

State Sequence Estimation

Suppose we want to find the most likely sequence of states given our observations. This means we should compute

Putting these equations gives us the Viterbi algorithm.

Kalman Filtering

Kalman Prediction Filter

Putting this into a concrete algorithm, we get the Kalman Prediction Filter.

Schmidt’s Modification of the Kalman Filter

This mimics the approach of the forward algorithm for Hidden Markov Models, which separated updates to the distribution using a time update and a measurement update. Using our innovation process,

For the time update,

Writing this as an algorithm,

Kalman Smoother

If we want to look at the error of this estimator, we see that

Writing all of this as an algorithm,

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