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sftrans.m

## Copyright (C) 1999 Paul Kienzle
##
## 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 2 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, write to the Free Software
## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

## usage: [Sz, Sp, Sg] = sftrans(Sz, Sp, Sg, W, 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).  Warning: gain Sg is returned as a complex value
## which may have some residual imaginary component. This matters for
## poles at extreme values in the s-plane, such as those created by odd
## order chebyshev type II low pass filters.
##
## 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:
##
## 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 ħ sqrt(b^2-FhFl)   pole: b ħ 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 ħ sqrt(b^2-FhFl)   pole: b ħ sqrt(b^2-FhFl)
##        S(Fh-Fl)   pole: ħsqrt(-FhFl)         zero: ħsqrt(-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
## ----------------  -------------------------  ------------------------
##
## 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:
##
##    H(S)=g prod(S-Xi)/prod(S-Xj)
##
## 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.

## Author: pkienzle@cs.indiana.edu

## 2000-03 pkienzle@kienzle.powernet.co.uk
##       leave transformed Sg as a complex value since cheby2 blows up
##       otherwise (but only for odd-order low-pass filters).  bilinear
##       will return Zg as real, so there is no visible change to the
##       user of the IIR filter design functions.

## TODO: Shoul return real(Sg) rather than complex.  Must fix cheby2 to be
## TODO:    more stable first
function [Sz, Sp, Sg] = sftrans(Sz, Sp, Sg, W, stop)

  if (nargin != 5)
    usage("[Sz, Sp, Sg] = sftrans(Sz, Sp, Sg, W, stop)");
  end;

  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");
  end

  if length(W)==2
    Fl = W(1);
    Fh = W(2);
    if stop
## ----------------  -------------------------  ------------------------
## Band Stop         zero: b ħ sqrt(b^2-FhFl)   pole: b ħ sqrt(b^2-FhFl)
##        S(Fh-Fl)   pole: ħsqrt(-FhFl)         zero: ħsqrt(-FhFl)
## S -> C --------   gain: -x                   gain: -1/x
##        S^2+FhFl   b=C/x (Fh-Fl)/2            b=C/x (Fh-Fl)/2
## ----------------  -------------------------  ------------------------
      Sg = Sg * prod(-Sz)/prod(-Sp);
      b = (C*(Fh-Fl)/2)./Sp;
      Sp = ;
      extend = ;
      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))];
      end
      end
    else
## ----------------  -------------------------  ------------------------
## Band Pass         zero: b ħ sqrt(b^2-FhFl)   pole: b ħ 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 = ;
      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))];
      end
      end
    end
  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
## ----------------  -------------------------  ------------------------
      Sg = Sg * prod(-Sz)/prod(-Sp);
      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)];
      end
      end
    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;
    end
  end
endfunction
      

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