Question

Hello, I'm taking signal systems course. please solve this question in matlab as soon as possbile please.

Question 1 a) Write a function that calculates the Continuous Time Fourier Transform of a periodic signal x() Syntax: [w, X] = CTFT(t, x) The outputs to the function are: w = the frequencies in rad/s, and X = the continuous Fourier transform of the signal The inputs to the function are: x-one period of the signal x(t), andt the time vector The continuous time Fourier transform of a signal x(t) is given by: b) Write a function that calculates the Inverse Continuous Time Fourier Transform. The inverse continuous-time Fourier transform is given by: Syntax: x-CTFT(t, w, X) The output to the function is: x = the signal x(t) The inputs to the function are: t- the time vector, w the frequencies in rad/s, and X -the continuous Fourier transform c) Test both of your functions CTFT and ICTFT with x(t) =sin(t), where t-50: 0.1: 50, and use subplot to plot each of x(Q.X(o), and the output of ICFT to verify its accuracy. Label all plots and axes appropriately

Answer 1

a)

function [w,X]=CTFT(t,x)
w=-50:.1:50; %create the w axis
for i=1:length(w)
X(i)=0;
for j=1:length(t)-1
X(i)=X(i)+x(j)*exp(-1i*w(i)*t(j))*(t(j+1)-t(j));
end
end

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b)

function x=ICTFT(t,w,X)
for i=1:length(t)
x(i)=0;
for j=1:length(w)-1
x(i)=x(i)+X(j)*exp(1i*w(j)*t(i))*(w(j+1)-w(j));
end
end
x=x/(2*pi);

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c) Run the following code in the MATLAB command window after saving the above two functions

t=-50:0.1:50;
x=sin(t);
[w,X]=CTFT(t,x);
figure;
subplot(3,1,1);plot(w,abs(X),'k');xlabel('\omega');ylabel('Magnitude');title('Magnitude of CTFT');
x1=ICTFT(t,w,X);
subplot(3,1,2);plot(t,x,'k');xlabel('time');ylabel('value');title('Time Domain Signal');
subplot(3,1,3);plot(t,x1,'k');xlabel('time');ylabel('value');title('Time Domain Signal after using ICTFT');

Magnitude of CTFT a 100 E 50 50 40 30 -20 -10 0 10 2030 40 50 Time Domain Signal 50 40 30 -20 -10 0 10 2030 40 50 time Time Domain Signal after using ICTFT 50 40 30 20 1010 20 30 40 50 time

The mismatch is because after applying ICTFT the signal x1 is not same as x as we cannot perform continuous integration. So, it will be a complex number.