1425 lines
52 KiB
Python
1425 lines
52 KiB
Python
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"""
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Discrete Fourier Transforms
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Routines in this module:
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fft(a, n=None, axis=-1, norm="backward")
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ifft(a, n=None, axis=-1, norm="backward")
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rfft(a, n=None, axis=-1, norm="backward")
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irfft(a, n=None, axis=-1, norm="backward")
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hfft(a, n=None, axis=-1, norm="backward")
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ihfft(a, n=None, axis=-1, norm="backward")
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fftn(a, s=None, axes=None, norm="backward")
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ifftn(a, s=None, axes=None, norm="backward")
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rfftn(a, s=None, axes=None, norm="backward")
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irfftn(a, s=None, axes=None, norm="backward")
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fft2(a, s=None, axes=(-2,-1), norm="backward")
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ifft2(a, s=None, axes=(-2, -1), norm="backward")
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rfft2(a, s=None, axes=(-2,-1), norm="backward")
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irfft2(a, s=None, axes=(-2, -1), norm="backward")
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i = inverse transform
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r = transform of purely real data
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h = Hermite transform
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n = n-dimensional transform
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2 = 2-dimensional transform
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(Note: 2D routines are just nD routines with different default
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behavior.)
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"""
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__all__ = ['fft', 'ifft', 'rfft', 'irfft', 'hfft', 'ihfft', 'rfftn',
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'irfftn', 'rfft2', 'irfft2', 'fft2', 'ifft2', 'fftn', 'ifftn']
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import functools
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from numpy.core import asarray, zeros, swapaxes, conjugate, take, sqrt
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from . import _pocketfft_internal as pfi
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from numpy.core.multiarray import normalize_axis_index
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from numpy.core import overrides
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array_function_dispatch = functools.partial(
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overrides.array_function_dispatch, module='numpy.fft')
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# `inv_norm` is a float by which the result of the transform needs to be
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# divided. This replaces the original, more intuitive 'fct` parameter to avoid
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# divisions by zero (or alternatively additional checks) in the case of
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# zero-length axes during its computation.
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def _raw_fft(a, n, axis, is_real, is_forward, inv_norm):
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axis = normalize_axis_index(axis, a.ndim)
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if n is None:
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n = a.shape[axis]
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fct = 1/inv_norm
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if a.shape[axis] != n:
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s = list(a.shape)
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index = [slice(None)]*len(s)
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if s[axis] > n:
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index[axis] = slice(0, n)
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a = a[tuple(index)]
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else:
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index[axis] = slice(0, s[axis])
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s[axis] = n
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z = zeros(s, a.dtype.char)
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z[tuple(index)] = a
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a = z
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if axis == a.ndim-1:
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r = pfi.execute(a, is_real, is_forward, fct)
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else:
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a = swapaxes(a, axis, -1)
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r = pfi.execute(a, is_real, is_forward, fct)
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r = swapaxes(r, axis, -1)
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return r
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def _get_forward_norm(n, norm):
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if n < 1:
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raise ValueError(f"Invalid number of FFT data points ({n}) specified.")
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if norm is None or norm == "backward":
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return 1
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elif norm == "ortho":
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return sqrt(n)
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elif norm == "forward":
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return n
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raise ValueError(f'Invalid norm value {norm}; should be "backward",'
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'"ortho" or "forward".')
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def _get_backward_norm(n, norm):
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if n < 1:
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raise ValueError(f"Invalid number of FFT data points ({n}) specified.")
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if norm is None or norm == "backward":
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return n
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elif norm == "ortho":
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return sqrt(n)
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elif norm == "forward":
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return 1
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raise ValueError(f'Invalid norm value {norm}; should be "backward", '
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'"ortho" or "forward".')
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_SWAP_DIRECTION_MAP = {"backward": "forward", None: "forward",
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"ortho": "ortho", "forward": "backward"}
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def _swap_direction(norm):
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try:
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return _SWAP_DIRECTION_MAP[norm]
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except KeyError:
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raise ValueError(f'Invalid norm value {norm}; should be "backward", '
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'"ortho" or "forward".') from None
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def _fft_dispatcher(a, n=None, axis=None, norm=None):
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return (a,)
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@array_function_dispatch(_fft_dispatcher)
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def fft(a, n=None, axis=-1, norm=None):
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"""
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Compute the one-dimensional discrete Fourier Transform.
