octave-signal 1.3.0-1 / usr / share / octave / packages / signal-1.3.0 / sftrans.m

## Copyright (C) 1999-2001 Paul Kienzle <pkienzle@users.sf.net>
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {[@var{Sz}, @var{Sp}, @var{Sg}] =} sftrans (@var{Sz}, @var{Sp}, @var{Sg}, @var{W}, @var{stop})
##
## Transform band edges of a generic lowpass filter (cutoff at W=1)
## represented in splane zero-pole-gain form.  W is the edge of the
## target filter (or edges if band pass or band stop). Stop is true for
## high pass and band stop filters or false for low pass and band pass
## filters. Filter edges are specified in radians, from 0 to pi (the
## nyquist frequency).
##
## Theory: Given a low pass filter represented by poles and zeros in the
## splane, you can convert it to a low pass, high pass, band pass or
## band stop by transforming each of the poles and zeros individually.
## The following table summarizes the transformation:
##
## @example
## Transform         Zero at x                  Pole at x
## ----------------  -------------------------  ------------------------
## Low Pass          zero: Fc x/C               pole: Fc x/C
## S -> C S/Fc       gain: C/Fc                 gain: Fc/C
## ----------------  -------------------------  ------------------------
## High Pass         zero: Fc C/x               pole: Fc C/x
## S -> C Fc/S       pole: 0                    zero: 0
##                   gain: -x                   gain: -1/x
## ----------------  -------------------------  ------------------------
## Band Pass         zero: b +- sqrt(b^2-FhFl)  pole: b +- sqrt(b^2-FhFl)
##        S^2+FhFl   pole: 0                    zero: 0
## S -> C --------   gain: C/(Fh-Fl)            gain: (Fh-Fl)/C
##        S(Fh-Fl)   b=x/C (Fh-Fl)/2            b=x/C (Fh-Fl)/2
## ----------------  -------------------------  ------------------------
## Band Stop         zero: b +- sqrt(b^2-FhFl)  pole: b +- sqrt(b^2-FhFl)
##        S(Fh-Fl)   pole: +-sqrt(-FhFl)        zero: +-sqrt(-FhFl)
## S -> C --------   gain: -x                   gain: -1/x
##        S^2+FhFl   b=C/x (Fh-Fl)/2            b=C/x (Fh-Fl)/2
## ----------------  -------------------------  ------------------------
## Bilinear          zero: (2+xT)/(2-xT)        pole: (2+xT)/(2-xT)
##      2 z-1        pole: -1                   zero: -1
## S -> - ---        gain: (2-xT)/T             gain: (2-xT)/T
##      T z+1
## ----------------  -------------------------  ------------------------
## @end example
##
## where C is the cutoff frequency of the initial lowpass filter, Fc is
## the edge of the target low/high pass filter and [Fl,Fh] are the edges
## of the target band pass/stop filter.  With abundant tedious algebra,
## you can derive the above formulae yourself by substituting the
## transform for S into H(S)=S-x for a zero at x or H(S)=1/(S-x) for a
## pole at x, and converting the result into the form:
##
## @example
##    H(S)=g prod(S-Xi)/prod(S-Xj)
## @end example
##
## The transforms are from the references.  The actual pole-zero-gain
## changes I derived myself.
##
## Please note that a pole and a zero at the same place exactly cancel.
## This is significant for High Pass, Band Pass and Band Stop filters
## which create numerous extra poles and zeros, most of which cancel.
## Those which do not cancel have a "fill-in" effect, extending the
## shorter of the sets to have the same number of as the longer of the
## sets of poles and zeros (or at least split the difference in the case
## of the band pass filter).  There may be other opportunistic
## cancellations but I will not check for them.
##
## Also note that any pole on the unit circle or beyond will result in
## an unstable filter.  Because of cancellation, this will only happen
## if the number of poles is smaller than the number of zeros and the
## filter is high pass or band pass.  The analytic design methods all
## yield more poles than zeros, so this will not be a problem.
##
## References:
##
## Proakis & Manolakis (1992). Digital Signal Processing. New York:
## Macmillan Publishing Company.
## @end deftypefn

