butter.m

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function [num, den, z, p] = butter(n, Wn, varargin)
%BUTTER Butterworth digital and analog filter design.
%   [B,A] = BUTTER(N,Wn) designs an Nth order lowpass digital
%   Butterworth filter and returns the filter coefficients in length 
%   N+1 vectors B (numerator) and A (denominator). The coefficients 
%   are listed in descending powers of z. The cut-off frequency 
%   Wn must be 0.0 < Wn < 1.0, with 1.0 corresponding to 
%   half the sample rate.
%
%   If Wn is a two-element vector, Wn = [W1 W2], BUTTER returns an 
%   order 2N bandpass filter with passband  W1 < W < W2.
%   [B,A] = BUTTER(N,Wn,'high') designs a highpass filter.
%   [B,A] = BUTTER(N,Wn,'stop') is a bandstop filter if Wn = [W1 W2].
%   
%   When used with three left-hand arguments, as in
%   [Z,P,K] = BUTTER(...), the zeros and poles are returned in
%   length N column vectors Z and P, and the gain in scalar K. 
%
%   When used with four left-hand arguments, as in
%   [A,B,C,D] = BUTTER(...), state-space matrices are returned.
%
%   BUTTER(N,Wn,'s'), BUTTER(N,Wn,'high','s') and BUTTER(N,Wn,'stop','s')
%   design analog Butterworth filters.  In this case, Wn can be bigger
%   than 1.0.
%
%   See also BUTTORD, BESSELF, CHEBY1, CHEBY2, ELLIP, FREQZ, FILTER.

%   Author(s): J.N. Little, 1-14-87
%   	   J.N. Little, 1-14-88, revised
%   	   L. Shure, 4-29-88, revised
%   	   T. Krauss, 3-24-93, revised
%   Copyright (c) 1988-98 by The MathWorks, Inc.
%   $Revision: 1.22 $  $Date: 1997/12/02 18:36:25 $

%   References:
%     [1] T. W. Parks and C. S. Burrus, Digital Filter Design,
%         John Wiley & Sons, 1987, chapter 7, section 7.3.3.

[btype,analog,errStr] = iirchk(Wn,varargin{:});
error(errStr)

if n>500
	error('Filter order too large.')
end

% step 1: get analog, pre-warped frequencies
if ~analog,
	fs = 2;
	u = 2*fs*tan(pi*Wn/fs);
else
	u = Wn;
end

Bw=[];
% step 2: convert to low-pass prototype estimate
if btype == 1	% lowpass
	Wn = u;
elseif btype == 2	% bandpass
	Bw = u(2) - u(1);
	Wn = sqrt(u(1)*u(2));	% center frequency
elseif btype == 3	% highpass
	Wn = u;
elseif btype == 4	% bandstop
	Bw = u(2) - u(1);
	Wn = sqrt(u(1)*u(2));	% center frequency
end

% step 3: Get N-th order Butterworth analog lowpass prototype
[z,p,k] = buttap(n);

% Transform to state-space
[a,b,c,d] = zp2ss(z,p,k);

% step 4: Transform to lowpass, bandpass, highpass, or bandstop of desired Wn
if btype == 1		% Lowpass
	[a,b,c,d] = lp2lp(a,b,c,d,Wn);

elseif btype == 2	% Bandpass
	[a,b,c,d] = lp2bp(a,b,c,d,Wn,Bw);

elseif btype == 3	% Highpass
	[a,b,c,d] = lp2hp(a,b,c,d,Wn);

elseif btype == 4	% Bandstop
	[a,b,c,d] = lp2bs(a,b,c,d,Wn,Bw);
end

% step 5: Use Bilinear transformation to find discrete equivalent:
if ~analog,
	[a,b,c,d] = bilinear(a,b,c,d,fs);
end

if nargout == 4
	num = a;
	den = b;
	z = c;
	p = d;
else	% nargout <= 3
% Transform to zero-pole-gain and polynomial forms:
	if nargout == 3
		[z,p,k] = ss2zp(a,b,c,d,1);
		z = buttzeros(btype,n,Wn,Bw,analog);
		num = z;
		den = p;
		z = k;
	else % nargout <= 2
		den = poly(a);
		num = buttnum(btype,n,Wn,Bw,analog,den);
		% num = poly(a-b*c)+(d-1)*den;

	end
end

%---------------------------------
function b = buttnum(btype,n,Wn,Bw,analog,den)
% This internal function returns more exact numerator vectors
% for the num/den case.
% Wn input is two element band edge vector
if analog
    switch btype
    case 1  % lowpass
        b = [zeros(1,n) n^(-n)];
        b = real( b*polyval(den,-j*0)/polyval(b,-j*0) );
    case 2  % bandpass
        b = [zeros(1,n) Bw^n zeros(1,n)];
        b = real( b*polyval(den,-j*Wn)/polyval(b,-j*Wn) );
    case 3  % highpass
        b = [1 zeros(1,n)];
        b = real( b*den(1)/b(1) );
    case 4  % bandstop
        r = j*Wn*((-1).^(0:2*n-1)');
        b = poly(r);
        b = real( b*polyval(den,-j*0)/polyval(b,-j*0) );
    end
else
    Wn = 2*atan2(Wn,4);
    switch btype
    case 1  % lowpass
        r = -ones(n,1);
        w = 0;
    case 2  % bandpass
        r = [ones(n,1); -ones(n,1)];
        w = Wn;
    case 3  % highpass
        r = ones(n,1);
        w = pi;
    case 4  % bandstop
        r = exp(j*Wn*( (-1).^(0:2*n-1)' ));
        w = 0;
    end
    b = poly(r);
    % now normalize so |H(w)| == 1:
    kern = exp(-j*w*(0:length(b)-1));
    b = real(b*(kern*den(:))/(kern*b(:)));
end

function z = buttzeros(btype,n,Wn,Bw,analog)
% This internal function returns more exact zeros.
% Wn input is two element band edge vector
if analog
    % for lowpass and bandpass, don't include zeros at +Inf or -Inf
    switch btype
    case 1  % lowpass
        z = zeros(0,1);
    case 2  % bandpass
        z = zeros(n,1);
    case 3  % highpass
        z = zeros(n,1);
    case 4  % bandstop
        z = j*Wn*((-1).^(0:2*n-1)');
    end
else
    Wn = 2*atan2(Wn,4);
    switch btype
    case 1  % lowpass
        z = -ones(n,1);
    case 2  % bandpass
        z = [ones(n,1); -ones(n,1)];
    case 3  % highpass
        z = ones(n,1);
    case 4  % bandstop
        z = exp(j*Wn*( (-1).^(0:2*n-1)' ));
    end
end