EECS 658_________________________PROBLEM SET #7_________________________Fall 1999

**ASSIGNED:** Nov. 11, 1999. **READ:** HANDOUTS: Levinson and Schur algorithms, Book chapter.

**DUE DATE:** Nov. 18, 1999. **THIS WEEK:** Levinson and Schur, wavelet matrix sparsification.

We wish to solve the linear system of equations
[x_{1},x_{2},x_{3},x_{4}]C=[0,4,-4,0] (see below)

where C is a *circulant* matrix whose first row is [1,4,3,2] (see below for C *transposed*).
- Solve this linear system of equations using the 4-point DFT.
- Use the 2-point DFT to solve the general 2×2 circulant system of equations.

A 0-mean WSS process has covariance R(k)=0.7071^{|k|}[6cos((pi)k/4)+2sin((pi)|k|/4)].
- Set up the extended Yule-Walker equations for computing the 3
^{rd}-order

autoregressive linear prediction filter for estimating x(n) from x(n-1),x(n-2),x(n-3).

Note that the Yule-Walker equations are a 4×4 Toeplitz linear system of equations.
Solve this using the *Levinson algorithm*. At each recursion:
- Specify the linear system of equations that has been solved at that recursion;
- Specify the matrix factorization that has been computed at that recursion.

Now apply the *Schur algorithm* to the above problem:
- Use the Schur algorithm to compute the reflection coefficients.

Also specify the matrix factorization that has been computed at each recursion.
- Write down and fill in the matrix equation (2.8) for this problem.

Equation (2.8) is from the book chapter I handed out in class.
- Using only one side of the covariance function leads to U
_{2}(z)=0 in the Schur algorithm.

Show that D_{2}(z) is the *modelling* or *shaping* filter for the random process.

We observe y(n)=h(n)*u(n),n=0,1,2,3,4,5,6,7 where impulse response h(n)=0.99^{|n|}.

We wish to *deconvolve* u(n) from observations y(n) for n=0,1,2,3,4,5,6,7.
- Formulate this deconvolution problem as a Toeplitz linear system of equations.
- Use the Haar discrete wavelet basis to transform this into a
*sparse* linear system.

A *sparse* linear system has a system matrix in which most (*not all*) elements are small.
- Small elements can be neglected, compared to large elements, in solving the system.

- HINT: y=Hu implies Wy=(WHW')(Wu) where W is the matrix given on Nov. 2.
- BUT: That matrix W is
*not* orthogonal. You must scale the rows of the matrix.
- GIVE: The full 8×8
*sparsified* matrix (for this h(n), this is easy to compute).