bilinear.m

GPS多路径效应的谱分析工具

function [zd, pd, kd, dd] = bilinear(z, p, k, fs, fp, fp1)
%BILINEAR Bilinear transformation with optional frequency prewarping.
%   [Zd,Pd,Kd] = BILINEAR(Z,P,K,Fs) converts the s-domain transfer
%   function specified by Z, P, and K to a z-transform discrete
%   equivalent obtained from the bilinear transformation:
%
%      H(z) = H(s) |
%                  | s = 2*Fs*(z-1)/(z+1)
%
%   where column vectors Z and P specify the zeros and poles, scalar
%   K specifies the gain, and Fs is the sample frequency in Hz.
%   [NUMd,DENd] = BILINEAR(NUM,DEN,Fs), where NUM and DEN are 
%   row vectors containing numerator and denominator transfer
%   function coefficients, NUM(s)/DEN(s), in descending powers of
%   s, transforms to z-transform coefficients NUMd(z)/DENd(z).
%   [Ad,Bd,Cd,Dd] = BILINEAR(A,B,C,D,Fs) is a state-space version.
%   Each of the above three forms of BILINEAR accepts an optional
%   additional input argument that specifies prewarping. For example,
%   [Zd,Pd,Kd] = BILINEAR(Z,P,K,Fs,Fp) applies prewarping before
%   the bilinear transformation so that the frequency responses
%   before and after mapping match exactly at frequency point Fp
%   (match point Fp is specified in Hz).
%
%   See also IMPINVAR.

%   Author(s): J.N. Little, 4-28-87 
%   	   J.N. Little, 5-5-87, revised

%   Gene Franklin, Stanford Univ., motivated the state-space
%   approach to the bilinear transformation.

[mn,nn] = size(z);
[md,nd] = size(p);

if (nd == 1 & nn < 2) & nargout ~= 4	% In zero-pole-gain form
	if mn > md
		error('Numerator cannot be higher order than denominator.')
	end
	if nargin == 5		% Prewarp
		fp = 2*pi*fp;
		fs = fp/tan(fp/fs/2);
	else
		fs = 2*fs;
	end
	z = z(finite(z));	 % Strip infinities from zeros
	pd = (1+p/fs)./(1-p/fs); % Do bilinear transformation
	zd = (1+z/fs)./(1-z/fs);
% real(kd) or just kd?
	kd = (k*prod(fs-z)./prod(fs-p));
	zd = [zd;-ones(length(pd)-length(zd),1)];  % Add extra zeros at -1

elseif (md == 1 & mn == 1) | nargout == 4 %
	if nargout == 4		% State-space case
		a = z; b = p; c = k; d = fs; fs = fp;
		error(abcdchk(a,b,c,d));
		if nargin == 6			% Prewarp
			fp = fp1;		% Decode arguments
			fp = 2*pi*fp;
			fs = fp/tan(fp/fs/2)/2;
		end
	else			% Transfer function case
		if nn > nd
			error('Numerator cannot be higher order than denominator.')
		end
		num = z; den = p;		% Decode arguments
		if nargin == 4			% Prewarp
			fp = fs; fs = k;	% Decode arguments
			fp = 2*pi*fp;
			fs = fp/tan(fp/fs/2)/2;
		else
			fs = k;			% Decode arguments
		end
		% Put num(s)/den(s) in state-space canonical form.  
		[a,b,c,d] = tf2ss(num,den);
	end
	% Now do state-space version of bilinear transformation:
	t = 1/fs;
	r = sqrt(t);
	t1 = eye(size(a)) + a*t/2;
	t2 = eye(size(a)) - a*t/2;
	ad = t2\t1;
	bd = t/r*(t2\b);
	cd = r*c/t2;
	dd = c/t2*b*t/2 + d;
	if nargout == 4
		zd = ad; pd = bd; kd = cd;
	else
		% Convert back to transfer function form:
		p = poly(ad);
		zd = poly(ad-bd*cd)+(dd-1)*p;
		pd = p;
	end
else
	error('First two arguments must have the same orientation.')
end