# 0.10 Constrained least squares (cls) problem  (Page 3/5)

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These two results together illustrate the importance of the transition bandwidth for a CLS design. Clearly one can decrease maximum error tolerances by widening the transition band. Yet finding the perfect balance between a transition bandwidth and a given tolerance can prove a difficult task, as will be shown in [link] . Hence the relevance of a CLS method that is not restricted by two types of specifications competing against each other. In principle, one should just determine how much error one can live with, and allow an algorithm to find the optimal transition band that meets such tolerance.

## Two problem solutions

[link] introduced some important remarks regarding the behavior of extrema points and transition bands in ${l}_{2}$ and ${l}_{\infty }$ filters. As one increases the constraints on an ${l}_{2}$ filter, the result is a filter whose frequency response looks more and more like an ${l}_{\infty }$ filter.

[link] introduced the frequency-varying problem and an IRLS-based method to solve it. It was also mentioned that, while the method does not solve the intended problem (but a similar one), it could prove to be useful for the CLS problem. As it turns out, in CLS design one is merely interested in solving an unweighted, constrained least squares problem. In this work, we achieve this by solving a sequence of weighted, unconstrained least squares problems, where the sole role of the weights is to "constraint" the maximum error of the frequency response at each iteration. In other words, one would like to find weights $w$ such that

$\begin{array}{cc}\underset{h}{\text{min}}\hfill & {\parallel D\left(\omega \right)-H\left(\omega ;h\right)\parallel }_{2}\hfill \\ \text{subject}\phantom{\rule{4.pt}{0ex}}\text{to}\hfill & {\parallel D\left(\omega \right)-H\left(\omega ;h\right)\parallel }_{\infty }\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}\tau \phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}\omega \in \left[0,{\omega }_{pb}\right]\cup \left[{\omega }_{sb},\pi \right]\hfill \end{array}$

is equivalent to

$\underset{h}{\text{min}}\phantom{\rule{0.277778em}{0ex}}{\parallel w\left(\omega \right)·\left(D\left(\omega \right)-H\left(\omega ;h\right)\right)\parallel }_{2}$

Hence one can revisit the frequency-varying design method and use it to solve the CLS problem. Assuming that one can reasonably approximate ${l}_{\infty }$ by using high values of $p$ , at each iteration the main idea is to use an ${l}_{p}$ weighting function only at frequencies where the constraints are exceeded. A formal formulation of this statement is

$w\left(ϵ,\left(,\omega ,\right)\right)=\left\{\begin{array}{cc}{|ϵ\left(\omega \right)|}^{\frac{p-2}{2}}\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}|ϵ\left(\omega \right)|\phantom{\rule{-0.166667em}{0ex}}>\phantom{\rule{-0.166667em}{0ex}}\tau \hfill \\ 1\hfill & \text{otherwise}\hfill \end{array}\right)$

Assuming a suitable weighting function existed such that the specified tolerances are related to the frequency response constraints, the IRLS method would iterate and assign rather large weights to frequencies exceeding the constraints, while inactive frequencies get a weight of one. As the method iterates, frequencies with large errors move the response closer to the desired tolerance. Ideally, all the active constraint frequencies would eventually meet the constraints. Therefore the task becomes to find a suitable weighting function that penalizes large errors in order to have all the frequencies satisfying the constraints; once this condition is met, we have reached the desired solution.

One proposed way to find adequate weights to meet constraints is given by a polynomial weighting function of the form

$w\left(\omega \right)=1+{\left|\frac{ϵ\left(\omega \right)}{\tau }\right|}^{\frac{p-2}{2}}$

where $\tau$ effectively serves as a threshold to determine whether a weight is dominated by either unity or the familiar ${l}_{p}$ weighting term. [link] illustrates the behavior of such a curve.

where we get a research paper on Nano chemistry....?
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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
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
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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?
what is a peer
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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