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# numpy.fft.fft

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

  Compute the one-dimensional discrete Fourier Transform.

  This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm [CT].

  Parameters : | a : array_like Input array, can be complex. n : int, optional Length of the transformed axis of the output. 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.
  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.
  Raises : | IndexError : if axes is larger than the last axis of a.

  See also

  `numpy.fft`
    for definition of the DFT and conventions used.
  `ifft`
    The inverse of `fft`.
  `fft2`
    The two-dimensional FFT.
  `fftn`
    The n-dimensional FFT.
  `rfftn`
    The n-dimensional FFT of real input.
  `fftfreq`
    Frequency bins for given FFT parameters.

  Notes

  FFT (Fast Fourier Transform) refers to a way the discrete Fourier Transform (DFT) can be calculated efficiently, by using symmetries in the calculated terms. The symmetry is highest when n is a power of 2, and the transform is therefore most efficient for these sizes.

  The DFT is defined, with the conventions used in this implementation, in the documentation for the `numpy.fft` module.

  References

  [CT] | Cooley, James W., and John W. Tukey, 1965, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19: 297-301.

  Examples

      >>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
      array([ -3.44505240e-16 +1.14383329e-17j,
               8.00000000e+00 -5.71092652e-15j,
               2.33482938e-16 +1.22460635e-16j,
               1.64863782e-15 +1.77635684e-15j,
               9.95839695e-17 +2.33482938e-16j,
               0.00000000e+00 +1.66837030e-15j,
               1.14383329e-17 +1.22460635e-16j,
               -1.64863782e-15 +1.77635684e-15j])

      >>> import matplotlib.pyplot as plt
      >>> t = np.arange(256)
      >>> sp = np.fft.fft(np.sin(t))
      >>> freq = np.fft.fftfreq(t.shape[-1])
      >>> plt.plot(freq, sp.real, freq, sp.imag)
      [<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
      >>> plt.show()

  In this example, real input has an FFT which is Hermitian, i.e., symmetric in the real part and anti-symmetric in the imaginary part, as described in the `numpy.fft` documentation.

  (Source code)

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