The quantity K2 is known as the Kalman gain. It is the optimum gain in the mean squared error sense. Substitute it into the expression for

The estimate of x based on N data points can then be made without reprocessing the first N, points. Their effect can be included simply by starting with pseudo observation which is equal to the estimate based on the first N points having a variance equal to the variance of the estimate based on N The same is true of the variance of the estimate based on N

The two middle terms are zero: fort>lo,n(t) and are uncorrelated becauset) is white(impulse correlation function) For=, n() has a finite effect on x()because n() is white. But the integral of a finite quantity over one point is zero

Solution in the Free Configuration Non-Real-Time Filter Case The applicable condition in this important case is:

System parameters are contained in,(t) and(t) Desired output is generated by taking the signal through the desired operator. The difference between the actual output and the desired output is the error, whose mean squared value we want to minimize We require a stable and realizable system. The error e(t) is given by:

so if t is the current time in a real time situation, we cannot compute ()for which is necessary since w, () is nonzero only for>0. But we showed earlier that

Last time: Ergodic processes An ergodic process is necessarily stationary. Example: Binary process

where is the linearized system matrix. But this requires the full(same number of equations as finite differencing). In =time when the nominal trajectory

Non-zero power at non-zero frequency If R(r) includes a sinusoidal component corresponding to the component x()=Asin(o41+6) where 0 is uniformly distributed over 2t, A is random independent of 0, that component will be

If a set of random variables X, having the multidimensional normal distribution is uncorrelated(the covariance matrix is diagonal, they are independent. The argument of the exponential becomes the sum over i of Thus, the distribution becomes a product of exponential