numpy.fft.rfft — NumPy v1.13 Manual (original) (raw)
numpy.fft. rfft(a, n=None, axis=-1, norm=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).
| 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. |
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| 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
For definition of the DFT and conventions used.
The inverse of rfft.
The one-dimensional FFT of general (complex) input.
The _n_-dimensional FFT.
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]) np.fft.rfft([0, 1, 0, 0]) array([ 1.+0.j, 0.-1.j, -1.+0.j])
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.