# How does nufft function work in matlab?

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Ghanasyam Remesh le 2 Mar 2021
Réponse apportée : Paul le 29 Juin 2024
I am trying to understand how to use nufft function. For this, I started by comparing the results of the fft and nufft transforms of a Gaussian to see if they yield the same result. I find that for a spacing in t space that is not unity, they yield different results. The exmple in MATLAB for nufft also has unit spacing between the t coordinates. Does nufft function not work for non-unit spacing? This would be a problem for me since the next step is to get the fourier transform of a gaussian with different spacings in the t space, i.e, dt varies for each element.
Thank you.
tmax = 12;
n = 2^11;
tau = 1; % Width of the gaussian pulse
dt = 2*tmax/n;
t = (-tmax:dt:tmax-dt);
fmax = 1/(2*dt);
df = 2*fmax/n;
f = -fmax:df:fmax-df;
Pulse = exp(-(t/tau).^2);
%Pulse = sin(t/tau);
fftPulse = ifftshift(fft(Pulse));
subplot(2,1,1)
plot(t,Pulse)
subplot(2,1,2)
plot(f,abs(fftPulse)); hold on;
nufftPulse = ifftshift(nufft(Pulse,t));
plot(f,abs(nufftPulse))
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Ghanasyam Remesh le 2 Mar 2021
tmax = 20;
n = 2^11;
tau = 0.5;
dt = 2*tmax/n;
t = (-tmax:dt:tmax-dt);
fmax = 1/(2*dt);
df = 2*fmax/n;
f = (-fmax:df:fmax-df);
Pulse = exp(-(t/tau).^2);
%Pulse = sin(t/tau);
fftPulse = ifftshift(fft(Pulse));
subplot(2,1,1)
plot(t,Pulse)
xlabel('t');ylabel('f(t)')
subplot(2,1,2)
plot(f,abs(fftPulse)); hold on;
nufftPulse = (nufft(Pulse,t*2)); % The 2 factor here and next line allows us to make the freqency range [0-2]
plot((f+fmax)*dt*2,abs(nufftPulse)) % PLEASE NOTE THAT WE HAVE TO SCALE AND SHIFT
xlabel('\nu');ylabel('f(\nu)')
evelyn le 29 Juin 2024
how to define the new time index, like 't*2', you mention?

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### Réponses (1)

Paul le 29 Juin 2024
To get fft and nufft to yield the same results ....
tmax = 12;
n = 2^11;
tau = 1; % Width of the gaussian pulse
dt = 2*tmax/n;
t = (-tmax:dt:tmax-dt);
fmax = 1/(2*dt);
df = 2*fmax/n;
f = -fmax:df:fmax-df;
Pulse = exp(-(t/tau).^2);
%fftPulse = ifftshift(fft(Pulse));
fftPulse = fftshift(fft(ifftshift(Pulse)));
nufftPulse = nufft(Pulse,t,f);
figure
subplot(2,1,1)
% plot the phase of the transform instead of the original signal
%plot(t,Pulse)
plot(f,angle(fftPulse),f,angle(nufftPulse));
subplot(2,1,2)
plot(f,abs(fftPulse)); hold on;
plot(f,abs(nufftPulse))
The hair on the angle plot is just numerical noise that results from taking the angle of complex number that has very small magnitude. At low frquencies where the magnitude isn't small we see that the angle is essentially zero as would be expected.
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