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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

and

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.

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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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