# 2.2 Least squared error design of fir filters  (Page 13/13)

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## Section conclusions

This section has derived the four basic ideal lowpass filters: the constant gain passband lowpass filter, the linearly increasing gain passbandlowpass filter, the differentiator with a lowpass filter, and the Hilbert transformer with a lowpass filter. It is shown that each of these can bemodified to allow a spline transition function by a simple weighting function.

Because of using an ${L}_{2}$ approximation error criterion and because of the orthogonality of the basis functions of the Fourier transform, it is shownthat an optimal multiband filter can be built from the linear combination of these optimal building blocks. This new filter design method has theflexibility of the Parks-McClellan algorithm but the simplicity of the windowing methods. It is extremely fast and has no numerical problems.Unlike the windowing methods, the new method allows explicit independent control of multiple transition band edges and gives an optimal design.Its only limitation is not allowing error weighting.

We then derived a second method that likewise allowed multiple pass, stop, and transition bands with arbitrary band edges, but also allowedindependent weighting of each frequency band. There are two limitations on this method. For long filters with wide transition bands with zeroweights and where $N\left({f}_{p}-{f}_{s}\right)>12$ , the equations that must be solved are ill conditioned. This can be partially addressed using optimal splinefunctions with small weights in the transition bands. The second problem is that solving a large number of simultaneous equations can be slow andrequire considerable memory. These problems might be addressed by using special Toeplitz or Toeplitz plus Hankel algorithms [link] or some iterative method.

When should these methods be used? The second method which minimizes the weighted integral squared error should be used anytime the originalproblem dictates a squared error criterion and the product of the length and transition band width is less than twelve, $N\phantom{\rule{0.166667em}{0ex}}\left({f}_{p}-{f}_{s}\right)<12$ . These conditions are often met because the squared error is a measure of thesignal or noise energy and one seldom wants a long filter and a wide transition band. Even though this method requires solution of a set ofsimultaneous equations and is, therefore, slower than the spline transition function method, it executes in a few seconds on a PC orworkstation and allows independent weighting of different frequency bands.

The first method which uses spline functions in the ideal response transition bands will design essentially arbitrarily long filters veryquickly but it will not allow any error weighting. Although artificial transition functions are used in the ideal response, the optimized splinefunctions are very close to the response actually obtained by the second method with zero weighting in the transition band. This means the optimalapproximation to the ideal response with spline functions transition bands is close to that obtained using the second numerical method. Comparisonsof these effects for a single band can be found in [link] . If a Chebyshev approximation is desired, the Parks-McClellan method should beused although it too has numerical problems for long filters with wide transition bands. If different error measures are wanted in differentbands, the iterative reweighted least squares (IRLS) algorithm [link] should be used. Recent research suggest that for many practical signal specifications, a mixture of Chebyshev and least squaresis appropriate with no explicit transition bands [link] .

If the equations that must be solved to obtain the optimal filter coefficients are ill-conditioned, an orthogonalization procedure can beused to improve the conditioning [link] .

## Characteristics of optimalFilters

Gibbs phenomenon, transition band, pole-zero plots, etc.

## ComplexAnd minimum phase approximation

Here we talk about which methods also solve the complex approximation problem. We talk about the minimum phase filter.

where we get a research paper on Nano chemistry....?
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
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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