# Examples of FINUFFT?
**URL:** https://discourse.julialang.org/t/examples-of-finufft/103708
**Category:** Numerics
**Tags:** nfft
**Created:** [September 9, 2023, 7:28pm UTC](https://discourse.julialang.org/t/examples-of-finufft/103708 "2023-09-09T19:28:35Z")
**Posts on this page:** 5
**Page:** 1
### Author:  [@erny123](https://discourse.julialang.org/u/erny123)
#### Post date: [September 9, 2023, 7:28pm UTC](https://discourse.julialang.org/t/examples-of-finufft/103708/1 "2023-09-09T19:28:35Z")
I’m trying to learn some basic Non-Uniform FFT.
I want to see if I can get a frequency spectrum of a laser pulse with a NUFFT type 1 using the FINUFFT.jl library.
Here’s the example:
```julia
using FFTW
using FINUFFT
c = 2.998e8; #m/s speed of light
wl = 525.0e-9; #m wavelength
pl = wl/c; #period
wlfreq = c/wl; #2pi*Hz frequency
fwhm = 500.0e-15; #temporal fwhm
tau0 = fwhm/sqrt(2*log(2)); #gaussian fwhm
tlbw = 0.44/fwhm #transform limited bandwidth
maxf = c/wl +tlbw
minf = c/wl-tlbw
df = (maxf - minf)/1.0e3
#simulation
dt = pl/5.0;
t0 = -2.0*fwhm
tend = 2.0*fwhm
ttot = ceil(Int64,(tend-t0)/dt)
E = [exp(-t^2/tau0^2)*cos(2.0*pi*wlfreq*t) for t in t0:dt:tend];
t = [i for i in t0:dt:tend];
lent = length(E)
Nt = length(t);
```
The pulse looks like:
Taking the fft:
```julia
freq = FFTW.fftshift(FFTW.fftfreq(lent,1/dt))
Ef = FFTW.fftshift(FFTW.fft(E));
If = abs2.(Ef); #intensity spectrum
```
Looks like:
Which is exactly what I expect.
Now I want to us a NUFFT type 1 according to:
[https://finufft.readthedocs.io/en/latest/math.html](https://finufft.readthedocs.io/en/latest/math.html)
I decide to take nonuniform points in the form of a tangent curve so as to get most of the FWHM of the pulse:
```julia
#nonuniform indices to extract sampling points
#using tangent curve
maxpi = pi*22.0/100.0;
dpi = 2.0*maxpi/(lent/2);
l = [i for i in -maxpi:dpi:maxpi];
tt = @. tan(1.95*l);
ttt = @. -tt[1]+tt;
ttt = @. ttt*lent/8.81+1;
t4 = @. ceil(Int64,ttt);
#extract nonuniform points
Enu = convert.(ComplexF64,E[t4]); #field at nonuniform points
tnu = t[t4]; #time at noniniform points
lentnu = length(Enu)
```
This looks like:
```julia
wmax = freq[end]; #max freq get from previous FFTW, around 1.5e15
Nw = length(freq); #number of total points|
dw = 2*wmax/Nw; # spacing of target k grid|
w = dw .* [i for i in ceil(Int64,-Nw/2):ceil(Int64,Nw/2-1)]; # % a particular uniform M-grid of this spacing|
#normalize
Enunorm = Enu ./ maximum(E)
fhat = FINUFFT.nufft1d1(tnu, Enunorm, 1, 1.0e-12, Nw); #% type 1, requesting M modes
```
And the intensity spectrum looks like:
Not sure what to make of it. The amplitudes change but by an extremely small bit?
The main pulse frequency is off by more than 2.
---
### Author:  [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)
#### Post date: [September 9, 2023, 8:34pm UTC](https://discourse.julialang.org/t/examples-of-finufft/103708/2 "2023-09-09T20:34:44Z")
Fyi, the code posted doesn’t run: what are `pl` and `wlfreq`?
---
### Author:  [@erny123](https://discourse.julialang.org/u/erny123)
#### Post date: [September 9, 2023, 9:18pm UTC](https://discourse.julialang.org/t/examples-of-finufft/103708/3 "2023-09-09T21:18:00Z")
Oops sorry, edited my original post
---
### Author:  [@erny123](https://discourse.julialang.org/u/erny123)
#### Post date: [September 9, 2023, 9:30pm UTC](https://discourse.julialang.org/t/examples-of-finufft/103708/4 "2023-09-09T21:30:09Z")
Ok, looks like I still need more sampling points to get a better NUFFT spectrum.
For example increasing the sampling frequency to 1/10 the period of the wavelength rather than 1/5 in the original signal:
```julia
#simulation
dt = pl/10.0;
t0 = -2.0*fwhm
tend = 2.0*fwhm
ttot = ceil(Int64,(tend-t0)/dt)
E = [exp(-t^2/tau0^2)*cos(2.0*pi*wlfreq*t) for t in t0:dt:tend];
t = [i for i in t0:dt:tend];
lent = length(E)
Nt = length(t);
```
And then changing the nonuniform tangent sampling to match the number of points from indices 1 to length(t):
```julia
#nonuniform points
maxpi = pi*22.0/100.0;
dpi = 2.0*maxpi/(lent/2);
l = [i for i in -maxpi:dpi:maxpi];
tt = @. tan(1.95*l);
ttt = @. -tt[1]+tt;
ttt = @. ttt*lent/8.82+1;
t4 = @. ceil(Int64,ttt);
```
and I had to normalize the timed samples and the pulse amplitudes;
```julia
wmax = freq[end]; #max freq get from previous FFTW, around 1.5e15
Nw = length(freq); #number of total points|
dw = 2*wmax/Nw; # spacing of target k grid|
w = dw .* [i for i in ceil(Int64,-Nw/2):ceil(Int64,Nw/2-1)]; # % a particular uniform M-grid of this spacing
#normalize
Enunorm = Enu ./ maximum(E)
tnunorm = tnu ./ tend
fhat = FINUFFT.nufft1d1(tnunorm, Enunorm, 1, 1.0e-12, Nw); #% type 1, requesting M modes
```
With the rest of the code unchanged, my spectrum is now much cleaner:
The only issue I have now is that the frequency isn’t on the center frequency of approximately 5.7e14
So I’m not sure whether the frequencies I chose are exactly the way to do it
---
### Author:  [@erny123](https://discourse.julialang.org/u/erny123)
#### Post date: [September 9, 2023, 9:51pm UTC](https://discourse.julialang.org/t/examples-of-finufft/103708/5 "2023-09-09T21:51:16Z")
Ok figured it out. The nufft1 takes time samples normalized from [-pi , pi] and the amplitudes need to be complex normalized to 1. So the time sampling needs to be normalized as follows:
```julia
#normalize
Enunorm = Enu ./ maximum(E)
tnunorm = tnu ./ tend .* (pi)
```
With this:
Which is extremely close to the actual value.