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# numpy.fft.fft2

fft.fft2(a, s=None, axes=(-2, -1), norm=None, out=None)

  Compute the 2-dimensional discrete Fourier Transform.

  This function computes the n-dimensional discrete Fourier Transform over any axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). By default, the transform is computed over the last two axes of the input array, i.e., a 2-dimensional FFT.

  Parameters:
    a array_like

      Input array, can be complex

    s sequence of ints, optional

      Shape (length of each transformed axis) of the output (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). This corresponds to `n` for `fft(x, n)`. Along each axis, if the given shape is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros.

      Changed in version 2.0: If it is `-1`, the whole input is used (no padding/trimming).

      If s is not given, the shape of the input along the axes specified by axes is used.

      Deprecated since version 2.0: If s is not `None`, axes must not be `None` either.

      Deprecated since version 2.0: s must contain only `int` s, not `None` values. `None` values currently mean that the default value for `n` is used in the corresponding 1-D transform, but this behaviour is deprecated.

    axes sequence of ints, optional

      Axes over which to compute the FFT. If not given, the last two axes are used. A repeated index in axes means the transform over that axis is performed multiple times. A one-element sequence means that a one-dimensional FFT is performed. Default: `(-2, -1)`.

      Deprecated since version 2.0: If s is specified, the corresponding axes to be transformed must not be `None`.

    norm{“backward”, “ortho”, “forward”}, optional

      Normalization mode (see `numpy.fft`). Default is “backward”. Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor.

      New in version 1.20.0: The “backward”, “forward” values were added.

    out complex ndarray, optional

      If provided, the result will be placed in this array. It should be of the appropriate shape and dtype for all axes (and hence only the last axis can have `s` not equal to the shape at that axis).

      New in version 2.0.0.

  Returns:
    out complex ndarray

      The truncated or zero-padded input, transformed along the axes indicated by axes, or the last two axes if axes is not given.

  Raises:
    ValueError

      If s and axes have different length, or axes not given and `len(s) != 2`.

    IndexError

      If an element of axes is larger than than the number of axes of a.

  See also

  `numpy.fft`

    Overall view of discrete Fourier transforms, with definitions and conventions used.

  `ifft2`

    The inverse two-dimensional FFT.

  `fft`

    The one-dimensional FFT.

  `fftn`

    The n-dimensional FFT.

  `fftshift`

    Shifts zero-frequency terms to the center of the array. For two-dimensional input, swaps first and third quadrants, and second and fourth quadrants.

  Notes

  `fft2` is just `fftn` with a different default for axes.

  The output, analogously to `fft`, contains the term for zero frequency in the low-order corner of the transformed axes, the positive frequency terms in the first half of these axes, the term for the Nyquist frequency in the middle of the axes and the negative frequency terms in the second half of the axes, in order of decreasingly negative frequency.

  See `fftn` for details and a plotting example, and `numpy.fft` for definitions and conventions used.

  Examples

      >>> import numpy as np
      >>> a = np.mgrid[:5, :5][0]
      >>> np.fft.fft2(a)
      array([[ 50.  +0.j        ,   0.  +0.j        ,   0.  +0.j        , # may vary
                0.  +0.j        ,   0.  +0.j        ],
             [-12.5+17.20477401j,   0.  +0.j        ,   0.  +0.j        ,
                0.  +0.j        ,   0.  +0.j        ],
             [-12.5 +4.0614962j ,   0.  +0.j        ,   0.  +0.j        ,
                0.  +0.j        ,   0.  +0.j        ],
             [-12.5 -4.0614962j ,   0.  +0.j        ,   0.  +0.j        ,
                0.  +0.j        ,   0.  +0.j        ],
             [-12.5-17.20477401j,   0.  +0.j        ,   0.  +0.j        ,
                0.  +0.j        ,   0.  +0.j        ]])