2. QR-RLS Algorithm Cy Shimabukuro EE 491D 05-13-05

3. Overview What is QR-RLS What is QR-RLS Different methods of Computation Different methods of Computation Simulation Simulation Results Results

4. QR-RLS? QR-RLS algorithm is used to solve linear least square problems. The decomposition is the basis for the QR algorithm. Algorithm is a procedure to produce eigenvalues of a matrix.

5. Advantage Using this QR method is not for speed, but the numerical stablility How? proceeds by orthogonal similarity transforms. works directly with data from decomp. eliminating the correlation matrix.

6. Computing QR Decomp. Gram-Schmidt Process Gram-Schmidt Process Householder Transformation Householder Transformation a.k.a Householder reflection a.k.a Householder reflection Givens Rotation Givens Rotation

7. Gram-Schmidt A method of orthogonalizing a set of vectors A method of orthogonalizing a set of vectors This method is numerically Unstable This method is numerically Unstable The vectors aren’t orthogonal due to rounding errors. The vectors aren’t orthogonal due to rounding errors. Loss of orthogonality is bad Loss of orthogonality is bad

8. Householder Used to calculate QR decompositions Used to calculate QR decompositions Reflection of a vector plane in 3-D space. Reflection of a vector plane in 3-D space. Hyperplane is a unit vector orthogonal to hyperplane Hyperplane is a unit vector orthogonal to hyperplane

9. Householder Used to zero out subdiagonal elements Used to zero out subdiagonal elements A is decomposed: where Q T =H n …H 2 H 1 is the orthogonal product of Householders and R is upper triangular. where Q T =H n …H 2 H 1 is the orthogonal product of Householders and R is upper triangular. Over determined system Ax=b is transformed into the easy-to-solve Over determined system Ax=b is transformed into the easy-to-solve

10. Householder Properties it follows: Properties it follows: Symmetrical : Q = Q^T Symmetrical : Q = Q^T it is orthogonal: Q^{-1}=Q^T it is orthogonal: Q^{-1}=Q^Torthogonal therefore it is also involutary: Q^2=I therefore it is also involutary: Q^2=I By using the Householder transformation method, it has more stability than the Gram-Schmidt By using the Householder transformation method, it has more stability than the Gram-Schmidt

11. Givens Rotation Another transformation to find Q matrix Another transformation to find Q matrix Method zeros out element in the matrix Method zeros out element in the matrix Most useful because: Most useful because: Don’t have to build a new matrix but just manipulating original Don’t have to build a new matrix but just manipulating original Less work and zeros out what is needed Less work and zeros out what is needed Much more easily parallelized Much more easily parallelized

12. The Matrix ‘c’ represents cos(θ), ‘s’ represents sin(θ)

13. Properties The cosine parameter c is always real, but the sine parameter s is complex when dealing with complex data. The cosine parameter c is always real, but the sine parameter s is complex when dealing with complex data. The parameters c and s are always constrained by trigonometric relation The parameters c and s are always constrained by trigonometric relation The Givens rotation is non-Hermitian The Givens rotation is non-Hermitian Givens rotation is unitary. Givens rotation is unitary. The Givens rotation is length preserving The Givens rotation is length preserving

14. How Givens Rotations Works Methodsome matrixoutput            =             =             =              = G m G m-1 G m-2... G 2 G 1 U = Upper triangular and Diagonal

15. QR-RLS Algorithm Data matrix: Data matrix: - M represents the number of FIR filter coefficients

16. Phi represents the correlation matrix Phi represents the correlation matrix The matrix here is the exponential weighting matrix. The matrix here is the exponential weighting matrix. Lambda is the exponential weighting factor Lambda is the exponential weighting factor

17. Simulations QR decomposition RLS adaptation algorithm QR decomposition RLS adaptation algorithm Program used: MATLAB Program used: MATLAB

18. Graph LMS

19. Graph RLS

20. Graph QR-decomposition

21. Summary QR decomposition is one of the best numerical procedures for solving the recursive lease squares estimation problem QR decomposition is one of the best numerical procedures for solving the recursive lease squares estimation problem QR decomposition operates on inputs only QR decomposition operates on inputs only QR decomposition involves the use of only numerically well behaved unitary rotations QR decomposition involves the use of only numerically well behaved unitary rotations

22. QR-RLS eliminates almost all the error QR-RLS eliminates almost all the error Has good numerical properties and good stability. Has good numerical properties and good stability. Reliable Reliable