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This function computes the one-dimensional *n*-point discrete Fourier
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Transform (DFT) with the efficient Fast Fourier Transform (FFT)
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algorithm [CT].
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Parameters
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----------
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a : array_like
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Input array, can be complex.
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n : int, optional
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Length of the transformed axis of the output.
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If `n` is smaller than the length of the input, the input is cropped.
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If it is larger, the input is padded with zeros. If `n` is not given,
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the length of the input along the axis specified by `axis` is used.
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axis : int, optional
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Axis over which to compute the FFT. If not given, the last axis is
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used.
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norm : {"backward", "ortho", "forward"}, optional
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.. versionadded:: 1.10.0
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Normalization mode (see `numpy.fft`). Default is "backward".
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Indicates which direction of the forward/backward pair of transforms
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is scaled and with what normalization factor.
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.. versionadded:: 1.20.0
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The "backward", "forward" values were added.
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Returns
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-------
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out : complex ndarray
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The truncated or zero-padded input, transformed along the axis
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indicated by `axis`, or the last one if `axis` is not specified.
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Raises
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------
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IndexError
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If `axis` is not a valid axis of `a`.
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See Also
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--------
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numpy.fft : for definition of the DFT and conventions used.
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ifft : The inverse of `fft`.
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fft2 : The two-dimensional FFT.
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fftn : The *n*-dimensional FFT.
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rfftn : The *n*-dimensional FFT of real input.
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fftfreq : Frequency bins for given FFT parameters.
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Notes
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-----
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FFT (Fast Fourier Transform) refers to a way the discrete Fourier
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Transform (DFT) can be calculated efficiently, by using symmetries in the
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calculated terms. The symmetry is highest when `n` is a power of 2, and
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the transform is therefore most efficient for these sizes.
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The DFT is defined, with the conventions used in this implementation, in
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the documentation for the `numpy.fft` module.
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References
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----------
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.. [CT] Cooley, James W., and John W. Tukey, 1965, "An algorithm for the
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machine calculation of complex Fourier series," *Math. Comput.*
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19: 297-301.
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Examples
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--------
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>>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
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array([-2.33486982e-16+1.14423775e-17j, 8.00000000e+00-1.25557246e-15j,
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2.33486982e-16+2.33486982e-16j, 0.00000000e+00+1.22464680e-16j,
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-1.14423775e-17+2.33486982e-16j, 0.00000000e+00+5.20784380e-16j,
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1.14423775e-17+1.14423775e-17j, 0.00000000e+00+1.22464680e-16j])
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In this example, real input has an FFT which is Hermitian, i.e., symmetric
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in the real part and anti-symmetric in the imaginary part, as described in
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the `numpy.fft` documentation:
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>>> import matplotlib.pyplot as plt
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>>> t = np.arange(256)
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>>> sp = np.fft.fft(np.sin(t))
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>>> freq = np.fft.fftfreq(t.shape[-1])
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>>> plt.plot(freq, sp.real, freq, sp.imag)
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[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
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>>> plt.show()
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"""
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a = asarray(a)
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if n is None:
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n = a.shape[axis]
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inv_norm = _get_forward_norm(n, norm)
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output = _raw_fft(a, n, axis, False, True, inv_norm)
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return output
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@array_function_dispatch(_fft_dispatcher)
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def ifft(a, n=None, axis=-1, norm=None):
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"""
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Compute the one-dimensional inverse discrete Fourier Transform.
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This function computes the inverse of the one-dimensional *n*-point
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discrete Fourier transform computed by `fft`. In other words,
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``ifft(fft(a)) == a`` to within numerical accuracy.
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For a general description of the algorithm and definitions,
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see `numpy.fft`.
