function [sys, Gp] = mapCtoD(sys_c,t,f0)%[sys, Gp] = mapCtoD(sys_c,t=[0 1],f0=0)%Map a MIMO continuous-time system to a SIMO discrete-time equivalent.%The criterion for equivalence is that the sampled pulse response%of the CT system must be identical to the impulse response of the DT system.%i.e. If yc is the output of the CT system with an input vc taken%from a set of DACs fed with a single DT input v, then y, the output%of the equivalent DT system with input v satisfies%y(n) = yc(n-) for integer n. The DACs are characterized by%rectangular impulse responses with edge times specified in the t matrix.%%Input% sys_c	The LTI description of the CT system.% t	The edge times of the DAC pulse used to make CT waveforms %	from DT inputs. Each row corresponds to one of the system%       inputs; [-1 -1] denotes a CT input. The default is [0 1],%       for all inputs except the first.% f0	The (normalized) frequency at which the Gp filters' gains are%	to be set to unity. Default 0 (DC).%%Output% sys	The LTI description for the DT equivalent% Gp	The mixed CT/DT prefilters which form the samples %	fed to each state for the CT inputs.%% Reference% R. Schreier and B. Zhang, "Delta-sigma modulators employing continuous-time% circuitry," IEEE Transactions on Circuits and Systems I, vol. 43, no. 4,% pp. 324-332, April 1996.% Handle the input argumentsif nargin <1    error('Insufficient arguments');endparameters = {'sys_c' 't' 'f0'};defaults = { NaN, [0 1], 0 };for i=1:length(defaults)    parameter = char(parameters(i));    if i>nargin | ( eval(['isnumeric(' parameter ') '])  &  ...     eval(['any(isnan(' parameter ')) | isempty(' parameter ') ']) )        eval([parameter '=defaults{i};'])    endendni = size(sys_c.b,2);if all(size(t) == [1 2])  &  (ni > 1)    t = [-1 -1; ones(ni-1,1)*t];endif any(size(t) ~= [ni 2])    error('The t argument has the wrong dimensions.');enddi = ones(1,ni);for i = 1:ni    if t(i,:)==[-1 -1]	di(i) = 0;    endenddi = logical(di);% c2d assumes t1=0, t2=1.% Also c2d often complains about poor scaling and can even produce % incorrect results.v = version;if v(1) >= '7'	set(sys_c,'InputDelay',0);endsys = c2d(sys_c,1,'zoh');Ac = sys_c.a;Bc1 = sys_c.b(:,~di);A = sys.a;	B = sys.b;	C = sys.c;	D = sys.d;B1 = B(:,~di);	D1 = D(:,~di);% Examine the discrete-time inputs to see how big the % augmented matrices need to be.n = size(A,1);t2 = ceil(t(di,2));esn = (t2 == t(di,2)) & (D(1,di)~=0)';	% extra states needed?np = n + max(t2-1+esn);% Augment A to np x np, B to np x 1, C to 1 x np.Ap = padb( padr(A,np), np );for i=n+2:np    Ap(i,i-1) = 1;endBp = zeros(np,1);if np>n    Bp(n+1)=1;endCp = padr(C,np);Dp = 0;% Add in the contributions from each DACfor i = find(di)    t1 = t(i,1);    t2 = t(i,2);    B2 = B(:,i);    D2 = D(:,i);    if t1==0 & t2==1 & D2==0	% No fancy stuff necessary	Bp = Bp + padb(B2,np);    else	n1 = floor(t1);	