Compute the onedimensional discrete Fourier Transform for real input.
This function computes the onedimensional npoint discrete Fourier Transform (DFT) of a realvalued array by means of an efficient algorithm called the Fast Fourier Transform (FFT).
Parameters :  a : array_like
n : int, optional
axis : int, optional


Returns :  out : complex ndarray

Raises :  IndexError :

See also
Notes
When the DFT is computed for purely real input, the output is Hermitesymmetric, i.e. the negative frequency terms are just the complex conjugates of the corresponding positivefrequency terms, and the negativefrequency 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), A[0] contains the zerofrequency term, which must be purely real due to the Hermite symmetry.
If n is even, A[1] contains the term for frequencies n/2 and n/2, and must also be purely real. If n is odd, A[1] contains the term for frequency A[(n1)/2], 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 nonnegative frequency terms.