# 0.10 Constrained least squares (cls) problem  (Page 2/5)

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## Two problem formulations

As mentioned in [link] , one can address problem [link] in two ways depending on how one views the role of the transition band in a CLS problem. The original problem posed by Adams in [link] can be written as follows,

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

where $0\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}{\omega }_{pb}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}{\omega }_{sb}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\pi$ . From a traditional standpoint this formulation feels familiar. It assigns fixed frequencies to the transition band edges as a number of filter design techniques do. As it turns out, however, one might not want to do this in CLS design.

An alternate formulation to [link] could implicitly introduce a transition frequency ${\omega }_{tb}$ (where ${\omega }_{pb}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}{\omega }_{tb}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}{\omega }_{sb}$ ); the user only specifies ${\omega }_{tb}$ . Consider

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

The algorithm at each iteration generates an induced transition band in order to satisfy the constraints in [link] . Therefore $\left\{{\omega }_{pb},{\omega }_{sb}\right\}$ vary at each iteration.

It is critical to point out the differences between [link] and [link] . [link] .a explains Adams' CLS formulation, where the desired filter response is only specified at the fixed pass and stop bands. At any iteration, Adams' method attempts to minimize the least squares error ( ${\epsilon }_{2}$ ) at both bands while trying to satisfy the constraint $\tau$ . Note that one could think of the constraint requirements in terms of the Chebishev error ${\epsilon }_{\infty }$ by writing [link] as follows,

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

In contrast, [link] .b illustrates our proposed problem [link] . The idea is to minimize the least squared error ${\epsilon }_{2}$ across all frequencies while ensuring that constraints are met in an intelligent manner. At this point one can think of the interval $\left({\omega }_{pb},{\omega }_{sb}\right)$ as an induced transition band, useful for the purposes of constraining the filter. [link] presents the actual algorithms that solve [link] , including the process of finding $\left\{{\omega }_{pb},{\omega }_{sb}\right\}$ .

It is important to note an interesting behavior of transition bands and extrema points in ${l}_{2}$ and ${l}_{\infty }$ filters. [link] shows ${l}_{2}$ and ${l}_{\infty }$ length-15 linear phase filters (designed using Matlab's firls and firpm functions); the transition band was specified at $\left\{{\omega }_{pb}=0.4/\pi ,{\omega }_{sb}=0.5/\pi \right\}$ . The dotted ${l}_{2}$ filter illustrates an important behavior of least squares filters: typically the maximum error of an ${l}_{2}$ filter is located at the transition band. The solid ${l}_{\infty }$ filter shows why minimax filters are important: despite their larger error across most of the bands, the filter shows the same maximum error at all extrema points, including the transition band edge frequencies. In a CLS problem then, typically an algorithm will attempt to reduce iteratively the maximum error (usually located around the transition band) of a series of least squares filters.

Another important fact results from the relationship between the transition band width and the resulting error amplitude in ${l}_{\infty }$ filters. [link] shows two ${l}_{\infty }$ designs; the transition bands were set at $\left\{0.4/\pi ,0.5/\pi \right\}$ for the solid line design, and at $\left\{0.4/\pi ,0.6/\pi \right\}$ for the dotted line one. One can see that by widening the transition band a decrease in error ripple amplitude is induced.

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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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Alexandre
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Rafiq
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Damian
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what is a peer
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LITNING
scanning tunneling microscope
Sahil
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Santosh
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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
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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
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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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