%%Problem A.1
u=@(t) 1.0*(t>=0);
x=@(t) u(t)-u(t-10);
h=@(t) u(t)-u(t-10);
dtau = 0.005; tau = 0:dtau:20;
ti = 0; tvec = 0:0.01:20;
z = 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);
    z(ti) = sum(xh.*dtau); % Trapezoidal approximation of convolution integral
end
    plot(tvec, z)
    title('z(t)=x(t)*x(t)');
    xlabel('t');
    ylabel('z(t)');
    figure;

%%Problem A.2
N = 100; PulseWidth = 10;
t = [0:1:(N-1)];
X = [ones(1,PulseWidth), zeros(1,N-PulseWidth)];
stairs(t,X); grid on; axis([-10,110,-0.1,1.1])
title('x(t)=pulse with width of 10');
xlabel('t');
ylabel('x(t)');
Xf=fft(X);
Zf=Xf.*Xf;
%%Problem A.3
figure;
f=[-(N/2):1:(N/2)-1]*(1/N);
w=2*pi*f;
subplot(211); plot(f,fftshift( abs(Zf))); grid on;
title('|Z(f)|');
xlabel('f');
ylabel('|Z(f)|');
subplot(212); plot(f,fftshift(angle(Zf))); grid on;
title('phase of Z(f)');
xlabel('f');
ylabel('phase of Z(f)');

%%problem A.4
figure;
N = 100; PulseWidth = 10;
t = [0:1:(N-1)];
x = [ones(1,PulseWidth), zeros(1,N-PulseWidth)];
f = [-(N/2):1:(N/2)-1]*(1/N);
z=conv(x,x);
T=[0:1:(2*N)-2];
plot(T,z)
axis([0 N -2 10])
title('time domain operation');
xlabel('f');
ylabel('z(t)');

figure;
xF=fft(x);
zF=xF.*xF;
Z=ifft(zF);
plot(t,Z)
title('frequency domain operation');
xlabel('f');
ylabel('z(t)');

%the Plots shown using the time-domain and frequency are the
%exact same, as was predicted by the property of the fourier transform
%which states the multiplication in the frequency domain is the same as
%convolution in the time domain

%% problem A.5
N = 100; PulseWidth = 5;
t = [0:1:(N-1)];
X = [ones(1,PulseWidth), zeros(1,N-PulseWidth)];
Zf=fft(X);
f=[-(N/2):1:(N/2)-1]*(1/N);
w=2*pi*f;
figure;
subplot(211); plot(f,fftshift( abs(Zf))); grid on;
title('magnitude of frequencies of x(t) with a width of 5');
xlabel('f');
ylabel('x(f)');
subplot(212); plot(f,fftshift(angle(Zf))); grid on;
title('phase plot of x(t) with a width of 5');
xlabel('f');
ylabel('x(f)');
figure;

N = 100; PulseWidth = 25;
t = [0:1:(N-1)];
X = [ones(1,PulseWidth), zeros(1,N-PulseWidth)];
Zf=fft(X);
f=[-(N/2):1:(N/2)-1]*(1/N);
w=2*pi*f;
subplot(211); plot(f,fftshift( abs(Zf))); grid on;
title('magnitude of frequencies of x(t) with a width of 25');
xlabel('f');
ylabel('x(f)');
subplot(212); plot(f,fftshift(angle(Zf))); grid on;
title('phase plot of x(t) with a width of 25');
xlabel('f');
ylabel('x(f)');
figure;

%the magnitude of frequency plot it scaled according to the
%time scale. in addition, the frequency plot of phase is also
%scaled proportional to the time scale.
%this is the same as the property which states: a time scale of a
%will be :(1/|a|)X(w/a)

%%problem A.6
N = 100; PulseWidth = 5;
t = [0:1:(N-1)];
X = [ones(1,PulseWidth), zeros(1,N-PulseWidth)];
f = [-(N/2):1:(N/2)-1]*(1/N);
ex1=exp(i*(pi/3).*t);
wplus=fft(ex1.*X);
subplot(211); plot(f,fftshift( abs(wplus))); grid on;
title('magnitude plot of x(t)*e^{j*(pi/3)*t}');
xlabel('f');
ylabel('|z(f)| with shift');
subplot(212); plot(f,fftshift(angle(wplus))); grid on;
title('phase plot of x(t)*e^{j*(pi/3)*t}');
xlabel('f');
ylabel('angle of z(f) with shift');

figure;
ex2=exp(-i*(pi/3).*t);
wminus=fft(ex2.*X);
subplot(211); plot(f,fftshift( abs(wminus))); grid on;
title('magnitude plot of x(t)*e^{-j*(pi/3)*t}');
xlabel('f');
ylabel('|z(f)| with shift');
subplot(212); plot(f,fftshift(angle(wminus))); grid on;
title('phase plot of x(t)*e^{-j*(pi/3)*t}');
xlabel('f');
ylabel('angle of z(f) with shift');

figure;
thing=cos(t*pi/3);
wc=fft(thing.*X);
subplot(211); plot(f,fftshift( abs(wc))); grid on;
title('magnitude plot of x(t)*cos((pi/3)*t)');
xlabel('f');
ylabel('');
subplot(212); plot(f,fftshift(angle(wc))); grid on;
title('phase plot of x(t)*cos((pi/3)*t)');
xlabel('f');
ylabel('');

%this final property is an example of a shift in the frequency domain
%if you multiply a function with a complex exponential, that corresponds to
%a horizontal shift in the frequency domain. this is clearly seen in the
%first 2 graphs for A.6. the last graph is just the real component of a
%complex exponential, which is why it is not a perfect shift.