Question: Moving Average. Suppose two initial numbers are given and successive numbers are computed as the average of last two. For example, with 1 and 3, we generate the sequence
1, 3, 2, 2½, 2¼,......
Note that the sequence seems to be converging, but not to the average of the original two numbers (which is 2).
(a) Formulate this process in state-space form.
(b) The system matrix A you obtained should be positive but not strictly positive. What is the Frobenius-Perron eigenvalue λ0?
(c) Is the system asymptotically stable?
(d) What are the right and left eigenvectors corresponding to λ0?
(e) What is the general formula for the limiting value of a sequence of this kind, expressed in terms of Its first two terms? For the example given, what is the limiting value?
(f) Now generalize to an nth-order average. That is, n initial numbers are given and successive numbers are the averages of the last n numbers. What is the formula for the limit?