IIRFilters.ChebyshevIAnalog Method

procedure ChebyshevIAnalog(Order: Integer; PassRipple: Double; z: TVec; p: TVec; out k: Double);

Design an analog Chebyshev type I lowpass prototype filter of order Order with PassRipple dB of equiripple in the passband, cutoff fixed at 1 rad/s. Returns zero-pole-gain (z empty, all zeros at infinity): |H(jomega)|^2=1/(1+varepsilon^2 T_n^2(omega)), varepsilon=sqrt(10^(R_p/10)-1), where T_n is the order-n Chebyshev polynomial and R_p=PassRipple. The magnitude is equiripple in $[0,1]$ and monotone beyond; at the passband edge omega=1 it equals 10^(-R_p/20). Domain: 1 <= Order <= MaxIirOrder, PassRipple>0 dB. Poles satisfy Re(p_k)<0 (stable). z and p must share precision or an exception is raised.

#	Name	Type	Description
1	Order	Integer
2	PassRipple	Double	scalar
3	z	TVec	source TVec
4	p	TVec	source TVec
5	k	Double

Result: stored in self (calling object)

Remarks:

Design analog Chebyshev type I lowpass prototype filter of order Order. Place the resulting transfer function in zero-pole form in Z (zeros), P (poles) and K (gain). PassRipple defines the ripple of the passband (dB). The cutoff frequency of the prototype filter is preset to 1 rad/sec, the unit circle. Chebyshevs type I filters are all-pole designs and are equiripple in the passband. The filter has all zeros in infinity. The design formulas are found in [1] p. 232:

Poles: p[k] = s[k] + j*W[k] s[k] = -sinh(Phi)sin((2*k-1)*Pi/(2*n)) W[k] = cosh(Phi)*cos((2*k-1)*Pi/(2*n)) sinh(phi) = 0.5(v - 1/v) cosh(phi) = 0.5*(v + 1/v) 1 + (1 + eps^2)^0.5 v = ( --------------------- )^(1/n) eps n - order of the filter k = 1,...,n

References:

[1] "Theory and application of digital signal processing, Lawrence R. Rabiner and Bernard Gold. Prentice-Hall, 1975".

Examples

uses MtxExpr, Math387, MtxVec, SignalUtils, MtxVecTee, MtxVecEdit, IirFilters,
LinearSystems;

procedure TForm1.Button1Click(Sender: TObject);
var z,p, num,den, FreqFr,Response: Vector;
Order: integer;
k,Wc,BW: double;
begin
    Order := 5; //design a fifth order filter.
    ChebyshevIAnalog(Order,0.2,z,p,k);  //design analog protype
    Wc := 2;
    LowpassToHighpass(z,p,k,WC);  //frequency transformation in s-domain
    ZeroPoleToTransferFun(num,den,z,p,k);
    FreqFr.Length := 1000;         //Define the frequency grid (logarithmic)
    LogRamp(FreqFr,-1,1); //between 0.1 (=10^(-1)) and 10 (=10^1) rad/sec
    FrequencyResponseS(num,den,FreqFr,Response); //Laplace
    DrawIt(Response); //Y axis linear, X axis logarithmic;
end;