# 0.14 L_p approximation

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Infinite Impulse Response (IIR) filters are important tools in signal processing. The flexibility they offer with the use of poles and zeros allows for relatively small filters meeting specifications that would require somewhat larger FIR filters. Therefore designing IIR filters in an efficient and robust manner is an inportant problem.

This section covers the design of a number of important ${l}_{p}$ IIR problems. The methods proposed are consistent with the methods presented for FIR filters, allowing one to build up on the lessons learned from FIR design problems. The complex ${l}_{p}$ IIR problem is first presented in [link] , being an essential tool for other relevant problems. The ${l}_{p}$ frequency-dependent IIR problem is also introduced in [link] . While the frequency-dependent formulation might not be practical in itself as a filter design formulation, it is fundamental for the more relevant magnitude ${l}_{p}$ IIR filter design problem, presented in [link] .

Some complications appear when designing IIR filters, among which the intrinsic least squares solving step clearly arises from the rest. Being a nonlinear problem, special handling of this step is required. It was detemined after thorough experimentation that the quasilinearization method of Soewito presented in [link] can be employed successfully to handle this issue.

## Complex and frequency-dependent ${l}_{p}$ Approximation

Chapter [link] introduced the problem of designing ${l}_{p}$ complex FIR filters. The complex ${l}_{p}$ IIR algorithm builds up on its FIR counterpart by introducing a nested structure that internally solves for an ${l}_{2}$ complex IIR problem. [link] illustrates this procedure in more detail. This method was first presented in [link] .

Compared to its FIR counterpart, the IIR method only replaces the weighted linear least squares problem for Soewito's quasilinearization algorithm. While this nesting approach might suggest an increase in computational expense, it was found in practice that after the initial ${l}_{2}$ iteration, in general the ${l}_{p}$ iterations only require from one to only a few internal weighted ${l}_{2}$ quasilinearization iterations, thus maintaining the algorithm efficiency. Figures [link] through [link] present results for a design example using a length-5 IIR filter with $p=100$ and transition edge frequencies of 0.2 and 0.24 (in normalized frequency).

[link] compares the ${l}_{2}$ and ${l}_{p}$ results and includes the desired frequency samples. Note that no transition band was specified. [link] illustrates the effect of increasing $p$ . The largest error for the ${l}_{2}$ solution is located at the transition band edges. As $p$ increases the algorithm weights the larger errors heavier; as a result the largest errors tend to decrease. In this case the magnitude of the frequency response went from 0.155 at the stopband edge (in the ${l}_{2}$ case) to 0.07 (for the ${l}_{p}$ design). [link] shows the error function for the ${l}_{p}$ design, illustrating the quasiequiripple behavior for large values of $p$ .

Another fact worth noting from [link] is the increase in the peak in the right hand side of the passband edge (around $f=0.22$ ). The ${l}_{p}$ solution increased the amplitude of this peak with respect to the corresponding ${l}_{2}$ solution. This is to be expected, since this peak occurs at frequencies not included in the specifications, and since the ${l}_{p}$ algorithm will move poles and zeros around in order to meet find the optimal ${l}_{p}$ solution (based on the frequencies included for the filter derivation). The addition of a specified transition band function (such as a spline) would allow for control of this effect, depending on the user's preferences.

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
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
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
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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?
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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
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