invchebyshev.html

invchebyshev.mws

syntfil[InvChebyshev] - compute the inverse Chebyshev approximation

syntfil[InvChebyshevB] - compute the inverse type B Chebyshev approximation

Calling sequence:

InvChebyshev(order, Os, ap, var)

Parameters:

order - order of the Chebyshev approximation [-]

Os - stopband frequency of normalized lowpass (NLP) [1/s]

ap - passband ripple [dB]

var - variable symbol in tranfer and characteristic function

Parameter order must be positive integer and for type B in addition even. Parameters Os and ap must be positive numbers where Os > 1 . Parameter var must be symbol .

Description:

This function returns the inverse transfer function

, characteristic function

, and one dimensional array of transfer function's zeros.

Inverse transfer function is rational function with polynomial in numerator and denominator in variable var . Numerator polynomial has degree order . Denominator polynomial has degree order for even order or order-1 for odd order. In case of type B the denominator polynomial has degree order-2 .

Characteristic function is also rational function, such that the denominator of

is identical with denominator of H^(-1)

, and the zeros of

lay in origin (zero) and poles of

lay on imaginary axis.

Inverse type B Chebyshev aproximation is computed from standard inverse Chebyshev aproximation (type A), which in case of even order eventuates to negative element value (L or C) at realization of LC ladder filter.

Info level:

Setting of variable infolevel[syntfil] can be used to get more detailed results.

infolevel[syntfil] =

2 - print polynomials of inverse transfer function characteristic function and zeros of transfer function on separate lines with description.

3 - as level 2 + print transfer function's poles and parameter .

4 - as level 3 + print ellipse parameter ( , ) on which lies poles of transfer function and secondary parameter .

5 - for type B - as level 4 + print transfer function's poles and zeros for type A.

Example:

> with(syntfil):

> G_a,Phi_a,zeros_a:=InvChebyshev(4,1.2,3,s);

> infolevel[syntfil]:=3:

> G_b,Phi_b,zeros_b:=InvChebyshevB(4,1.2,3,s);

epsilon = .9976283451

Poles of H:

[-.2636496923+1.018799929*I, -.2636496923-1.018799929*I, -1.217556347+.7169388538*I, -1.217556347-.7169388538*I]

InvChebyshevB:

G = (1.829360748+.8273937923*s^4+2.451081364*s^3+3.630556516*s^2+3.102333590*s)/(s^2+1.829360748)

Phi = .8273937923*s^4/(s^2+1.829360748)

Zeros = [1.352538631*I, -1.352538631*I]

Magnitude frequency response for type A and B .

> plot([MagnitudeHdB(1/G_a)(omega),MagnitudeHdB(1/G_b)(omega)],omega=0..8,-40..0,color=[red,green]);

DroppNLP, TestCharEqn, sortzeros, MagnitudeH, MagnitudeHdB, PhaseH, GroupDelayH
in addition to inverse Chebyshev approximation the following approximations can be used
Butterworth, Chebyshev, Cauer, CauerB, CauerC