If the system reaches a statistical steady state the covariance matrix would be constant. The system would have to be invariant

This means that in the cascade of Ho(s) with F(s), the effect of the RHP zero on the amplitude of the product is cancelled out, but the effect on the phase is not In fact, the phase lag due to the RHP zero is doubled in the product. Note:

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

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

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

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

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

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

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

The Poisson approximation to the binomial distribution The binomial distribution, like the Poisson, is that of a random variable taking only positive integral values. Since it involves factorials, the binomial distribution is not very convenient for numerical application