QR-RLS AlgorithmQR-RLS Algorithm

Cy ShimabukuroCy Shimabukuro

EE 491DEE 491D05-13-0505-13-05

OverviewOverview

What is QR-RLSWhat is QR-RLS

Different methods of ComputationDifferent methods of Computation

SimulationSimulation

ResultsResults

QR-RLS?QR-RLS?

QR-RLS algorithm is used to solve QR-RLS algorithm is used to solve linear least square problems.linear least square problems.

The decomposition is the basis for the The decomposition is the basis for the QR algorithm.QR algorithm.

Algorithm is a procedure to produce Algorithm is a procedure to produce eigenvalues of a matrix. eigenvalues of a matrix.

AdvantageAdvantage

Using this QR method is not for speed, but Using this QR method is not for speed, but the numerical stablilitythe numerical stablility

How? How? proceeds by orthogonal similarity proceeds by orthogonal similarity

transforms.transforms.works directly with data from works directly with data from

decomp.decomp.eliminating the correlation matrix.eliminating the correlation matrix.

Computing QR Decomp.Computing QR Decomp. Gram-Schmidt ProcessGram-Schmidt Process

Householder TransformationHouseholder Transformationa.k.a Householder reflectiona.k.a Householder reflection

Givens RotationGivens Rotation

Gram-SchmidtGram-Schmidt

A method of orthogonalizing a set of A method of orthogonalizing a set of vectorsvectors

This method is numerically UnstableThis method is numerically Unstable The vectors aren’t orthogonal due to The vectors aren’t orthogonal due to

rounding errors.rounding errors. Loss of orthogonality is badLoss of orthogonality is bad

HouseholderHouseholder

Used to calculate QR decompositions Used to calculate QR decompositions

Reflection of a vector plane in 3-D Reflection of a vector plane in 3-D space.space.

Hyperplane is a unit vector Hyperplane is a unit vector orthogonal to hyperplaneorthogonal to hyperplane

Householder Householder Used to zero out Used to zero out

subdiagonal elementssubdiagonal elementsAA is decomposed: is decomposed:

where where QQTT==HHnn……HH22HH11 is is the orthogonal product the orthogonal product of Householders and of Householders and RR is upper triangular.is upper triangular.

Over determined Over determined system system Ax=bAx=b is is transformed into the transformed into the easy-to-solveeasy-to-solve

0

or 0

RQAAQQ

RAQ TT

bQxR T

0

HouseholderHouseholder

Properties it follows:Properties it follows: Symmetrical : Q = Q^T Symmetrical : Q = Q^T it is it is orthogonalorthogonal: Q^{-1}=Q^T : Q^{-1}=Q^T therefore it is also involutary: Q^2=I therefore it is also involutary: Q^2=I

By using the Householder By using the Householder transformation method, it has more transformation method, it has more stability than the Gram-Schmidtstability than the Gram-Schmidt

Givens RotationGivens Rotation Another transformation to find Q matrixAnother transformation to find Q matrix

Method zeros out element in the matrixMethod zeros out element in the matrix

Most useful because:Most useful because: Don’t have to build a new matrix but just Don’t have to build a new matrix but just

manipulating originalmanipulating original Less work and zeros out what is neededLess work and zeros out what is needed Much more easily parallelizedMuch more easily parallelized

The MatrixThe Matrix

‘c’ represents cos(θ), ‘s’ represents sin(θ)

PropertiesProperties The cosine parameter c is always real, The cosine parameter c is always real,

but the sine parameter s is complex but the sine parameter s is complex when dealing with complex data.when dealing with complex data.

The parameters c and s are always The parameters c and s are always constrained by trigonometric relationconstrained by trigonometric relation

The Givens rotation is non-HermitianThe Givens rotation is non-Hermitian Givens rotation is unitary.Givens rotation is unitary. The Givens rotation is length The Givens rotation is length

preservingpreserving

How Givens Rotations WorksHow Givens Rotations Works

Method some matrix output

=

=

=

=

Gm Gm-1 Gm-2 ... G2 G1 U = Upper triangular and Diagonal

QR-RLS AlgorithmQR-RLS Algorithm

Data matrix:Data matrix:

- M represents the number of FIR filter coefficients

Phi represents the correlation matrixPhi represents the correlation matrix

The matrix here is the exponential The matrix here is the exponential weighting matrix. weighting matrix.

Lambda is the exponential weighting factorLambda is the exponential weighting factor

SimulationsSimulations

QR decomposition RLS adaptation QR decomposition RLS adaptation algorithmalgorithm

Program used: MATLABProgram used: MATLAB

Graph LMSGraph LMS

Graph RLSGraph RLS

Graph QR-decompositionGraph QR-decomposition

SummarySummary

QR decomposition is one of the best QR decomposition is one of the best numerical procedures for solving the numerical procedures for solving the recursive lease squares estimation recursive lease squares estimation problemproblem

QR decomposition operates on inputs onlyQR decomposition operates on inputs only QR decomposition involves the use of only QR decomposition involves the use of only

numerically well behaved unitary rotationsnumerically well behaved unitary rotations

QR-RLS eliminates almost all the QR-RLS eliminates almost all the errorerror

Has good numerical properties and Has good numerical properties and good stability.good stability.

ReliableReliable