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This graph contains two waveforms. One wave is identified by a thin solid line and the other is identified by a dotted line. The x-axis is labeled ω/π and the y-axis is labeled |H(ω)|. Both wave forms follow the same general path. They start high on the y-axis and proceed to the right. The waves drop drastically to the x-axis and bounce along the x-axis until the end of the graph. The upper portion of the waves are bound vertically by two sets of horizontal line. Solid line has a greated amplitude and contained within these bounds are two dotted lines which mark the bounds of the dotted line. The dotted line has a smaller amplitude. After the drastic descent, the upper bound of the solid line wave is marked by a solid line. The dotted line wave's upper bound is marked by a horizontal dotted line. The amplitude of the solid line is greater than the dotted line.
Effects of transition bands in l filters.

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 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 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

min h D ( ω ) - H ( ω ; h ) 2 subject to D ( ω ) - H ( ω ; h ) τ ω [ 0 , ω p b ] [ ω s b , π ]

is equivalent to

min h w ( ω ) · ( D ( ω ) - H ( ω ; h ) ) 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 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 ϵ ( ω ) = | ϵ ( ω ) | p - 2 2 if | ϵ ( ω ) | > τ 1 otherwise

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.

This image contains three graphs. The first graph represents Polynomial weighting (p=100). The second graph represents Linear-log view (p=100), and the third graph represents Linear-log view (p=500). The x-axis is labeled ε for all of these graphs, and the y-axis is labeled w(ε).
CLS polynomial weighting function.

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

w ( ω ) = 1 + ϵ ( ω ) τ p - 2 2

where τ 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.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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 .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Iterative design of l_p digital filters. OpenStax CNX. Dec 07, 2011 Download for free at http://cnx.org/content/col11383/1.1
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