EECS 658_________________________PROBLEM SET #8_________________________Fall 1999
ASSIGNED: Nov. 18, 1999. READ: Finish the book chapter on the Levinson and Schur algorithms.
DUE DATE: Dec. 02, 1999. THIS WEEK: Close-to-Toeplitz matrices, split Levinson and Schur.
    1. Schur algorithm and reconstruction of layered media from impulse responses:
      A layered medium is probed with an discrete-time impulse of unit height.
      The goal is to reconstruct the reflection coefficients of the layered medium.
    1. The reflection response to the impulse is r(n)=0.9n for n=0,1,2...
      Compute the reflection coefficients of the medium using the Schur algorithm.
      Confirm your result by solving a nested set of Toeplitz systems of equations.
    2. Now the reflection response to the impulse is r(n)=0.9n2 for n=0,1,2...
      Compute the reflection coefficients of the medium using the Schur algorithm.
      Confirm your result by solving a nested set of Toeplitz systems of equations.
    3. Use the Schur algorithm to compute the spectral factorization of this sequence:
      r(n)={7,33,44,66,100,66,44,33,7}. We want minimum phase a(n) such that a(n)*a(-n)=r(n).
      HINT: All coefficients of the spectral factors are integers.
    4. Confirm the Schur algorithm in (c) is factoring a nested set of Toeplitz matrices
      whose columns are approaching the spectral factor as the matrix size increases.
    2. The split Levinson and split Schur algorithms:
    1. Use the split Levinson algorithm to solve Problem #2 of Problem Set #7.
    2. Use the split Schur algorithm to compute the "potential" required in (a).
    3. Displacement rank and close-to-Toeplitz matrices:
      Recall displacement rank d(R)=RANK[R-ZRZ']. Define dd(R)=RANK[R-Z'RZ].
    1. Prove that d(R-1)=dd(R) for any nonsingular matrix R.
      Use this to prove that the inverse of a Toeplitz matrix has d=2.
      HINT: RANK[I-AB]=rank[I-BA] for any square matrices A and B.
    2. Prove that d(A+B)<=d(A)+d(B). HINT: RANK[A+B]<=RANK[A]+RANK[B].
    3. Prove that d(AB)<=d(A)+d(B). HINT: RANK[AB]<=MIN{RANK[A],RANK[B]}.
      HINT: Simplify A(B-ZBZ')+(A-ZAZ')(ZBZ') and note Z'Z=I. Assume A,B full rank.
    4. In ARMA linear prediction, the system matrix has the form R=[Toeplitz]+[A 0; 0 0].
      Show that the matrix R has Toeplitz distance at most the size of A.
    5. Using the above, show the product of Toeplitz matrices is close-to-Toeplitz, and the
      set of close-to-Toeplitz matrices is closed under addition, multiplication, inversion.