numpy.fft.rfft

numpy.fft.rfft(a, n=None, axis=-1, norm=None)

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).

Parameters:

a : array_like

Input array

n : 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.

axis : int, optional

Axis over which to compute the FFT. If not given, the last axis is used.

norm : {None, “ortho”}, optional

New in version 1.10.0.

Normalization mode (see numpy.fft). Default is None.

Returns:

out : complex ndarray

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.

Raises:

IndexError

If axis is larger than the last axis of a.

See also

numpy.fft

For definition of the DFT and conventions used.

irfft

The inverse of rfft.

fft

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

fftn

The n-dimensional 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 = 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.

If the input a contains an imaginary part, it is silently discarded.

Examples

>>> np.fft.fft([0, 1, 0, 0])
array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j]) # may vary
>>> np.fft.rfft([0, 1, 0, 0])
array([ 1.+0.j,  0.-1.j, -1.+0.j]) # may vary

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