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

Special case: [(P)] is rational: In this solution formula we can carry out the indicated integrations in literal form in the case in which [(P)] is rational. In our work, we deal in practical way only with rational,sn, and, so this function will be rational if D(p)is rational This will be true of every desired operation except predictor. Thus except in the case of prediction, the above

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

For the differential system characterized by its equations of state,specialization to invariance means that the system matrices A, B,C are constants

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:

Note n, s enter the system at the same place F is fixed. We design(and perhaps B). We must stabilize F if it is given as unstable

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

Note that in the case of repeated roots of the denominator, a pole of multiple order contributes only a single residue. To evaluate F(s)ds by integrating around a closed contour enclosing the entire left half plane, note that if F()0 faster than-for large s, the integral along the curved part of the contour is zero

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

Prof vander Velde Use error between early and late indicator to lock onto signal. Error is a linear function of shift, within the range (-T,T) Return to the 1 st example process and take the case where the change points are Poisson distributed