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.

*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.

- The reflection response to the impulse is r(n)=0.9
^{n} 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.

- Now the reflection response to the impulse is r(n)=0.9
^{n2} 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.

- 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.

- 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.

*The split Levinson and split Schur algorithms:*
- Use the
*split* Levinson algorithm to solve Problem #2 of Problem Set #7.
- Use the
*split* Schur algorithm to compute the "potential" required in (a).

*Displacement rank and close-to-Toeplitz matrices:*

Recall displacement rank d(R)=RANK[R-ZRZ']. Define dd(R)=RANK[R-Z'RZ].

- 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.

- Prove that d(A+B)<=d(A)+d(B).
**HINT:** RANK[A+B]<=RANK[A]+RANK[B].

- 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.

- 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.

- 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.