EECS 658_________________________PROBLEM SET #7_________________________Fall 1999

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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.

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1. We wish to solve the linear system of equations [x₁,x₂,x₃,x₄]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).
   1. Solve this linear system of equations using the 4-point DFT.
   2. Use the 2-point DFT to solve the general 2×2 circulant system of equations.

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2. A 0-mean WSS process has covariance R(k)=0.7071^|k|[6cos((pi)k/4)+2sin((pi)|k|/4)].
   1. 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.
   2. Solve this using the Levinson algorithm. At each recursion:
      1. Specify the linear system of equations that has been solved at that recursion;
      2. Specify the matrix factorization that has been computed at that recursion.

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3. Now apply the Schur algorithm to the above problem:
   1. Use the Schur algorithm to compute the reflection coefficients.
      Also specify the matrix factorization that has been computed at each recursion.
   2. 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.
   3. Using only one side of the covariance function leads to U₂(z)=0 in the Schur algorithm.
      Show that D₂(z) is the modelling or shaping filter for the random process.

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4. 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.
   1. Formulate this deconvolution problem as a Toeplitz linear system of equations.
   2. Use the Haar discrete wavelet basis to transform this into a sparse linear system.

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   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.

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   • 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).

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