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The input should be ordered in the same way as is returned by `fft`,
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i.e.,
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* ``a[0]`` should contain the zero frequency term,
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* ``a[1:n//2]`` should contain the positive-frequency terms,
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* ``a[n//2 + 1:]`` should contain the negative-frequency terms, in
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increasing order starting from the most negative frequency.
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For an even number of input points, ``A[n//2]`` represents the sum of
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the values at the positive and negative Nyquist frequencies, as the two
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are aliased together. See `numpy.fft` for details.
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Parameters
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----------
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a : array_like
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Input array, can be complex.
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n : int, optional
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Length of the transformed axis of the output.
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If `n` is smaller than the length of the input, the input is cropped.
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If it is larger, the input is padded with zeros. If `n` is not given,
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the length of the input along the axis specified by `axis` is used.
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See notes about padding issues.
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axis : int, optional
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Axis over which to compute the inverse DFT. If not given, the last
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axis is used.
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norm : {"backward", "ortho", "forward"}, optional
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.. versionadded:: 1.10.0
|
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Normalization mode (see `numpy.fft`). Default is "backward".
|
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Indicates which direction of the forward/backward pair of transforms
|
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is scaled and with what normalization factor.
|
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|
|
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.. versionadded:: 1.20.0
|
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The "backward", "forward" values were added.
|
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Returns
|
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-------
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out : complex ndarray
|
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The truncated or zero-padded input, transformed along the axis
|
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indicated by `axis`, or the last one if `axis` is not specified.
|
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|
|
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Raises
|
||
|
|
------
|
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|
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IndexError
|
||
|
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If `axis` is not a valid axis of `a`.
|
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|
|
|
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|
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See Also
|
||
|
|
--------
|
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|
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numpy.fft : An introduction, with definitions and general explanations.
|
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fft : The one-dimensional (forward) FFT, of which `ifft` is the inverse
|
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ifft2 : The two-dimensional inverse FFT.
|
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ifftn : The n-dimensional inverse FFT.
|
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|
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|
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|
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Notes
|
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|
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-----
|
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|
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If the input parameter `n` is larger than the size of the input, the input
|
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|
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is padded by appending zeros at the end. Even though this is the common
|
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|
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approach, it might lead to surprising results. If a different padding is
|
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|
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desired, it must be performed before calling `ifft`.
|
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|
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|
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|
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Examples
|
||
|
|
--------
|
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|
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>>> np.fft.ifft([0, 4, 0, 0])
|
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array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j]) # may vary
|
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|
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Create and plot a band-limited signal with random phases:
|
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>>> import matplotlib.pyplot as plt
|
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>>> t = np.arange(400)
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>>> n = np.zeros((400,), dtype=complex)
|
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>>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,)))
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>>> s = np.fft.ifft(n)
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>>> plt.plot(t, s.real, label='real')
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[<matplotlib.lines.Line2D object at ...>]
|
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>>> plt.plot(t, s.imag, '--', label='imaginary')
|
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[<matplotlib.lines.Line2D object at ...>]
|
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>>> plt.legend()
|
||
|
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<matplotlib.legend.Legend object at ...>
|
||
|
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>>> plt.show()
|
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|
|
|
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|
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"""
|
||
|
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a = asarray(a)
|
||
|
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if n is None:
|
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|
|
n = a.shape[axis]
|
||
|
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inv_norm = _get_backward_norm(n, norm)
|
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|
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output = _raw_fft(a, n, axis, False, False, inv_norm)
|
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|
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return output
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fft_dispatcher)
|
||
|
|
def rfft(a, n=None, axis=-1, norm=None):
|
||
|
|
"""
|
||
|
|
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 : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
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 not a valid axis of `a`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
numpy.fft : For definition of the DFT and conventions used.
|
||
|
|
irfft : The inverse of `rfft`.
|
||
|
|
fft : The one-dimensional FFT of general (complex) input.
|
||
|
|
fftn : The *n*-dimensional FFT.
