function f = time_constant(T1,T2,k)

%simulate the response of the canonical first order LTI for 2 different
%values of time constants: T1 and T2 (T1<T2)

close all

t = linspace(0,5*T2,10000);


num = [k];


den = [T1 1];
y1 = step(num,den,t);
plot([-1 0 t],[0 0 y1'],'LineWidth',2)
hold on
%dx = y1(2)-y1(1);
plot([0 T1],[0 k],'b--','LineWidth',1)
plot([T1 T1],[k 0],'b--','LineWidth',1)

den = [T2 1];
y2 = step(num,den,t);
plot([-1 0 t],[0 0 y2'],'c','LineWidth',2)
plot([0 T2],[0 k],'c--','LineWidth',1)
plot([T2 T2],[k 0],'c--','LineWidth',1)

plot([-1 0 t],[0 0 ones(1,length(t))],'r','LineWidth',2)

set(gca,'Xlim',[-1 max(t)],'Ylim',sort([0 k+sign(k)*1-eps]),'YTick',sort([0 1-eps k]), ...
    'XTick',[0 T1 T2 2.5*T2 5*T2])
set(gca,'FontSize',16,'FontWeight','b','FontAngle','i')
grid on

xlabel('Time [s]')
th1 = text(6.75,1.25,'u(t)=h(t)','Color','r');
th2 = text(6.75,k*1.05,'y(t)','Color','b');
th3 = text(6.75,k*0.90,'y(t)','Color','c');
th4 = text(8.0,k*1.05,['T = ' num2str(T1)],'Color','b');
th5 = text(8.0,k*0.9,['T = ' num2str(T2)],'Color','c');
set([th1 th2 th3 th4 th5],'FontSize',16,'FontWeight','b','FontAngle','i')