frep2tf
=======

transfer function realization from frequency response



Calling Sequence
~~~~~~~~~~~~~~~~


::

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




See Also
~~~~~~~~


+ `imrep2ss`_ state-space realization of an impulse response
+ `arl2`_ SISO model realization by L2 transfer approximation
+ `time_id`_ SISO least square identification
+ `armax`_ armax identification
+ `frfit`_ frequency response fit


.. _frfit: frfit.html
.. _imrep2ss: imrep2ss.html
.. _arl2: arl2.html
.. _time_id: time_id.html
.. _armax: armax.html