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scipy.fft.

# rfft

scipy.fft.rfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, *, plan=None)

  Compute the 1-D discrete Fourier Transform for real input.

  This function computes the 1-D n-point discrete Fourier Transform (DFT) of a real-valued array by means of an efficient algorithm called the Fast Fourier Transform (FFT).

  Parameters:
    x 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{“backward”, “ortho”, “forward”}, optional

      Normalization mode (see `fft`). Default is “backward”.

    overwrite_x bool, optional

      If True, the contents of x can be destroyed; the default is False. See `fft` for more details.

    workers int, optional

      Maximum number of workers to use for parallel computation. If negative, the value wraps around from `os.cpu_count()`. See `fft` for more details.

    plan object, optional

      This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

      Added in version 1.5.0.

  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

  `irfft`

    The inverse of `rfft`.

  `fft`

    The 1-D FFT of general (complex) input.

  `fftn`

    The N-D FFT.

  `rfft2`

    The 2-D FFT of real input.

  `rfftn`

    The N-D 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 `X = rfft(x)` and fs is the sampling frequency, `X[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.

  Array API Standard Support

  `rfft` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ✅ | ✅
  Dask | ⚠️ computes graph | n/a

  See Support for the array API standard for more information.

  Examples

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