Exponential Apodization

In this example, we will use an exponential function to perform a Lorentzian convolution to an example dataset. The exponential function used for this apodization is defined as follows

(39)\[f(x) = e^{-\sigma \pi |x|}\]

where \(\sigma\) is parametrized by the the full width at half maximum as follows

(40)\[\sigma = \frac{\text{FWHM}}{2 \sqrt{2 \ln{2}}}\]

Below we import the necessary modules

import csdmpy as cp
import numpy as np
from mrsimulator import signal_processor as sp

First we create processor, an instance of the SignalProcessor class. The required attribute of the SignalProcessor class, operations, is a list of operations to which we add a Exponential object sandwiched between two Fourier transformations.

processor = sp.SignalProcessor(
    operations=[
        sp.IFFT(),
        sp.apodization.Exponential(FWHM="75 Hz"),
        sp.FFT(),
    ]
)

Next we create a CSDM object with a test dataset which our signal processor will operate on. Here, the dataset spans 500 Hz with a delta function centered at 0 Hz.

test_data = np.zeros(500)
test_data[250] = 1
csdm_object = cp.CSDM(
    dependent_variables=[cp.as_dependent_variable(test_data)],
    dimensions=[cp.LinearDimension(count=500, increment="1 Hz", complex_fft=True)],
)

Now to apply the processor to the CSDM object, use the apply_operations() method as follows

processed_dataset = processor.apply_operations(dataset=csdm_object).real

To see the results of the exponential apodization, we create a simple plot using the matplotlib library.

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 2, figsize=(8, 3.5), subplot_kw={"projection": "csdm"})
ax[0].plot(csdm_object, color="black", linewidth=1)
ax[0].set_title("Before")
ax[1].plot(processed_dataset.real, color="black", linewidth=1)
ax[1].set_title("After")
plt.tight_layout()
plt.show()