n2 = ceil(t2)-n1-1;	t1 = t1-n1;	t2 = t2-n2-n1;	if t2==1 & D2~=0	    n2 = n2+1;	    extraStateNeeded = 1;	else	    extraStateNeeded = 0;	end	nt = n+n1+n2;	if n2>0 & t2~=1	    Ap(1:n,nt) = Ap(1:n,nt) + B2formula(Ac,0,t2,B2);	end	if n2>0			% pulse extends to the next period	    Btmp = B2formula(Ac,t1,1,B2);	else			% pulse ends in this period	    Btmp = B2formula(Ac,t1,t2,B2);	end	if n1>0	    Ap(1:n,n+n1) = Ap(1:n,n+n1) + Btmp;	else	    Bp = Bp + padb(Btmp,np);	end	if n2>0	    Cp = Cp + padr([zeros(size(D2,1),n+n1) D2*ones(1,n2)],np);	end    endendsys = ss(Ap,Bp,Cp,Dp,1);if any(di==0)    if nargout>1	% Compute the prefilters and add in the CT feed-ins.	% Gp = inv(sI-Ac) * (zI-A)/z *Bc1	[n m] = size(Bc1);	Gp = cell(n,m);	ztf = cell(size(Bc1));	% !!Make this like stf: an array of zpk objects	% Compute the z-domain portions of the filters	ABc1 = A*Bc1;	for h = 1:m	    for i = 1:n		if Bc1(i,h)==0		    ztf{i,h}= zpk([],0,-ABc1(i,h),1);		else		    ztf{i,h}= zpk(ABc1(i,h)/Bc1(i,h),0,Bc1(i,h),1);		end	    end	end	% Compute the s-domain portions of each of the filters	stf = zpk(ss(Ac,eye(n),eye(n),zeros(n,n)));%Doesn't do pole-zero cancelation	for h=1:m	    for i=1:n		k=1;		for j=1:n		   if stf.k(i,j)~=0 & ztf{j}.k~=0	% A non-zero term		       Gp{i,h}(k).Hs = stf(i,j);		       Gp{i,h}(k).Hz = ztf{j,h};		       k = k+1;		   end		end	    end	end	if f0 ~= 0 	% Need to correct the gain terms calculated by c2d	    %B1 = gains of Gp @f0;	    for h=1:m		for i=1:n		    B1(i,h) = evalMixedTF(Gp{i,h},f0);		    % abs() used because ss() whines if B has complex entries...		    % This is clearly incorrect.		    % I've fudged the complex stuff by including a sign....		    B1(i,h) = abs(B1(i,h)) * sign(real(B1(i,h)));		    if abs(B1(i,h)) < 1e-9			B1(i,h) = 1e-9;	% This prevents NaN in line 174		    end		end	    end	end	% Adjust the gains of the pre-filters	for h=1:m	    for i=1:n		for j=1:length(Gp{i,h})		    Gp{i,h}(j).Hs.k = Gp{i,h}(j).Hs.k/B1(i,h);	%This is line 174		end	    end	end	set(sys,'b',[padb(B1,np) sys.b],'d',[D1 sys.d]);    else % Cheat and just dublicate the B1 terms	set(sys,'b',[padb(Bc1,np) sys.b],'d',[D1 sys.d]);    endendfunction term = B2formula(Ac,t1,t2,B2);    if t1==0 & t2==0	term = B2;	return;    end    n = size(Ac,1);    tmp = eye(n)-expm(-Ac);    if cond(tmp) < 1e6	term = ( (expm(-Ac*t1)-expm(-Ac*t2)) * inv(tmp)) *B2;	return;    end    % Numerical trouble. Perturb slightly and check the result    ntry = 0;    k = sqrt(eps);    Ac0 = Ac;    while ntry <2	Ac = Ac0 + k*rand(n,n);	tmp = eye(n)-expm(-Ac);	if cond(tmp) < 1/sqrt(eps) 	    ntry = ntry+1;	    if ntry == 1		term = ( (expm(-Ac*t1)-expm(-Ac*t2)) * inv(tmp)) *B2;	    else		term1 = ( (expm(-Ac*t1)-expm(-Ac*t2)) * inv(tmp)) *B2;	    end	end	k = k*sqrt(2);    end    if norm(term1-term) > 1e-3	fprintf(2, 'Warning: Inaccurate calculation in mapCtoD.\n');    endreturn;