dpnp.fft.rfft

dpnp.fft.rfft(a, n=None, axis=-1, norm=None, out=None)[source]

Compute the one-dimensional discrete Fourier Transform for real input.

This function computes the one-dimensional n-point discrete Fourier Transform (DFT) of a real-valued array by means of an efficient algorithm called the Fast Fourier Transform (FFT).

For full documentation refer to numpy.fft.rfft.

Parameters:

  • a ({dpnp.ndarray, usm_ndarray}) -- Input array, taken to be real.

  • n ({None, int}, optional) -- Number of points along transformation axis in the input to use. If n is smaller than the length of the input, the input is cropped. If it is larger, the input is padded with zeros. If n is not given, the length of the input along the axis specified by axis is used. Default: None.

  • axis (int, optional) -- Axis over which to compute the FFT. If not given, the last axis is used. Default: -1.

  • norm ({None, "backward", "ortho", "forward"}, optional) -- Normalization mode (see dpnp.fft). Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. None is an alias of the default option "backward". Default: "backward".

  • out ({None, dpnp.ndarray or usm_ndarray of complex dtype}, optional) -- If provided, the result will be placed in this array. It should be of the appropriate shape and dtype. Default: None.

Returns:

out -- The truncated or zero-padded input, transformed along the axis indicated by axis, or the last one if axis is not specified. If n is even, the length of the transformed axis is (n/2)+1. If n is odd, the length is (n+1)/2.

Return type:

dpnp.ndarray of complex dtype

See also

dpnp.fft

For definition of the DFT and conventions used.

dpnp.fft.irfft

The inverse of dpnp.fft.rfft.

dpnp.fft.fft

The one-dimensional FFT of general (complex) input.

dpnp.fft.fftn

The N-dimensional FFT.

dpnp.fft.rfftn

The N-dimensional FFT of real input.

Notes

When the DFT is computed for purely real input, the output is Hermitian-symmetric, i.e. the negative frequency terms are just the complex conjugates of the corresponding positive-frequency terms, and the negative-frequency terms are therefore redundant. This function does not compute the negative frequency terms, and the length of the transformed axis of the output is therefore n//2 + 1.

When A = dpnp.fft.rfft(a) and fs is the sampling frequency, A[0] contains the zero-frequency term 0*fs, which is real due to Hermitian symmetry.

If n is even, A[-1] contains the term representing both positive and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely real. If n is odd, there is no term at fs/2; A[-1] contains the largest positive frequency (fs/2*(n-1)/n), and is complex in the general case.

Examples

>>> import dpnp as np
>>> a = np.array([0, 1, 0, 0])
>>> np.fft.fft(a)
array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j]) # may vary
>>> np.fft.rfft(a)
array([ 1.+0.j,  0.-1.j, -1.+0.j]) # may vary

Notice how the final element of the dpnp.fft.fft output is the complex conjugate of the second element, for real input. For dpnp.fft.rfft, this symmetry is exploited to compute only the non-negative frequency terms.