One-dimensional (1D) Fourier transformation

In this notebook, we are going to transform time-domain data into 1D or 2D spectra using SpectroChemPy processing tools

[1]:

import spectrochempy as scp

SpectroChemPy's API - v.0.8.2.dev7
©Copyright 2014-2025 - A.Travert & C.Fernandez @ LCS

FFT of 1D NMR spectra

First we open read some time domain data. Here is a NMD free induction decay (FID):

[2]:

path = scp.preferences.datadir / "nmrdata" / "bruker" / "tests" / "nmr" / "topspin_1d"
fid = scp.read_topspin(path)
fid

Running on GitHub Actions
MPL Configuration directory: /home/runner/.config/matplotlib
Stylelib directory: /home/runner/.config/matplotlib/stylelib

[2]:

NDDataset: [complex128] pp (size: 12411)[topspin_1d expno:1 procno:1 (FID)]

Summary

name

:

topspin_1d expno:1 procno:1 (FID)

author

:

runner@fv-az2211-104

created

:

2025-04-27 01:47:21+00:00

Data

title

:

intensity

values

:

...

R[ 1078 2284 ... 0.2342 -0.1008] pp
I[ -1037 -2200 ... 0.06203 -0.05273] pp

size

:

12411 (complex)

Dimension `x`

size

:

12411

title

:

F1 acquisition time

coordinates

:

[ 0 4 ... 4.964e+04 4.964e+04] µs

The type of the data is complex:

[3]:

fid.dtype

[3]:

dtype('complex128')

We can represent both real and imaginary parts on the same plot using the show_complex parameter.

[4]:

prefs = scp.preferences
prefs.figure.figsize = (6, 3)
fid.plot(show_complex=True, xlim=(0, 15000))
print("td = ", fid.size)

td =  12411

../../_images/userguide_processing_fourier_9_1.png

Now we perform a Fast Fourier Transform (FFT):

[5]:

spec = scp.fft(fid)
spec.plot(xlim=(100, -100))
print("si = ", spec.size)
spec

si =  16384

[5]:

NDDataset: [complex128] pp (size: 16384)[topspin_1d expno:1 procno:1 (FID)]

Summary

name

:

topspin_1d expno:1 procno:1 (FID)

author

:

runner@fv-az2211-104

created

:

2025-04-27 01:47:21+00:00

history

:

2025-04-27 01:47:22+00:00> `zf_size` shift performed on dimension `x` with parameters: {'size': 16384}
2025-04-27 01:47:22+00:00> Fft applied on dimension x
2025-04-27 01:47:22+00:00> Inplace binary op: imul with `[ 0.61+0.793j 0.61+0.793j ... 0.813+0.582j 0.813+0.582j]`
2025-04-27 01:47:22+00:00> `pk` applied to dimension `x` with parameters: {'phc0': 52.43836, 'phc1': -16.8366, 'pivot': 0.0, 'exptc': 0.0}

Data

title

:

intensity

values

:

...

R[ -1093 -1014 ... -1031 -1079] pp
I[ -229.7 -126.8 ... 187.2 216.1] pp

size

:

16384 (complex)

Dimension `x`

size

:

16384

title

:

$\delta\ ^{1}H$

coordinates

:

[ 253.1 253.1 ... -246.8 -246.8] ppm

../../_images/userguide_processing_fourier_11_2.png

Alternative notation

[6]:

k = 1024
spec = fid.fft(size=32 * k)
spec.plot(xlim=(100, -100))
print("si = ", spec.size)

si =  32768

../../_images/userguide_processing_fourier_13_1.png

[7]:

newfid = spec.ifft()
# x coordinate is in second (base units) so lets transform it
newfid.plot(show_complex=True, xlim=(0, 15000))

[7]:

../../_images/userguide_processing_fourier_14_1.png

Let’s compare fid and newfid. There differs as a rephasing has been automatically applied after the first FFT (with the parameters found in the original fid metadata: PHC0 and PHC1).

First point in the time domain of the real part is at the maximum.

[8]:

newfid.real.plot(c="r", label="fft + ifft")
ax = fid.real.plot(clear=False, xlim=(0, 5000), ls="--", label="original real part")
_ = ax.legend()

../../_images/userguide_processing_fourier_16_0.png

First point in the time domain of the imaginary part is at the minimum.

[9]:

fid.imag.plot(ls="--", label="original imaginary part")
ax = newfid.imag.plot(clear=False, xlim=(0, 5000), c="r", label="fft + ifft")
_ = ax.legend(loc="lower right")

../../_images/userguide_processing_fourier_18_0.png

Preprocessing

Line broadening

Often before applying FFT, some exponential multiplication emor other broadening filters such as gm or sp are applied. See the dedicated apodization tutorial.

[10]:

fid2 = fid.em(lb="50. Hz")
spec2 = fid2.fft()
spec2.plot()
spec.plot(
    clear=False, xlim=(10, -5), c="r"
)  # superpose the non broadened spectrum in red and show expansion.

[10]:

../../_images/userguide_processing_fourier_20_1.png

Zero-filling

[11]:

print("td = ", fid.size)

td =  12411

[12]:

td = 64 * 1024  # size: 64 K
fid3 = fid.zf_size(size=td)
print("new td = ", fid3.x.size)

new td =  65536

[13]:

spec3 = fid3.fft()
spec3.plot(xlim=(100, -100))
print("si = ", spec3.size)

si =  65536

../../_images/userguide_processing_fourier_24_1.png

Time domain baseline correction

See the dedicated Time domain baseline correction tutorial.

Magnitude calculation

[14]:

ms = spec.mc()
ms.plot(xlim=(10, -10))
spec.plot(clear=False, xlim=(10, -10), c="r")

[14]:

../../_images/userguide_processing_fourier_27_1.png

Power spectrum

[15]:

mp = spec.ps()
(mp / mp.max()).plot()
(spec / spec.max()).plot(
    clear=False, xlim=(10, -10), c="r"
)  # Here we have normalized the spectra at their max value.

[15]:

../../_images/userguide_processing_fourier_29_1.png

Real Fourier transform

In some case, it might be interesting to perform real Fourier transform . For instance, as a demonstration, we will independently transform real and imaginary part of the previous fid, and recombine them to obtain the same result as when performing complex fourier transform on the complex dataset.

[16]:

lim = (-20, 20)
spec3.plot(xlim=lim)
spec3.imag.plot(xlim=lim)

[16]:

../../_images/userguide_processing_fourier_32_1.png

../../_images/userguide_processing_fourier_32_2.png

[17]:

Re = fid3.real.astype("complex64")
fR = Re.fft()
fR.plot(xlim=lim, show_complex=True)
Im = fid3.imag.astype("complex64")
fI = Im.fft()
fI.plot(xlim=lim, show_complex=True)

[17]:

../../_images/userguide_processing_fourier_33_1.png

../../_images/userguide_processing_fourier_33_2.png

Recombination:

[18]:

(fR - fI.imag).plot(xlim=lim)
(fR.imag + fI).plot(xlim=lim)

[18]:

../../_images/userguide_processing_fourier_35_1.png

../../_images/userguide_processing_fourier_35_2.png