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LinearSystems.MatchedZTransform Method
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Transform the zeros and poles of a filter in s-domain to z-domain.

Syntax
C#
Visual Basic
public static void MatchedZTransform([In] TVec z, [In] TVec p, double FS);
Remarks

Transform the zeros Z and poles P of a filter from s-domain to z-domain, where FS is the sampling frequency. The transformation is defined as ([1], p. 224):

s + a --> 1 - z^(-1)*e^(-a/FS) FS - sampling frequency a - pole or zero

If the analog system has zeros with center frequencies greater then half the sampling frequency, their z-plane positions will be greatly aliased [1]. The transformation has the advantage of not affecting the phase response of the original transfer function. 

References:  

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

See Also
Example

A bessel analog lowpass filter is converted to z-domain by using the matched Z transform. The analog filter has a normalized cutoff frequency at 1 rad/sec.

using Dew.Math; using Dew.Math.Editors; using Dew.Math.Units; using Dew.Signal; using Dew.Signal.Units; using Dew.Math.Tee; using Dew.Signal.Tee; private void button1_Click(object sender, EventArgs e) { Vector z = new Vector(0); Vector p = new Vector(0); Vector num = new Vector(0); Vector den = new Vector(0); Vector Response = new Vector(0); Vector FreqFr = new Vector(0); double k,Wc,BW; double FS = 2; int Order = 5; //design a fifth order filter. IIRFilters.BesselAnalog(Order,z,p,out k); //design analog protype Wc = 0.5; LinearSystems.LowpassToLowpass(z,p,ref k,Wc); //frequency transformation in s-domain LinearSystems.MatchedZTransform(z, p, FS); k = k/LinearSystems.ComputeGain(z,p,1); z.Size(p.Length,false); z.SetVal(-1); //add missing zeros at -1 LinearSystems.ZeroPoleToTransferFun(num,den,z,p,k); SignalUtils.FrequencyResponse(num,den,Response,64,false,TSignalWindowType.wtRectangular,0); //zero padding set to 64 MtxVecTee.DrawIt(20*MtxExpr.Log10(MtxExpr.Abs(Response)),"Magnitude",false); MtxVecTee.DrawIt(MtxExpr.PhaseSpectrum(Response)*(180/Math.PI),"Phase",false); }
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