frep2tf transfer function realization from frequency response

## Syntax

                [h, err] = frep2tf(frq, repf, dg)
                [h, err] = frep2tf(frq, repf, dg, dom, tols, weight)

## Arguments

frq

  vector of frequencies in Hz.

repf

  vector of frequency response

dg

  degree of linear system

dom

  time domain (`'c'` or `'d'` or `dt`)

tols

  a vector of size 3 giving the relative and absolute tolerance and the maximum number of iterations (default values are `rtol=1.e-2; atol=1.e-4, N=10`).

weight

  vector of weights on frequencies

h

  SISO transfer function

err

  error (for example if `dom='c'` `sum(abs(h(2i*pi*frq) - rep)^2)/size(frq,*)`)

## Description

Frequency response to transfer function conversion. The order of `h` is a priori given in `dg` which must be provided. The following linear system is solved in the least square sense.

     weight(k)*(n( phi_k) - d(phi_k)*rep_k)=0, k=1,..,n

where `phi_k= 2*%i*%pi*frq` when `dom='c'` and `phi_k=exp(2*%i*%pi*dom*frq` if not. If the `weight` vector is not given a default penalization is used (when `dom='c'`).

A stable and minimum phase system can be obtained by using function `factors`.

## Examples

    s=poly(0,'s');
    h=syslin('c',(s-1)/(s^3+5*s+20))
    frq=0:0.05:3;
    repf=repfreq(h,frq);
    clean(frep2tf(frq,repf,3))

    Sys=ssrand(1,1,10);
    frq=logspace(-3,2,200);
    [frq,rep]=repfreq(Sys,frq);  //Frequency response of Sys
    [Sys2,err]=frep2tf(frq,rep,10);
    Sys2=clean(Sys2) //Sys2 obtained from freq. resp of Sys

    [frq,rep2]=repfreq(Sys2,frq); //Frequency response of Sys2
    clf();
    bode(frq,[rep;rep2])   //Responses of Sys and Sys2
    [gsort(spec(Sys('A'))), gsort(roots(Sys2('den')))] //poles

Sys=ssrand(1,1,10); frq=logspace(-3,2,200); [frq,rep]=repfreq(Sys,frq); [Sys2,err]=frep2tf(frq,rep,10); Sys2=clean(Sys2) [frq,rep2]=repfreq(Sys2,frq); bode(frq,[rep;rep2]) [gsort(spec(Sys('A'))), gsort(roots(Sys2('den')))]

    dom = 1/1000; // Sampling time
    z = poly(0,'z');
    h = syslin(dom,(z^2+0.5)/(z^3+0.1*z^2-0.5*z+0.08))
    frq = (0:0.01:0.5)/dom;
    repf = repfreq(h,frq);
    [Sys2,err] = frep2tf(frq,repf,3,dom);
    [frq,rep2] = repfreq(Sys2,frq); //Frequency response of Sys2
    clf();
    plot2d(frq, abs([repf;rep2])')

dom=1/1000; z=poly(0,'z'); h=syslin(dom,(z^2+0.5)/(z^3+0.1*z^2-0.5*z+0.08)) frq=(0:0.01:0.5)/dom; repf=repfreq(h,frq); [Sys2,err]=frep2tf(frq,repf,3,dom); [frq,rep2]=repfreq(Sys2,frq); plot2d(frq, abs([repf;rep2])')

## See also

- imrep2ss
- arl2
- time_id
- armax
- frfit