how to make inverse fourrier transform in the correct way?

as you can see in my code, i made a fft of a simple signal but when i try to reconstruct it using ifft i can only see it in the ABS way, meaning that the ifft that i made is incorrect, please let me know what i did wrong

clear all
close all
clc

Ts=0.001;
t=0:Ts:10;
omega1=2*pi*15;
omega2=2*pi*40;
x=sin(omega1*t)+sin(omega2*t);

figure(1)
subplot(4,1,1)
plot(t,x)
grid on
xlim([0 0.1*pi])
title('\bf x=sin(\omega_1t)+sin(\omega_2t)')
xlabel('t[sec]')
ylabel('x(t)')

X_dft=fftshift(fft(x))/length(x);
Fs=1/Ts;
f=linspace(-Fs/2,Fs/2,length(t));
subplot(4,1,2)
plot(f,abs(X_dft))
grid on
xlim([-60 60])
title('\bf |Fourrier(x)|')
xlabel('f[Hz]')
ylabel('|X(f)|')

subplot(4,1,3)
X_phase=angle(X_dft);
plot(f,X_phase)
grid on
xlim([-60 60])
title('\bf Phase(x)')
xlabel('f[Hz]')
ylabel('Angle (\theta) [Radians]')

subplot(4,1,4)
x_ifft=abs(ifft(X_dft))*length(x);
plot(t,x_ifft)
grid on
xlim([0 0.1*pi])
title('\bf |x=sin(\omega_1t)+sin(\omega_2t)|')
xlabel('t[sec]')
ylabel('|x(t)|')

[Adam]

Well, you have an 'abs' in front of the ifft so of course the abs result is all you see.

Also, I never use fftshift myself, but I would assume that if you do a shift of the fft then you would want to reverse the shift when you do the ifft unless you are particularly looking for the effect of shifting the frequency components when you do the inverse.

[Kobi]

the fft resault is complex so i must select real imag or abs, the closest resault to the original signal is the abs. about the fftshift it used to display the signal in the frequancy plane in the coreect way, i will do another fftshift and then ifft to get a better resault but still about the ifft resault, how can i get the original signal restored?

[Adam]

The original signal is the real part of the ifft if you literally just do an fft and then do an ifft on that without any intermediate editing. Even if you do editing it is always the real part of the returned signal that is in the same domain as the real signal you started with.

[Kobi]

the real part shows me some weird signal with a lot of fluctuations....

[Adam]

That probably comes from doing an fftshift - this is un-necessary and indeed gives incorrect results when applying an ifft to its result.

Answer 1

[Thorsten]

Simple as this:

u = fft(x);
x2 = ifft(u);

The difference is within the tolerance of machine precision:

 max(abs(x - x2))
 ans =

      3.10862446895044e-15