|
||
|
|
rfftn : 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]) # may vary
|
||
|
|
>>> np.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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
a = asarray(a)
|
||
|
|
if n is None:
|
||
|
|
n = a.shape[axis]
|
||
|
|
inv_norm = _get_forward_norm(n, norm)
|
||
|
|
output = _raw_fft(a, n, axis, True, True, inv_norm)
|
||
|
|
return output
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fft_dispatcher)
|
||
|
|
def irfft(a, n=None, axis=-1, norm=None):
|
||
|
|
"""
|
||
|
|
Computes the inverse of `rfft`.
|
||
|
|
|
||
|
|
This function computes the inverse of the one-dimensional *n*-point
|
||
|
|
discrete Fourier Transform of real input computed by `rfft`.
|
||
|
|
In other words, ``irfft(rfft(a), len(a)) == a`` to within numerical
|
||
|
|
accuracy. (See Notes below for why ``len(a)`` is necessary here.)
|
||
|
|
|
||
|
|
The input is expected to be in the form returned by `rfft`, i.e. the
|
||
|
|
real zero-frequency term followed by the complex positive frequency terms
|
||
|
|
in order of increasing frequency. Since the discrete Fourier Transform of
|
||
|
|
real input is Hermitian-symmetric, the negative frequency terms are taken
|
||
|
|
to be the complex conjugates of the corresponding positive frequency terms.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : array_like
|
||
|
|
The input array.
|
||
|
|
n : int, optional
|
||
|
|
Length of the transformed axis of the output.
|
||
|
|
For `n` output points, ``n//2+1`` input points are necessary. If the
|
||
|
|
input is longer than this, it is cropped. If it is shorter than this,
|
||
|
|
it is padded with zeros. If `n` is not given, it is taken to be
|
||
|
|
``2*(m-1)`` where ``m`` is the length of the input along the axis
|
||
|
|
specified by `axis`.
|
||
|
|
axis : int, optional
|
||
|
|
Axis over which to compute the inverse FFT. If not given, the last
|
||
|
|
axis is used.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axis
|
||
|
|
indicated by `axis`, or the last one if `axis` is not specified.
|
||
|
|
The length of the transformed axis is `n`, or, if `n` is not given,
|
||
|
|
``2*(m-1)`` where ``m`` is the length of the transformed axis of the
|
||
|
|
input. To get an odd number of output points, `n` must be specified.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
IndexError
|
||
|
|
If `axis` is not a valid axis of `a`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
numpy.fft : For definition of the DFT and conventions used.
|
||
|
|
rfft : The one-dimensional FFT of real input, of which `irfft` is inverse.
|
||
|
|
fft : The one-dimensional FFT.
|
||
|
|
irfft2 : The inverse of the two-dimensional FFT of real input.
|
||
|
|
irfftn : The inverse of the *n*-dimensional FFT of real input.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Returns the real valued `n`-point inverse discrete Fourier transform
|
||
|
|
of `a`, where `a` contains the non-negative frequency terms of a
|
||
|
|
Hermitian-symmetric sequence. `n` is the length of the result, not the
|
||
|
|
input.
|
||
|
|
|
||
|
|
If you specify an `n` such that `a` must be zero-padded or truncated, the
|
||
|
|
extra/removed values will be added/removed at high frequencies. One can
|
||
|
|
thus resample a series to `m` points via Fourier interpolation by:
|
||
|
|
``a_resamp = irfft(rfft(a), m)``.
|
||
|
|
|
||
|
|
The correct interpretation of the hermitian input depends on the length of
|
||
|
|
the original data, as given by `n`. This is because each input shape could
|
||
|
|
correspond to either an odd or even length signal. By default, `irfft`
|
||
|
|
assumes an even output length which puts the last entry at the Nyquist
|
||
|
|
frequency; aliasing with its symmetric counterpart. By Hermitian symmetry,
|
||
|
|
the value is thus treated as purely real. To avoid losing information, the
|
||
|
|
correct length of the real input **must** be given.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> np.fft.ifft([1, -1j, -1, 1j])
|
||
|
|
array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]) # may vary
|
||
|
|
>>> np.fft.irfft([1, -1j, -1])
|
||
|
|
array([0., 1., 0., 0.])
