# 5.8 Practical issues in wiener filter implementation

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The weiner-filter, ${W}_{\mathrm{opt}}=R^{(-1)}P$ , 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 $({r}_{\mathrm{xx}}(l))\approx \frac{1}{N}\sum_{k=0}^{N-1} {x}_{k}{x}_{k+l}$ $({r}_{\mathrm{xd}}(l))\approx \frac{1}{N}\sum_{k=0}^{N-1} {d}_{k}{x}_{k-l}$ then solve $({W}_{\mathrm{opt}})=(R^{(-1)})=(P)$

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.

$({r}_{\mathrm{xx}}(l))^{k}=\frac{1}{N}\sum_{m=0}^{N-1} {x}_{k-m}{x}_{k-m-l}$
$({r}_{\mathrm{dx}}(l))^{k}=\frac{1}{N}\sum_{m=0}^{N-1} {x}_{k-m-l}{d}_{k-m}$ and ${W}_{\mathrm{opt}}^{k}\approx ({R}_{k})^{(-1)}({P}_{k})$

How can $({r}_{\mathrm{xx}}^{k}(l))$ be computed efficiently?

Recursively! ${r}_{\mathrm{xx}}^{k}(l)={r}_{\mathrm{xx}}^{k-1}(l)+{x}_{k}{x}_{k-l}-{x}_{k-N}{x}_{k-N-l}$ This is critically stable, so people usually do $(1-)({r}_{\mathrm{xx}}(l)^{k}={r}_{\mathrm{xx}}^{k-1}(l)+{x}_{k}{x}_{k-l})$

how does one choose N?

Larger $N$ more accurate estimates of the correlation valuesbetter $({W}_{\mathrm{opt}})$ . 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.

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what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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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?
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What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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