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sftrans

PURPOSE ^

% error: [Sz, Sp, Sg] = sftrans(Sz, Sp, Sg, W, stop)

SYNOPSIS ^

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

DESCRIPTION ^

% error: [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).
%
% 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 � 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
% ----------------  -------------------------  ------------------------
%
% 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.

CROSS-REFERENCE INFORMATION ^

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