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Quadratic minimization problems

The least squares optimal filter design problem is quadratic in the filter coefficients: ε 2 r dd 0 2 P W W R W If R is positive definite, the error surface ε 2 w 0 w 1 w M - 1 is a unimodal "bowl" in N .

The problem is to find the bottom of the bowl. In an adaptive filter context, the shape and bottom of the bowl maydrift slowly with time; hopefully slow enough that the adaptive algorithm can track it.

For a quadratic error surface, the bottom of the bowl can be found in one step by computing R P . Most modern nonlinear optimization methods (which are used, for example, to solve the L P optimal IIR filter design problem!) locally approximate a nonlinear function with a second-order(quadratic) Taylor series approximation and step to the bottom of this quadratic approximation on each iteration. However, anolder and simpler appraoch to nonlinear optimaztion exists, based on gradient descent .

Contour plot of ε-squared

The idea is to iteratively find the minimizer by computingthe gradient of the error function: E w i ε 2 . The gradient is a vector in M pointing in the steepest uphill direction on the error surface at a given point W i , with having a magnitude proportional to the slope of the error surface inthis steepest direction.

By updating the coefficient vector by taking a step opposite the gradient direction : W i 1 W i μ i , we go (locally) "downhill" in the steepest direction, which seems to be a sensible way to iterativelysolve a nonlinear optimization problem. The performance obviously depends on μ ; if μ is too large, the iterations could bounce back and forth up out of thebowl. However, if μ is too small, it could take many iterations to approach thebottom. We will determine criteria for choosing μ later.

In summary, the gradient descent algorithm for solving the Weiner filter problem is: Guess  W 0 do  i 1 , i 2 P 2 R W i W i 1 W i μ i repeat W opt W The gradient descent idea is used in the LMS adaptive fitleralgorithm. As presented, this alogrithm costs O M 2 computations per iteration and doesn't appear very attractive, but LMS only requires O M computations and is stable, so it is very attractive when computation is an issue, even thought it converges moreslowly then the RLS algorithms we have discussed so far.

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Adaptive filters. OpenStax CNX. May 12, 2005 Download for free at http://cnx.org/content/col10280/1.1
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