Dynamical Signal Models

Recall the simple DC signal estimation problem.

Where AA is the unknown DC level and w n w n is the white Gaussian noise. AA could represent the voltage of a DC power supply. We know how to find several good estimators of AA given the measurements x 0 … x N - 1 x 0 … x N - 1 .

In practical situations this model may be too simplistic. the load on the power supply may charge over time and there will be other variations due to temperature and component aging.

To account for these variations we can employ a more accurate measurement model:

where the voltage A n A n is the true voltage at time nn.

Now the estimation problem is significantly more complicated since we must estimate A 0 … A N - 1 A 0 … A N - 1 . Suppose that the true voltage A n A n does not vary too rapidly over time. Then successive samples of A n A n will not be too different, suggesting that the voltage signal displays a high degree of correlation.

This reasoning suggests that it may be reasonable to regard the sequence A 0 … A N - 1 A 0 … A N - 1 , as a realization of a correlated (not white) random process. Adopting a random process model for A n A n allows us to pursue a Bayesian approach to the estimation problem (Figure 1).

Using the model in Equation 2, it is easy to verify that the maximum likelihood and MVUB esitmators are given by Our estimate is simply the noisy measurements! No averaging takes place, so there is no noise reduction.

Let's look at the example again, Figure 2.

The voltage A n A n is varying about an average value of 10V. Assume this average value is known and write Where y n y n is a zero-mean random process. Now a simple model for y n y n which allows us to specify the correlation between samples is the first-order Gauss-Markov prcoess model: Where u n ∼0 σ u 2 u n 0 σ u 2 iid (white Gaussian noise process). To initialize the process we take y 0 y 0 to be the realization of a Gaussian random variable: y 0 ∼0 σ y 2 y 0 0 σ y 2 . u n u n is called the driving or excitation noise. The model in Equation 5 is called the dynamical or state model. The current output y n y n depends only on the state of the system at the previous time, or y n - 1 y n - 1 , and the current input u n u n (Figure 3). y 1 =a y 0 + u 0 y 1 a y 0 u 0 ⋮ ⋮ y n =an+1 y 0 +∑k=1nak u n - k y n a n 1 y 0 k 1 n a k u n - k Correlation: If m>n m n , then If |a|>1 a 1 , then it's obvious that the process diverges ( variance→∞ variance ). This is equivalent to having a pole outside the unit circle shown in Figure 4. So, let's assume |a|<1 a 1 and hence a stable system. Thus as mm and nn get large am+n+2 σ y 2→0 a m n 2 σ y 2 0 Now let m−n=τ m n τ . Then for mm and nn large we have This shows us how correlated the process is: |a|→1⇒heavily correlated (or anticorrelated) a 1 heavily correlated (or anticorrelated) |a|→0⇒weakly correlated a 0 weakly correlated

How can we use this model to help us in our estimation problem?

The Kalman Filter

Let's look at a more general formulation of the problem at hand. Suppose that we have a vector-valued dynamical equation

Where yn y n is p×1 p 1 dimensional, A A is p×p p p , and b b is p×1 p 1 . The initial state vector is Y0∼0R0 Y 0 0 R 0 , where R0 R 0 is the covariance matrix and u n ∼0 σ u 2 u n 0 σ u 2 iid (white Gaussian excitation noise). This reduces to the case we just looked at when p=1 p 1 . This model could represent a p th p th order Gauss-Markov process: Define Then, Here A A is the state transition matrix. Since yn y n is a linear combination of Gaussian vectors: We know that yn y n is also Gaussian distributed with mean and covariance Rn=EynynT R n y n y n , Yn∼Rn Y n R n . The covariance can be recursively computed from the basic state equation: Assume that measurements of the state are available: Where w n ∼0 σ w 2 w n 0 σ w 2 iid independant of u n u n (white Gaussian observation noise).

For example, if C=0…01T C 0 … 0 1 , then

Where x n x n is the observation, y n y n is the signal, and w n w n is the noise. Since our model for the signal is Gaussian as well as the observation noise, it follows that x n ∼0 σ n 2 x n 0 σ n 2 , where σ n 2=CTRnC+ σ w 2 σ n 2 C R n C σ w 2 (Figure 5).

Kalman first posed the problem of estimating the state of yn y n from the sequence of measurements xn= x 0 ⋮ x n x n x 0 ⋮ x n To derive the Kalman filter we will call upon the Gauss-Markov Theorem.

First note that the conditional distribution of yn y n given xn x n is Gaussian: yn | xn ∼ŷ n | n P n | n y n | x n y n | n P n | n Where ŷ n | n y n | n is the conditional mean and P n | n P n | n is the covariance.

We know that this is the form of the conditional distribution because yn y n and xn x n are jointly Gaussian distributed.

This is all well and good, but we need to know what the conditional mean and covariance are explicitly. So the problem is now to find/compute ŷ n | n y n | n and P n | n P n | n . We can take advantage of the recursive state equation to obtain a recursive procedure for this calculation. To begin, consider the "predictor" ŷ n | n - 1 y n | n - 1 : yn | x n - 1 ∼ŷ n | n - 1 P n | n - 1 y n | x n - 1 y n | n - 1 P n | n - 1 Where yn y n is the signal samples, y n … y n - p + 1 y n … y n - p + 1 , x n - 1 x n - 1 is the observations x n - 1 … x n - p x n - 1 … x n - p , and ŷ n | n - 1 y n | n - 1 is the best min MSE estimator of yn y n given x n - 1 x n - 1 . Although we don't know what forms ŷ n | n - 1 y n | n - 1 and P n | n - 1 P n | n - 1 have, we do know two important facts: