How can i use the ifft() function properly?

I have a siganl xt, defined between -2 and 1, so I use fft(xt, 20*length(xt)) to get the signal xt in frecuency, but when i use ifft, i cannot plot again the signal properly, because it is not in the original limits between -2 and 1.

clear
close all

ts=0.01; 
fs=1/ts;
t=-10:ts:10;
xt=((t>=-2)&(t<=1)).*((2/(3))*(t+2));
plot(t,xt, 'b','LineWidth',2), grid;
title("\fontsize{15} x(t)"), xlim([-4 4]),
xlabel("\fontsize{15} t"), ylabel("\fontsize{15} x(t)");

xf=fft(xt, 20*length(xt));
xf=fftshift(xf);
f=linspace(-0.5*fs, 0.5*fs,length(xf));
plot(f,abs(xf)/fs,'b','LineWidth',2), grid, title("\fontsize{15} fft(x(t))");
xlim([-3 3]), xlabel("f"), ylabel("| x(f) |"), ylim([0 3])

xt_ifft = fftshift(ifft(ifftshift(xf)));
xt_ifft=abs(xt_ifft);
t1=linspace(-5, 5,length(xt_ifft));
plot(t1, xt_ifft ,'b','LineWidth',2), grid ;
title(" x(t) from ifft ");

from ifft i get

but i need it between -2 and 1

[Julius Muschaweck]

why do you call fft with (20*length(xt), which will just add a lot of zeros at the end of your xt array?

[Andres Diago Matta]

I call fft with (20*length(xt), because this way the function fft in matlab performe a better calculation, so i can do a better test of the signal, when i use filters or i add noise to the signal

Answer 1

[David Goodmanson]

Hi Andres,

Shifting a function in time by t0 produces a factor of exp(2 pi i f t0) in the frequency domain. In this case it's best to let that phase factor keep track of any time shift.

ts=0.01; 
fs=1/ts;
t=-10:ts:10;
xt=((t>=-2)&(t<=1)).*((2/(3))*(t+2));
xtnew = ifft(fft(xt));
figure(1)
plot(t,xt,t,xtnew+.1)   % add .1, otherwise you have an overlay

Of course this has to work since ifft(fft(z)) always gives back what you started with.

If you want to pad the time function before doing the fft (for example to get finer spacing in the frequency domain), then you can still let fft and ifft keep track of the shifts. ffts and iffts are oriented toward the first element of their respective arrays. You just need a new time array of the correct spacing, and that starts at -10 just like the old one:

xf = fft(xt, 20*length(xt));   % fft pads out xt function
xtnew = ifft(xf);
tnew = (0:length(xf)-1)*ts -10;  % tnew(1) = -10  
figure(2)
plot(t,xt,tnew,(ifft(xf))+.1)
xlim([-10 10])

[Andres Diago Matta]

Hi David, thanks a lot, that's just what i was looking for.

So, there´s no need to use fftshift before fft?, because that don´t allow the phase factor to keep a track to time shifts rigth?