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Recall the Weiner filter problem

x k , d k jointly wide sense stationary

Find W minimizing k 2 k d k y k d k i M 1 0 w i x k - i d k X k W k X k x k x k - 1 x k - M + 1 W k w 0 k w 1 k w M - 1 k The superscript denotes absolute time, and the subscript denotes time or a vector index.

the solution can be found by setting the gradient 0

k W k 2 2 k X k -2 d k X k W k X k 2 d k X k X k X k W -2 P 2 R W
W opt R P Alternatively, W opt can be found iteratively using a gradient descent technique W k + 1 W k k In practice, we don't know R and P exactly, and in an adaptive context they may be slowly varying with time.

To find the (approximate) Wiener filter, some approximations are necessary. As always, the key is to make the right approximations!

Approximate R and P :RLS methods, as discussed last time.
Approximate the gradient! k W k 2
Note that k 2 itself is a very noisy approximation to k 2 . We can get a noisy approximation to the gradient by finding the gradient of k 2 ! Widrow and Hoff first published the LMS algorithm, based on this clever idea, in 1960. k W k 2 2 k W d k W k X k 2 k X k 2 k X k This yields the LMS adaptive filter algorithm

The lms adaptive filter algorithm

  • y k W k X k i 0 M 1 w i k x k - i
  • k d k y k
  • W k + 1 W k k W k -2 k X k W k 2 k X k ( w i k + 1 w i k 2 k x k - i )
Got questions? Get instant answers now!

The LMS algorithm is often called a stochastic gradient algorithm, since k is a noisy gradient. This is by far the most commonly used adaptive filtering algorithm, because

  • it was the first
  • it is very simple
  • in practice it works well (except that sometimes it converges slowly)
  • it requires relatively litle computation
  • it updates the tap weights every sample, so it continually adapts the filter
  • it tracks slow changes in the signal statistics well

Computational cost of lms

To Compute y k k W k + 1 = Total
multiplies M 0 M 1 2 M 1
adds M 1 1 M 2 M

So the LMS algorithm is O M per sample. In fact, it is nicely balanced in that the filter computation and the adaptation require the sameamount of computation.

Note that the parameter plays a very important role in the LMS algorithm. It can also be varied with time, but usually a constant ("convergence weight facor") is used, chosen after experimentation for a givenapplication.

Tradeoffs

large : fast convergence, fast adaptivity

small : accurate W less misadjustment error, stability

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
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RAW Reply
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Damian
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
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LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
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Introduction about quantum dots in nanotechnology
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian 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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