update

%3.9(a)

t=linspace(0,20,1000);

x=cos(t);

b=1;

a=[1 1]; % RC=1

y1=lsim(b,a,x,t);

plot(t,x,'r');

hold on;

plot(t,y1,'g');

xlabel('t');

ylabel('x(t) and y(t)');

xlim([10,20]);

legend('x(t)','y(t)');

saveas(gcf, "plots/P3_9_a_out.png");

close;

%3.9(b)

x2=cos(t); 

x2(x2>0)=ones(size(x2(x2>0))); 

x2(x2<0)=-ones(size(x2(x2<0)));

y2=lsim(b,a,x2,t);

plot(t,y2,'g');

xlim([10,20]);

xlabel('t');

ylabel('y_2(t)');

saveas(gcf, "plots/P3_9_b_out.png");

close;

%3.9(c)

plot(t,x2,'g');

hold on;

apos_k=[2/pi -2/(3*pi) 2/(5*pi) -2/(7*pi) 2/(9*pi)];

aneg_k=[2/pi -2/(3*pi) 2/(5*pi) -2/(7*pi) 2/(9*pi)];

s1=apos_k(1)*exp(j*t)+aneg_k(1)*exp(-j*t);

s2=apos_k(2)*exp(3j*t)+aneg_k(2)*exp(-3j*t);

s3=apos_k(3)*exp(5j*t)+aneg_k(3)*exp(-5j*t);

s4=apos_k(4)*exp(7i*t)+aneg_k(4)*exp(-7j*t);

s5=apos_k(5)*exp(9j*t)+aneg_k(5)*exp(-9j*t);

sn=s1+s2+s3+s4+s5;

plot(t,sn,'r');

xlabel('t');

legend('x_2(t)','sum');

saveas(gcf, "plots/P3_9_c_out.png");

close;

%3.9(d)

ys1=lsim(b,a,s1,t);

ys2=lsim(b,a,s2,t);

ys3=lsim(b,a,s3,t);

ys4=lsim(b,a,s4,t);

ys5=lsim(b,a,s5,t);

ysn=lsim(b,a,sn,t);

ysum=ys1+ys2+ys3+ys4+ys5;

subplot(2,1,1);

plot(t,ysum);

xlabel('t');

title('response to sum of y1 to y5');

subplot(2,1,2);

plot(t,ysn);

xlabel('t');

title('response to ssum');

saveas(gcf, "plots/P3_9_d_out.png");

close;

%3.9(e)

ysn=lsim(b,a,sn,t);

y2=lsim(b,a,x2,t);

plot(t,ysn,'r--');

hold on;

plot(t,y2,'g-');

xlabel('t');

title('response to ssum and x_2(t)');

legend('y_ssum(t)','y_2(t)');

saveas(gcf, "plots/P3_9_e_out.png");

close;

%3.9(f)

% y1

y1=(1/(1+j))*apos_k(1)*exp(j*t)+(1/(1+j))*aneg_k(1)*exp(-j*t);

ys1=lsim(b,a,s1,t);

subplot(5,2,1);

plot(t,y1)

xlim([10,20]);

xlabel('t');

title('analytically determined y1');

subplot(5,2,2);

plot(t,ys1)

xlim([10,20]);

xlabel('t');

title('simulated y1');

% y2

y2=(1/(1+3j))*apos_k(2)*exp(3j*t)+(1/(1-3j))*aneg_k(2)*exp(-3j*t);

ys2=lsim(b,a,s2,t);

subplot(5,2,3);

plot(t,y2)

xlim([10,20]);

xlabel('t');

title('analytically determined y2');

subplot(5,2,4);

plot(t,ys2)

xlim([10,20]);

xlabel('t');

title('simulated y2');

% y3

y3=(1/(1+5j))*apos_k(3)*exp(5j*t)+(1/(1-5j))*aneg_k(3)*exp(-5j*t);

ys3=lsim(b,a,s3,t);

subplot(5,2,5);

plot(t,y3)

xlim([10,20]);

xlabel('t');

title('analytically determined y3');

subplot(5,2,6);

plot(t,ys3)

xlim([10,20]);

xlabel('t');

title('simulated y3');

% y4

y4=(1/(1+7j))*apos_k(4)*exp(7j*t)+(1/(1-7j))*aneg_k(4)*exp(-7j*t);

ys4=lsim(b,a,s4,t);

subplot(5,2,7);

plot(t,y4)

xlim([10,20]);

xlabel('t');

title('analytically determined y4');

subplot(5,2,8);

plot(t,ys4)

xlim([10,20]);

xlabel('t');

title('simulated y4');

% y5

y5=(1/(1+9j))*apos_k(5)*exp(9j*t)+(1/(1-9j))*aneg_k(5)*exp(-9j*t);

ys5=lsim(b,a,s5,t);

subplot(5,2,9);

plot(t,y5)

xlim([10,20]);

xlabel('t');

title('analytically determined y5');

subplot(5,2,10);

plot(t,ys5)

xlim([10,20]);

xlabel('t');

title('simulated y5');

saveas(gcf, "plots/P3_9_f_out.png");

close;