# Getting back the original x axis after IFFT-FFT in Python

I have an analitically generated spectrum $$I(\omega)$$, x axis represents angular frequency, y represents intensity. The spectrum is centered around some frequency value, which is often called central frequency of the signal. I want to perform IFFT on the data to time domain $$I(\tau)$$, cut its useful part with a gaussian curve, then FFT back to the original domain $$I(\omega)$$. My problem is that after $$F(F^{-1}(I(\omega)))$$ the central frequency is lost. I get back the spectrum by shape, but it's always centered around 0. Currently my solution to this is quite bad: I cache the original x axis and I restore it upon FFT calls. This obviously has many downsides, and I want to improve it. Below I included a small demo which demonstrates the problem. My question is: can this be solved in a more elegant way? Am I missing something?

import numpy as np
from scipy.fftpack import fft, ifft, fftshift, fftfreq
from scipy.interpolate import interp1d
import matplotlib.pyplot as plt

C_LIGHT = 299.793

def generate_data(start, stop, center, delay, GD=0, resolution=0.1):
window = 8 * np.log(2) / 50
lamend = 2 * np.pi * C_LIGHT / start
lamstart = 2 * np.pi * C_LIGHT/stop
lam = np.arange(lamstart, lamend + resolution, resolution)
omega = 2 * np.pi * C_LIGHT / lam
relom = omega - center
_i = np.exp(-(relom) ** 2 / window)
i = 2 * _i + 2 * np.cos(relom * GD + (omega * delay)) * np.sqrt(_i * _i)
return omega, i

if __name__ == '__main__':

# Generate data
x, y = generate_data(1, 3, 2, 800, GD=0)

# Linearly interpolate to be evenly spaced
xs = np.linspace(x, x[-1], len(x))
intp = interp1d(x, y, kind='linear')
ys = intp(xs)
x, y = xs, ys
plt.plot(x, y, label='original')

# IFFT
xt = fftfreq(len(x), d=(x-x)/(2*np.pi))
yt = ifft(y)
# plt.plot(xt, np.abs(yt))

# FFT back
xf = fftshift(fftfreq(len(xt), d=(xt-xt)/(2*np.pi)))
yf = fft(yt)
plt.plot(xf, np.abs(yf), label='after transforms')
plt.legend()
plt.grid()
plt.show()