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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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Adin Reply
?
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
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for screen printed electrodes ?
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What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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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