Scilab Function
Last update : 15/01/2004

frep2tf - transfer function realization from frequency response

### Calling Sequence

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

### Parameters

- `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
    xbasc();bode(frq,[rep;rep2])   //Responses of Sys and Sys2
    [sort(spec(Sys('A'))),sort(roots(Sys2('den')))] //poles

    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
    xbasc();plot2d1("onn",frq',abs([repf;rep2])');

### See Also

`imrep2ss`, `arl2`, `time_id`, `armax`, `frfit`,