function [Sz, Sp, Sg] = sftrans(Sz, Sp, Sg, W, stop)

  if (nargin != 5)
    print_usage;
  endif

  C = 1;
  p = length(Sp);
  z = length(Sz);
  if z > p || p == 0
    error("sftrans: must have at least as many poles as zeros in s-plane");
  endif

  if length(W)==2
    Fl = W(1);
    Fh = W(2);
    if stop
## ----------------  -------------------------  ------------------------
## Band Stop         zero: b ± sqrt(b^2-FhFl)   pole: b ± sqrt(b^2-FhFl)
##        S(Fh-Fl)   pole: ±sqrt(-FhFl)         zero: ±sqrt(-FhFl)
## S -> C --------   gain: -x                   gain: -1/x
##        S^2+FhFl   b=C/x (Fh-Fl)/2            b=C/x (Fh-Fl)/2
## ----------------  -------------------------  ------------------------
      if (isempty(Sz))
        Sg = Sg * real (1./ prod(-Sp));
      elseif (isempty(Sp))
        Sg = Sg * real(prod(-Sz));
      else
        Sg = Sg * real(prod(-Sz)/prod(-Sp));
      endif
      b = (C*(Fh-Fl)/2)./Sp;
      Sp = [b+sqrt(b.^2-Fh*Fl), b-sqrt(b.^2-Fh*Fl)];
      extend = [sqrt(-Fh*Fl), -sqrt(-Fh*Fl)];
      if isempty(Sz)
        Sz = [extend(1+rem([1:2*p],2))];
      else
        b = (C*(Fh-Fl)/2)./Sz;
        Sz = [b+sqrt(b.^2-Fh*Fl), b-sqrt(b.^2-Fh*Fl)];
        if (p > z)
          Sz = [Sz, extend(1+rem([1:2*(p-z)],2))];
        endif
      endif
    else
## ----------------  -------------------------  ------------------------
## Band Pass         zero: b ± sqrt(b^2-FhFl)   pole: b ± sqrt(b^2-FhFl)
##        S^2+FhFl   pole: 0                    zero: 0
## S -> C --------   gain: C/(Fh-Fl)            gain: (Fh-Fl)/C
##        S(Fh-Fl)   b=x/C (Fh-Fl)/2            b=x/C (Fh-Fl)/2
## ----------------  -------------------------  ------------------------
      Sg = Sg * (C/(Fh-Fl))^(z-p);
      b = Sp*((Fh-Fl)/(2*C));
      Sp = [b+sqrt(b.^2-Fh*Fl), b-sqrt(b.^2-Fh*Fl)];
      if isempty(Sz)
        Sz = zeros(1,p);
      else
        b = Sz*((Fh-Fl)/(2*C));
        Sz = [b+sqrt(b.^2-Fh*Fl), b-sqrt(b.^2-Fh*Fl)];
        if (p>z)
          Sz = [Sz, zeros(1, (p-z))];
        endif
      endif
    endif
  else
    Fc = W;
    if stop
## ----------------  -------------------------  ------------------------
## High Pass         zero: Fc C/x               pole: Fc C/x
## S -> C Fc/S       pole: 0                    zero: 0
##                   gain: -x                   gain: -1/x
## ----------------  -------------------------  ------------------------
      if (isempty(Sz))
        Sg = Sg * real (1./ prod(-Sp));
      elseif (isempty(Sp))
        Sg = Sg * real(prod(-Sz));
      else
        Sg = Sg * real(prod(-Sz)/prod(-Sp));
      endif
      Sp = C * Fc ./ Sp;
      if isempty(Sz)
        Sz = zeros(1,p);
      else
        Sz = [C * Fc ./ Sz];
        if (p > z)
          Sz = [Sz, zeros(1,p-z)];
        endif
      endif
    else
## ----------------  -------------------------  ------------------------
## Low Pass          zero: Fc x/C               pole: Fc x/C
## S -> C S/Fc       gain: C/Fc                 gain: Fc/C
## ----------------  -------------------------  ------------------------
      Sg = Sg * (C/Fc)^(z-p);
      Sp = Fc * Sp / C;
      Sz = Fc * Sz / C;
    endif
  endif

endfunction