# 5.8 Practical issues in wiener filter implementation

 Page 1 / 1

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?

## Tradeoffs

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.

#### Questions & Answers

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
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of signal processing' conversation and receive update notifications?

 By By By By