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 Chebyshev polynomials to develop the key formulas used for FIR filter sizing. This appendix provides a very brief review of these polynomials and the equations used to generate them. 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 out that the Russian engineer Chebyshev developed these polynomials as part of design effort which required minimizing the maximum lateral excursion of a 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 stated criteria 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 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 . To see that they describe the same polynomials as seen in , consider for values of |x| between -1 and 1. For such values cos-1x ranges between π and 0. Thus M·cos-1x ranges between Mπ and 0, and cos[M·cos-1x] cycles between -1 or 1 and 1, hiting M+1 extrema on the way, counting the endpoints. Similar analysis shows that equation grows monotonically in magnitude as |x| does. In fact it is easy to show that |cosh[M·cosh-1x]| 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 functional form appearing frequently in frequency-domain representations of filters. In light of this a final twist might be noted. is in fact superfluous given . To see this, consider evaluating for |x|=2. It initially appears that this won't work, since arccosine cannot be evaluated for arguments 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 see that the cosine of jx 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 jMx, which is the hyperbolic cosine of Mx. Thus, if imaginary arguments are permitted, then suffices to describe all of the Chebyshev polynomials.