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

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

Mean squared value: Higher order distribution and density functions. You can define these distributions of any order

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

Direct determination of the joint probability density of several functions o several ra andom variables Suppose we have the joint probability density function of several random variables x, Y, Z, and we wish the joint density of several other random variables defined as functions xyz