# 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?

## 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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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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