afe

% A1 , A2
R = [1e4, 1e4, 1e4];
C = [1e-6, 1e-6];
%Determine coefficients for characteristic equation:
A = [1, (1/R(1)+1/R(2)+1/R(3))/C(2), 1/(R(1)*R(2)*C(1)*C(2))];
%Determine characteristic roots:
lambda1 = roots(A);
poly(lambda1);

f1 = @(t) exp(lambda1(1)*t) + exp(lambda1(2)*t);
t = (0:0.0005:0.1);
grid;
plot(t,f1(t));

% A3
figure;
R = [1e4, 1e4, 1e4];
C = [1e-9, 1e-6];
%Determine coefficients for characteristic equation:
A = [1, (1/R(1)+1/R(2)+1/R(3))/C(2), 1/(R(1)*R(2)*C(1)*C(2))];
%Determine characteristic roots:
lambda2 = roots(A);
f2 = @(t) exp(lambda2(1)*t) + exp(lambda2(2)*t);
t = (0:0.0005:0.1);
grid;
plot(t,f2(t));

% B1
figure;
figure(1) % Create figure window and make visible on screen
u = @(t) 1.0*(t>=0);
x = @(t) 1.5*sin(pi*t).*(u(t)-u(t-1));
h = @(t) 1.5*(u(t)-u(t-1.5))-u(t-2)+u(t-2.5);
dtau = 0.005; tau = -1:dtau:4;
ti = 0; tvec = -.25:.1:3.75;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x(t-tau).*h(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,h(tau),'k-',tau,x(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end

% B2
figure;
u = @(t) 1.0*(t>=0);
x = @(t) 1*(u(t)-u(t-2));
h = @(t) (t+1).*(u(t+1)-u(t));
dtau = 0.005; tau = -1:dtau:4;
ti = 0; tvec = -.25:.1:3.75;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x(t-tau).*h(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,h(tau),'k-',tau,x(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end

% B3 a)
figure;
u = @(t) 1.0*(t>=0);
x1 = @(t) 1*(u(t-4)-u(t-6));
x2 = @(t) 2*(u(t+5)-u(t+4));
dtau = 0.005; tau = -7:dtau:4;
ti = 0; tvec = -.25:.1:3.75;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x1(t-tau).*x2(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,x2(tau),'k-',tau,x1(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end

% B3 b)
figure;
u = @(t) 1.0*(t>=0);
x1 = @(t) 1*(u(t-3)-u(t-5));
x2 = @(t) 2*(u(t+5)-u(t+3));
dtau = 0.005; tau = -7:dtau:4;
ti = 0; tvec = -.25:.1:3.75;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x1(t-tau).*x2(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,x2(tau),'k-',tau,x1(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end

% B3 h)
figure;
u = @(t) 1.0*(t>=0);
x1 = @(t) exp(t).*(u(t+2)-u(t));
x2 = @(t)  exp(-2*t).*(u(t)-u(t-1));
dtau = 0.005; tau = -3:dtau:4;
ti = 0; tvec = -.25:.1:3.75;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x1(t-tau).*x2(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,x2(tau),'k-',tau,x1(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end

%C1
t = (-1:0.001:5);
figure;
u = @(t) 1.0*(t>=0);
h1 = @(t) exp(t/5).*u(t);
plot(t,h1(t));
xlabel('t');
ylabel('h1(t)');
grid;


figure;
u = @(t) 1.0*(t>=0);
h2 = @(t) 4*exp(-t/5).*u(t);
plot(t,h2(t));
xlabel('t');
ylabel('h2(t)');
grid;

figure;
u = @(t) 1.0*(t>=0);
h3 = @(t) 4*exp(-t).*u(t);
plot(t,h3(t));
xlabel('t');
ylabel('h3(t)');
grid;

figure;
u = @(t) 1.0*(t>=0);
h4 = @(t) 4*(exp(-t/5) - exp(-t)).*u(t);
plot(t,h4(t));
xlabel('t');
ylabel('h4(t)');
grid;

%C2

% h1(t): lambda = 1/5
% h2(t): lambda = -1/5
% h3(t): lambda = -1
% h4(t): lambda = -1/5 , -1

%C3

figure;
u = @(t) 1.0*(t>=0);
h1 = @(t) exp(t/5).*u(t);
x = @(t) u(t) - u(t-3);
dtau = 0.005; tau = 0:dtau:20;
ti = 0; tvec = 0:.1:20;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x(t-tau).*h1(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,h1(tau),'k-',tau,x(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end

figure;
u = @(t) 1.0*(t>=0);
h2 = @(t) 4*exp(-t/5).*u(t);
x = @(t) u(t) - u(t-3);
dtau = 0.005; tau = 0:dtau:20;
ti = 0; tvec = 0:.1:20;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x(t-tau).*h2(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,h2(tau),'k-',tau,x(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end

figure;
u = @(t) 1.0*(t>=0);
h3 = @(t) 4*exp(-t).*u(t);
x = @(t) u(t) - u(t-3);
dtau = 0.005; tau = 0:dtau:20;
ti = 0; tvec = 0:.1:20;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x(t-tau).*h3(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,h3(tau),'k-',tau,x(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end

figure;
u = @(t) 1.0*(t>=0);
h4 = @(t) 4*(exp(-t/5) - exp(-t)).*u(t);
x = @(t) u(t) - u(t-3);
dtau = 0.005; tau = 0:dtau:20;
ti = 0; tvec = 0:.1:20;
y = NaN*zeros(1,length(tvec)); % Pre-allocate memory
for t = tvec,
ti = ti+1; % Time index
xh = x(t-tau).*h4(tau); lxh = length(xh);
y(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
subplot(2,1,1),plot(tau,h4(tau),'k-',tau,x(t-tau),'k--',t,0,'ok');
axis([tau(1) tau(end) -2.0 2.5]);
patch([tau(1:end-1);tau(1:end-1);tau(2:end);tau(2:end)],...
[zeros(1,lxh-1);xh(1:end-1);xh(2:end);zeros(1,lxh-1)],...
[.8 .8 .8],'edgecolor','none');
xlabel('\tau'); title('h(\tau) [solid], x(t-\tau) [dashed], h(\tau)x(t-\tau) [gray]');
c = get(gca,'children'); set(gca,'children',[c(2);c(3);c(4);c(1)]);
subplot(2,1,2),plot(tvec,y,'k',tvec(ti),y(ti),'ok');
xlabel('t'); ylabel('y(t) = \int h(\tau)x(t-\tau) d\tau');
axis([tau(1) tau(end) -1.0 2.0]); grid;
drawnow;
end