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IIRFilters.BesselAnalog Function
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Design analog Bessel type IIR prototype filter.

Pascal
procedure BesselAnalog(Order: integer; const z: TVec; const p: TVec; out k: double); overload;
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Description

Design analog Bessel prototype filter of order Order. Place the resulting transfer function in zero-pole form in Z (zeros), P (poles) and K (gain). The cutoff frequency of the prototype filter is preset to 1 rad/sec. The filter has all zeros in infinity. The transfer function is defined as([1], p. 230): 


          d0
H(s) = --------
         Bn(s)

      (2*n)!               n
d0 = ------- ,    Bn(s) = Sum(d[k]*s^k),  k = 0,...,n
      2^n*n!              k=0

          (2*n-k)!
d[k] = --------------    , n = order of the filter
        2^(n-k)*(n-k)!


Filter poles must be scaled with d0^(1/n)

 

Roots of the Bessel polynomial Bn(s) are found with the PolyRoots routine. Bessel lowpass filters are charachterized by the property that the group delay is maximally flat at the origing of the s-plane. ([1], p. 228). 

References:  

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

See Also
Example

Design an analog lowpass filter with cutoff at 1.1 rad/sec. 

 

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.
    BesselAnalog(Order,z,p,k);  //design analog protype
    Wc := 1.1;
    LowpassToLowpass(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;

 

    #include "MtxExpr.hpp"
#include "MtxVecEdit.hpp"
#include "MtxVecTee.hpp"
#include "SignalUtils.hpp"
#include "IirFilters.hpp"
#include "LinearSystems.hpp"

void __fastcall TForm1::BitBtn1Click(TObject *Sender)
{
    sVector z,p, num,den, FreqFr, Response;
    int Order;
    double k,Wc,BW;

    Order = 5; //design a fifth order filter.
    BesselAnalog(Order,z,p,k);  //design analog protype
    Wc = 1.1;
    LowpassToLowpass(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;
}
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