|
||
|
|
|
||
|
|
Notice how the last term in the input to the ordinary `ifft` is the
|
||
|
|
complex conjugate of the second term, and the output has zero imaginary
|
||
|
|
part everywhere. When calling `irfft`, the negative frequencies are not
|
||
|
|
specified, and the output array is purely real.
|
||
|
|
|
||
|
|
"""
|
||
|
|
a = asarray(a)
|
||
|
|
if n is None:
|
||
|
|
n = (a.shape[axis] - 1) * 2
|
||
|
|
inv_norm = _get_backward_norm(n, norm)
|
||
|
|
output = _raw_fft(a, n, axis, True, False, inv_norm)
|
||
|
|
return output
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fft_dispatcher)
|
||
|
|
def hfft(a, n=None, axis=-1, norm=None):
|
||
|
|
"""
|
||
|
|
Compute the FFT of a signal that has Hermitian symmetry, i.e., a real
|
||
|
|
spectrum.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : array_like
|
||
|
|
The input array.
|
||
|
|
n : int, optional
|
||
|
|
Length of the transformed axis of the output. For `n` output
|
||
|
|
points, ``n//2 + 1`` input points are necessary. If the input is
|
||
|
|
longer than this, it is cropped. If it is shorter than this, it is
|
||
|
|
padded with zeros. If `n` is not given, it is taken to be ``2*(m-1)``
|
||
|
|
where ``m`` is the length of the input along the axis specified by
|
||
|
|
`axis`.
|
||
|
|
axis : int, optional
|
||
|
|
Axis over which to compute the FFT. If not given, the last
|
||
|
|
axis is used.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axis
|
||
|
|
indicated by `axis`, or the last one if `axis` is not specified.
|
||
|
|
The length of the transformed axis is `n`, or, if `n` is not given,
|
||
|
|
``2*m - 2`` where ``m`` is the length of the transformed axis of
|
||
|
|
the input. To get an odd number of output points, `n` must be
|
||
|
|
specified, for instance as ``2*m - 1`` in the typical case,
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
IndexError
|
||
|
|
If `axis` is not a valid axis of `a`.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
rfft : Compute the one-dimensional FFT for real input.
|
||
|
|
ihfft : The inverse of `hfft`.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
|
||
|
|
opposite case: here the signal has Hermitian symmetry in the time
|
||
|
|
domain and is real in the frequency domain. So here it's `hfft` for
|
||
|
|
which you must supply the length of the result if it is to be odd.
|
||
|
|
|
||
|
|
* even: ``ihfft(hfft(a, 2*len(a) - 2)) == a``, within roundoff error,
|
||
|
|
* odd: ``ihfft(hfft(a, 2*len(a) - 1)) == a``, within roundoff error.
|
||
|
|
|
||
|
|
The correct interpretation of the hermitian input depends on the length of
|
||
|
|
the original data, as given by `n`. This is because each input shape could
|
||
|
|
correspond to either an odd or even length signal. By default, `hfft`
|
||
|
|
assumes an even output length which puts the last entry at the Nyquist
|
||
|
|
frequency; aliasing with its symmetric counterpart. By Hermitian symmetry,
|
||
|
|
the value is thus treated as purely real. To avoid losing information, the
|
||
|
|
shape of the full signal **must** be given.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> signal = np.array([1, 2, 3, 4, 3, 2])
|
||
|
|
>>> np.fft.fft(signal)
|
||
|
|
array([15.+0.j, -4.+0.j, 0.+0.j, -1.-0.j, 0.+0.j, -4.+0.j]) # may vary
|
||
|
|
>>> np.fft.hfft(signal[:4]) # Input first half of signal
|
||
|
|
array([15., -4., 0., -1., 0., -4.])
|
||
|
|
>>> np.fft.hfft(signal, 6) # Input entire signal and truncate
|
||
|
|
array([15., -4., 0., -1., 0., -4.])
