torch.fft.fftshift

torch.fft.fftshift(input, dim=None) → Tensor

Reorders n-dimensional FFT data, as provided by fftn(), to have negative frequency terms first.

This performs a periodic shift of n-dimensional data such that the origin (0, ..., 0) is moved to the center of the tensor. Specifically, to input.shape[dim] // 2 in each selected dimension.

Note

By convention, the FFT returns positive frequency terms first, followed by the negative frequencies in reverse order, so that f[-i] for all $0 < i \leq n/2$ in Python gives the negative frequency terms. fftshift() rearranges all frequencies into ascending order from negative to positive with the zero-frequency term in the center.

Note

For even lengths, the Nyquist frequency at f[n/2] can be thought of as either negative or positive. fftshift() always puts the Nyquist term at the 0-index. This is the same convention used by fftfreq().

Parameters

• input (Tensor) – the tensor in FFT order

• dim (int, Tuple[int], optional) – The dimensions to rearrange. Only dimensions specified here will be rearranged, any other dimensions will be left in their original order. Default: All dimensions of input.

Example

>>> f = torch.fft.fftfreq(4)
>>> f
tensor([ 0.0000,  0.2500, -0.5000, -0.2500])

>>> torch.fft.fftshift(f)
tensor([-0.5000, -0.2500,  0.0000,  0.2500])

Also notice that the Nyquist frequency term at f[2] was moved to the beginning of the tensor.

This also works for multi-dimensional transforms:

>>> x = torch.fft.fftfreq(5, d=1/5) + 0.1 * torch.fft.fftfreq(5, d=1/5).unsqueeze(1)
>>> x
tensor([[ 0.0000,  1.0000,  2.0000, -2.0000, -1.0000],
        [ 0.1000,  1.1000,  2.1000, -1.9000, -0.9000],
        [ 0.2000,  1.2000,  2.2000, -1.8000, -0.8000],
        [-0.2000,  0.8000,  1.8000, -2.2000, -1.2000],
        [-0.1000,  0.9000,  1.9000, -2.1000, -1.1000]])

>>> torch.fft.fftshift(x)
tensor([[-2.2000, -1.2000, -0.2000,  0.8000,  1.8000],
        [-2.1000, -1.1000, -0.1000,  0.9000,  1.9000],
        [-2.0000, -1.0000,  0.0000,  1.0000,  2.0000],
        [-1.9000, -0.9000,  0.1000,  1.1000,  2.1000],
        [-1.8000, -0.8000,  0.2000,  1.2000,  2.2000]])

fftshift() can also be useful for spatial data. If our data is defined on a centered grid ([-(N//2), (N-1)//2]) then we can use the standard FFT defined on an uncentered grid ([0, N)) by first applying an ifftshift().

>>> x_centered = torch.arange(-5, 5)
>>> x_uncentered = torch.fft.ifftshift(x_centered)
>>> fft_uncentered = torch.fft.fft(x_uncentered)

Similarly, we can convert the frequency domain components to centered convention by applying fftshift().

>>> fft_centered = torch.fft.fftshift(fft_uncentered)

The inverse transform, from centered Fourier space back to centered spatial data, can be performed by applying the inverse shifts in reverse order:

>>> x_centered_2 = torch.fft.fftshift(torch.fft.ifft(torch.fft.ifftshift(fft_centered)))
>>> torch.testing.assert_close(x_centered.to(torch.complex64), x_centered_2, check_stride=False)