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The weiner-filter, W opt R P , is ideal for many applications. But several issues must be addressed to use it in practice.

In practice one usually won't know exactly the statistics of x k and d k (i.e. R and P ) needed to compute the Weiner filter.

How do we surmount this problem?

Estimate the statistics r xx l 1 N k 0 N 1 x k x k + l r xd l 1 N k 0 N 1 d k x k - l then solve W opt R P

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In many applications, the statistics of x k , d k vary slowly with time.

How does one develop an adaptive system which tracks these changes over time to keep the system nearoptimal at all times?

Use short-time windowed estiamtes of the correlation functions.

r xx l k 1 N m N 1 0 x k - m x k - m - l
r dx l k 1 N m N 1 0 x k - m - l d k - m and W opt k R k P k

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How can r xx k l be computed efficiently?

Recursively! r xx k l r xx k - 1 l x k x k - l x k - N x k - N - l This is critically stable, so people usually do 1 r xx l k r xx k - 1 l x k x k - l

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Larger N more accurate estimates of the correlation valuesbetter W opt . However, larger N leads to slower adaptation.

The success of adaptive systems depends on x , d being roughly stationary over at least N samples, N M . That is, all adaptive filtering algorithms require that the underlying system varies slowly withrespect to the sampling rate and the filter length (although they can tolerate occasional step discontinuities in theunderlying system).

Computational considerations

As presented here, an adaptive filter requires computing a matrix inverse at each sample. Actually, since the matrix R is Toeplitz, the linear system of equations can be sovled with O M 2 computations using Levinson's algorithm, where M is the filter length. However, in many applications this may be too expensive, especiallysince computing the filter output itself requires O M computations. There are two main approaches to resolving the computation problem

  • Take advantage of the fact that R k 1 is only slightly changed from R k to reduce the computation to O M ; these algorithms are called Fast Recursive Least Squareds algorithms; all methods proposed so farhave stability problems and are dangerous to use.
  • Find a different approach to solving the optimization problem that doesn't require explicit inversion of thecorrelation matrix.

Adaptive algorithms involving the correlation matrix are called Recursive least Squares (RLS) algorithms. Historically, they were developed after the LMSalgorithm, which is the slimplest and most widely used approach O M . O M 2 RLS algorithms are used in applications requiring very fast adaptation.

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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 .?
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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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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absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
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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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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