|
||
|
|
|
||
|
|
|
||
|
|
>>> signal = np.array([[1, 1.j], [-1.j, 2]])
|
||
|
|
>>> np.conj(signal.T) - signal # check Hermitian symmetry
|
||
|
|
array([[ 0.-0.j, -0.+0.j], # may vary
|
||
|
|
[ 0.+0.j, 0.-0.j]])
|
||
|
|
>>> freq_spectrum = np.fft.hfft(signal)
|
||
|
|
>>> freq_spectrum
|
||
|
|
array([[ 1., 1.],
|
||
|
|
[ 2., -2.]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
a = asarray(a)
|
||
|
|
if n is None:
|
||
|
|
n = (a.shape[axis] - 1) * 2
|
||
|
|
new_norm = _swap_direction(norm)
|
||
|
|
output = irfft(conjugate(a), n, axis, norm=new_norm)
|
||
|
|
return output
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fft_dispatcher)
|
||
|
|
def ihfft(a, n=None, axis=-1, norm=None):
|
||
|
|
"""
|
||
|
|
Compute the inverse FFT of a signal that has Hermitian symmetry.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : array_like
|
||
|
|
Input array.
|
||
|
|
n : int, optional
|
||
|
|
Length of the inverse FFT, the 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 inverse FFT. If not given, the last
|
||
|
|
axis is used.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
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.
|
||
|
|
The length of the transformed axis is ``n//2 + 1``.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
hfft, irfft
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
`hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
|
||
|
|
opposite case: here the signal has Hermitian symmetry in the time
|
||
|
|
domain and is real in the frequency domain. So here it's `hfft` for
|
||
|
|
which you must supply the length of the result if it is to be odd:
|
||
|
|
|
||
|
|
* even: ``ihfft(hfft(a, 2*len(a) - 2)) == a``, within roundoff error,
|
||
|
|
* odd: ``ihfft(hfft(a, 2*len(a) - 1)) == a``, within roundoff error.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> spectrum = np.array([ 15, -4, 0, -1, 0, -4])
|
||
|
|
>>> np.fft.ifft(spectrum)
|
||
|
|
array([1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.+0.j]) # may vary
|
||
|
|
>>> np.fft.ihfft(spectrum)
|
||
|
|
array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j]) # may vary
|
||
|
|
|
||
|
|
"""
|
||
|
|
a = asarray(a)
|
||
|
|
if n is None:
|
||
|
|
n = a.shape[axis]
|
||
|
|
new_norm = _swap_direction(norm)
|
||
|
|
output = conjugate(rfft(a, n, axis, norm=new_norm))
|
||
|
|
return output
|
||
|
|
|
||
|
|
|
||
|
|
def _cook_nd_args(a, s=None, axes=None, invreal=0):
|
||
|
|
if s is None:
|
||
|
|
shapeless = 1
|
||
|
|
if axes is None:
|
||
|
|
s = list(a.shape)
|
||
|
|
else:
|
||
|
|
s = take(a.shape, axes)
|
||
|
|
else:
|
||
|
|
shapeless = 0
|
||
|
|
s = list(s)
|
||
|
|
if axes is None:
|
||
|
|
axes = list(range(-len(s), 0))
|
||
|
|
if len(s) != len(axes):
|
||
|
|
raise ValueError("Shape and axes have different lengths.")
|
||
|
|
if invreal and shapeless:
|
||
|
|
s[-1] = (a.shape[axes[-1]] - 1) * 2
|
||
|
|
return s, axes
|
||
|
|
|
||
|
|
|
||
|
|
def _raw_fftnd(a, s=None, axes=None, function=fft, norm=None):
|
||
|
|
a = asarray(a)
|
||
|
|
s, axes = _cook_nd_args(a, s, axes)
|
||
|
|
itl = list(range(len(axes)))
|
||
|
|
itl.reverse()
|
||
|
|
for ii in itl:
|
||
|
|
a = function(a, n=s[ii], axis=axes[ii], norm=norm)
|
||
|
|
return a
|
||
|
|
|
||
|
|
|
||
|
|
def _fftn_dispatcher(a, s=None, axes=None, norm=None):
|
||
|
|
return (a,)
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fftn_dispatcher)
|
||
|
|
def fftn(a, s=None, axes=None, norm=None):
|
||
|
|
"""
|
||
|
|
Compute the N-dimensional discrete Fourier Transform.
|
||
|
|
|
||
|
|
This function computes the *N*-dimensional discrete Fourier Transform over
|
||
|
|
any number of axes in an *M*-dimensional array by means of the Fast Fourier
|
||
|
|
Transform (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 any 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.
|
||
|
|
if `s` is not given, the shape of the input along the axes specified
|
||
|
|
by `axes` is used.
