Time for action – shifting frequencies

We will create a signal, transform it, and then shift the signal. Shift the frequencies with the following steps:

  1. Create a cosine wave with 30 points:
    x =  np.linspace(0, 2 * np.pi, 30)
wave = np.cos(x)
  2. Transform the cosine wave with the fft() function:
    transformed = np.fft.fft(wave)
  3. Shift the signal with the fftshift() function:
    shifted = np.fft.fftshift(transformed)
  4. Reverse the shift with the ifftshift() function. This should undo the shift. Check with the following code snippet:
    print(np.all((np.fft.ifftshift(shifted) - transformed) < 10 ** -9))

    The result appears as follows:

    True
  5. Plot the signal and transform it with matplotlib:
    plt.plot(transformed, lw=2, label="Transformed")
plt.plot(shifted, '--', lw=3, label="Shifted")
plt.title('Shifted and transformed cosine wave')
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.grid()
plt.legend(loc='best')
plt.show()

    The following diagram shows the effect of the shift and the FFT:

    Time for action – shifting frequencies

What just happened?

We applied the fftshift() function to a cosine wave. After applying the ifftshift() function, we got our signal back (see fouriershift.py):

import numpy as np
import matplotlib.pyplot as plt


x =  np.linspace(0, 2 * np.pi, 30)
wave = np.cos(x)
transformed = np.fft.fft(wave)
shifted = np.fft.fftshift(transformed)
print(np.all(np.abs(np.fft.ifftshift(shifted) - transformed) < 10 ** -9))

plt.plot(transformed, lw=2, label="Transformed")
plt.plot(shifted, '--', lw=3, label="Shifted")
plt.title('Shifted and transformed cosine wave')
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.grid()
plt.legend(loc='best')
plt.show()
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