Shift Theorem In Discrete Fourier Transform

I'm trying to solve a problem with python+numpy in which I've some functions of type that I need to convolve with another function . In order to optimize code, I performed the fft

Solution 1:

The problem in my code was both on input line 6, due to an incorrect (my fault) interpretation of the return value of np.fft.fftfreq(), and on the necessity to pad arrays in order to obtain sound results.

The following code works great and could be extended to multidimension.

In [1]: import numpy as np
In [2]: shift = 1
In [3]: dx = 0.5
In [4]: pad = 20
In [5]: x = np.arange(-10, 11, dx)
In [6]: y = np.cos(x)
In [7]: y = np.pad(y, (0,pad), 'constant')
In [8]: y_shift = np.cos(x-shift)
In [9]: y_fft = np.fft.fft(y)
In [10]: w = np.fft.fftfreq(y.size, dx)
In [11]: phase = np.exp(-2.0*np.pi*1.0j*w*shift)
In [12]: test = phase * y_fft
In [13]: # we use np.real since the resulting inverse fft has small imaginary part values that are zero
In [14]: inv_test = np.real(np.fft.ifft(test))
In [15]: np.allclose(y[:-pad-2],inv_test[2:-pad])
Out[15]: True

Solution 2:

Nice, thank you very much for sharing! I've implemented something in this lines some time ago, but couldn't really grasp the mathematics of it (I've blindly ported a whitepaper that described the algorithm). FWIW, this it it: https://github.com/creaktive/flare/blob/master/nrf905_demod.c#L376