632 lab 4 AB- Final

%% Part A
%problem 1
xn=[1 6/7 5/7 4/7 3/7 2/7 1/7 zeros(1,121)];                               %declaring xn as an array
n=0:127;                                                                   %delclaring n
xf=fft(xn);                                                                %finding DTFT of xn
xf=fftshift(xf);                                                           %shifting xf to be centered at 0
plot(n.*2*pi/128,abs(xf));                                                 %plotting from -pi to pi
title("fft calculated magnitude plot");                                    %setting a title


%problem 2
disp('For A2: The DTFT was calculated using the following equation: Sum (from n=-infinity to infinity) of x[n]*e^(-j*omega*n). The result for our given x[n] is: exp(-0i.*o) + (6/7)*exp(-i.*o) + (5/7)*exp(-2i.*o) + (4/7)*exp(-3i.*o) + (3/7)*exp(-4i.*o) + (2/7)*exp(-5i.*o) + (1/7)*exp(-6i.*o)')
xomega=@(o) exp(-0i.*o) + (6/7)*exp(-i.*o) + (5/7)*exp(-2i.*o) + (4/7)*exp(-3i.*o) + (3/7)*exp(-4i.*o) + (2/7)*exp(-5i.*o) + (1/7)*exp(-6i.*o); %creating the function for the hand calculated DTFT
figure;
omega=n.*2*pi/128;                                                         %setting an index for plotting
xf2=xomega(omega);                                                         %instantiating the function
xf2=fftshift(xf2);                                                         %shifting xf2 to be centered at 0
plot(omega,abs(xf2));                                                      %plotting
title("Manually calculated magnitude plot");                               %setting a title

%this part of the code is creating a graph comparing parts 1 and 2
figure;
plot(omega,abs(xf2));
hold on;
plot(omega,abs(xf),'-.g');
hold off;
title("comparison of manually calculated spectrum and fft spectrum");


%Problem 3
Xn=ifft(fftshift(xf));                                                     %preforming the Inverse DTFT
figure;
stem(n,Xn);                                                                %plotting the result
title("original signal for part 3 ");                                      %giving it a title
disp("Part A3: yes, the original signal is the exact same, due to the fact that we are finding the spectrum of the signal, then without any changes, recovereing the signal using the inverse DTFT");


%% Part B
un= @(n) 1.0.*(n>=0);                                                      %declaring a general step function
und= @(n) un(n).*(mod(n,1)==0);                                            %declaring a discrete step function
n=[0:15];                                                                  %declaring n for making the function
xn=sin(2*pi.*n/10).*(und(n)-und(n-10));                                    %creating the function xn
omega=linspace(-pi,pi,1001);                                               %doing the DTFT
W_omega1=exp(-j).^((0:length(xn)-1)'*omega);
X1=(xn*W_omega1);
plot(omega, abs(X1));                                                      %plotting the DTFT
title("magnitude of x[n] FT");                                             %adding a title
figure;

hn=und(n)-und(n-10);                                                       %creating the function h[n]
W_omega2=exp(-j).^((0:length(hn)-1)'*omega);                               %preforming the DTFT
X2=(hn*W_omega2);
plot(omega, abs(X2));                                                      %plotting the DTFT
title("magnitude of h[n] FT");                                             %adding a title
figure;

yf=X1.*X2;                                                                 %multiplying the spectrum of xn and hn
plot(omega,abs(yf));                                                       %plotting the result
title("magnitude of X[\Omega]H[\Omega]");                                  %adding a title
figure;

yn=conv(xn,hn);                                                            %preforming a convolution on the time domain functions of xn and hn
W_omega3=exp(-j).^((0:length(yn)-1)'*omega);                               %preforming the DTFT
X3=(yn*W_omega3);
plot(omega,abs(X3));                                                       %plotting the result
title("magnitude of y[n]");                                                %adding a title

%creating a graph to compart parts 3 and 5
figure;
plot(omega,abs(yf));
hold on;
plot(omega, abs(X3), '-.g');
hold off;
title("comparison of part 3 and part 5");
disp("Part B6 answer: convolution in the time domain is the same thing as multiplication in the frequency domain, so the results of the 2 operations should be the same");