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QR-RLS Algorithm

QR-RLS Algorithm. Cy Shimabukuro EE 491D 05-13-05. Overview. What is QR-RLS Different methods of Computation Simulation Results. QR-RLS?. QR-RLS algorithm is used to solve linear least square problems. The decomposition is the basis for the QR algorithm.

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QR-RLS Algorithm

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

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

  3. 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.

  4. 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.

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

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

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

  8. Householder • Used to zero out subdiagonal elements A is decomposed: where QT=Hn…H2H1 is the orthogonal product of Householders and R is upper triangular. • Over determined system Ax=b is transformed into the easy-to-solve

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

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

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

  12. Properties • 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 Givens rotation is non-Hermitian • Givens rotation is unitary. • The Givens rotation is length preserving

  13. 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 a b 0 0 0 0 0 0 g d 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 a b 0 0 0 0 0 0 g d 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 a b 0 0 0 0 0 0 g d 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 a b 0 0 0 0 0 0 g d * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * 0 * * * * * * * 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * 0 * * * * * * * 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * 0 * * * * * * * 0 0 * * * * * * = = = = How Givens Rotations Works Method some matrix output Gm Gm-1 Gm-2 ... G2 G1 U = Upper triangular and Diagonal

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

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

  16. Simulations • QR decomposition RLS adaptation algorithm • Program used: MATLAB

  17. Graph LMS

  18. Graph RLS

  19. Graph QR-decomposition

  20. Summary • 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 involves the use of only numerically well behaved unitary rotations

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

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