## 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: [Zz, Zp, Zg] = bilinear(Sz, Sp, Sg, T) ## [Zb, Za] = bilinear(Sb, Sa, T) ## ## Transform a s-plane filter specification into a z-plane ## specification. Filters can be specified in either zero-pole-gain or ## transfer function form. The input form does not have to match the ## output form. T is the sampling frequency represented in the z plane. ## ## Theory: Given a piecewise flat filter design, you can transform it ## from the s-plane to the z-plane while maintaining the band edges by ## means of the bilinear transform. This maps the left hand side of the ## s-plane into the interior of the unit circle. The mapping is highly ## non-linear, so you must design your filter with band edges in the ## s-plane positioned at 2/T tan(w*T/2) so that they will be positioned ## at w after the bilinear transform is complete. ## ## The following table summarizes the transformation: ## ## Transform Zero at x Pole at x ## ---------------- ------------------------- ------------------------ ## 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 ## ---------------- ------------------------- ------------------------ ## ## With 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(Z)=g prod(Z-Xi)/prod(Z-Xj) ## ## Please note that a pole and a zero at the same place exactly cancel. ## This is significant since the bilinear transform creates 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. 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 function [Zz, Zp, Zg] = bilinear(Sz, Sp, Sg, T) if nargin==3 T = Sg; [Sz, Sp, Sg] = tf2zp(Sz, Sp); elseif nargin!=4 usage("[Zz, Zp, Zg]=bilinear(Sz,Sp,Sg,T) or [Zb, Za]=blinear(Sb,Sa,T)"); end; p = length(Sp); z = length(Sz); if z > p || p==0 error("bilinear: must have at least as many poles as zeros in s-plane"); end ## ---------------- ------------------------- ------------------------ ## 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 ## ---------------- ------------------------- ------------------------ Zg = real(Sg * prod((2-Sz*T)/T) / prod((2-Sp*T)/T)); Zp = (2+Sp*T)./(2-Sp*T); if isempty(Sz) Zz = -ones(size(Zp)); else Zz = [(2+Sz*T)./(2-Sz*T)]; Zz = postpad(Zz, p, -1); end if nargout==2, [Zz, Zp] = zp2tf(Zz, Zp, Zg); endif endfunction

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