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Some notes on chebyshev polynomials

The section "The Derivation of the Formula" from the module titled "Filter Sizing" used some of the properties of the Chebyshevpolynomials to develop the key formulas used for FIR filter sizing. This appendix provides a very brief review of these polynomials and theequations used to generate them.

[link] shows a set of polynomials which have the property that, for values of x between -1 and 1, the polynomial has peak magnitude of unity. A footnote in The section "The Derivation of the Formula" from the module titled "Filter Sizing" pointed outthat the Russian engineer Chebyshev developed these polynomials as part of design effort which required minimizing the maximum lateral excursion ofa locomotive drive rod. For each polynomial order, say M , the objective is to choose the polynomial's coefficients so that that it “ripples" between x = - 1 and x = 1 and then proceeds off proportional to | x | M for values of | x | > 1 . Not only did Chebyshev find such polynomials, he found that one exists for each positive value of M , and that they are related thorugh a recursion equation, that is, the polynomial for M can directly obtained for the polynomial for M -1.

Graphs of Chebyshev Polynomials of Orders 0 through 4

Consider the following recursion expression:

P M ( x ) = 2 x · P M - 1 ( x ) - P M - 2 ( x ) ,

with initial conditions of

P 0 = 1


P 1 = x

Note that both of these initial conditions meet (if trivially) the statedcriteria for being Chebyshev polynomials.

Using this recursion expression we find, for M from 0 to 5, that:

P 0 ( x ) = 1 P 1 ( x ) = x P 2 ( x ) = 2 x 2 - 1 P 3 ( x ) = 4 x 3 - 3 x P 4 ( x ) = 8 x 4 - 8 x 2 + 1 P 5 ( x ) = 16 x 5 - 20 x 3 + 5 x

These polynomials are plotted in [link] and it may be confirmed by inspection that they meet the stated criteria.

A surprising result is that there is yet another way to present these polynomials. This method is given by the following equations:

P M ( x ) = c o s [ M · c o s - 1 ( x ) ] , f o r | x | 1 , a n d
P M ( x ) = c o s h [ M · c o s h - 1 ( x ) ] , f o r | x | > 1 .

Analytically it can be confirmed that these equations satisfy the recursion seen in equation [link] . To see that they describe the same polynomials as seen in [link] , consider [link] for values of | x | between -1 and 1. For such values c o s - 1 x ranges between π and 0. Thus M · c o s - 1 x ranges between M π and 0, and c o s [ M · c o s - 1 x ] cycles between -1 or 1 and 1, hiting M + 1 extrema on the way, counting the endpoints. Similar analysis shows that equation [link] grows monotonically in magnitude as | x | does. In fact it is easy to show that | c o s h [ M · c o s h - 1 x ] | assymptotically approaches | x | M as | x | gets much greater than one.

This second form of the definition for Chebyshev polynomials is very useful since it is a closed form and because it involves cosines, a functionalform appearing frequently in frequency-domain representations of filters. In light of this a final twist might be noted. [link] is in fact superfluous given [link] . To see this, consider evaluating [link] for | x | = 2 . It initially appears that this won't work, since arccosine cannot be evaluated forarguments greater than unity. In fact it can, it's just that the result is purely imaginary. It is easy, using Euler's definition of the cosine, to seethat the cosine of j x is the same as the hyberbolic cosine of x . Thus the arccosine of 2 is j times the inverse hyperbolic cosine of 2, that is, j · 1 . 31 . Multiplying by M and taking the cosine of the product yields the cosine of j M x , which is the hyperbolic cosine of M x . Thus, if imaginary arguments are permitted, then [link] suffices to describe all of the Chebyshev polynomials.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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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
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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.
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I think
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Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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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
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Damian Reply
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Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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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, Notes on the design of optimal fir filters. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10553/1.3
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