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
Axes over which to compute the FFT. If not given, the last ``len(s)``
|
||
|
|
axes are used, or all axes if `s` is also not specified.
|
||
|
|
Repeated indices in `axes` means that the transform over that axis is
|
||
|
|
performed multiple times.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : complex ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` and `a`,
|
||
|
|
as explained in the parameters section above.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
ValueError
|
||
|
|
If `s` and `axes` have different length.
|
||
|
|
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.
|
||
|
|
ifftn : The inverse of `fftn`, the inverse *n*-dimensional FFT.
|
||
|
|
fft : The one-dimensional FFT, with definitions and conventions used.
|
||
|
|
rfftn : The *n*-dimensional FFT of real input.
|
||
|
|
fft2 : The two-dimensional FFT.
|
||
|
|
fftshift : Shifts zero-frequency terms to centre of array
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The output, analogously to `fft`, contains the term for zero frequency in
|
||
|
|
the low-order corner of all axes, the positive frequency terms in the
|
||
|
|
first half of all axes, the term for the Nyquist frequency in the middle
|
||
|
|
of all axes and the negative frequency terms in the second half of all
|
||
|
|
axes, in order of decreasingly negative frequency.
|
||
|
|
|
||
|
|
See `numpy.fft` for details, definitions and conventions used.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> a = np.mgrid[:3, :3, :3][0]
|
||
|
|
>>> np.fft.fftn(a, axes=(1, 2))
|
||
|
|
array([[[ 0.+0.j, 0.+0.j, 0.+0.j], # may vary
|
||
|
|
[ 0.+0.j, 0.+0.j, 0.+0.j],
|
||
|
|
[ 0.+0.j, 0.+0.j, 0.+0.j]],
|
||
|
|
[[ 9.+0.j, 0.+0.j, 0.+0.j],
|
||
|
|
[ 0.+0.j, 0.+0.j, 0.+0.j],
|
||
|
|
[ 0.+0.j, 0.+0.j, 0.+0.j]],
|
||
|
|
[[18.+0.j, 0.+0.j, 0.+0.j],
|
||
|
|
[ 0.+0.j, 0.+0.j, 0.+0.j],
|
||
|
|
[ 0.+0.j, 0.+0.j, 0.+0.j]]])
|
||
|
|
>>> np.fft.fftn(a, (2, 2), axes=(0, 1))
|
||
|
|
array([[[ 2.+0.j, 2.+0.j, 2.+0.j], # may vary
|
||
|
|
[ 0.+0.j, 0.+0.j, 0.+0.j]],
|
||
|
|
[[-2.+0.j, -2.+0.j, -2.+0.j],
|
||
|
|
[ 0.+0.j, 0.+0.j, 0.+0.j]]])
|
||
|
|
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> [X, Y] = np.meshgrid(2 * np.pi * np.arange(200) / 12,
|
||
|
|
... 2 * np.pi * np.arange(200) / 34)
|
||
|
|
>>> S = np.sin(X) + np.cos(Y) + np.random.uniform(0, 1, X.shape)
|
||
|
|
>>> FS = np.fft.fftn(S)
|
||
|
|
>>> plt.imshow(np.log(np.abs(np.fft.fftshift(FS))**2))
|
||
|
|
<matplotlib.image.AxesImage object at 0x...>
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
return _raw_fftnd(a, s, axes, fft, norm)
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fftn_dispatcher)
|
||
|
|
def ifftn(a, s=None, axes=None, norm=None):
|
||
|
|
"""
|
||
|
|
Compute the N-dimensional inverse discrete Fourier Transform.
|
||
|
|
|
||
|
|
This function computes the inverse of the N-dimensional discrete
|
||
|
|
Fourier Transform over any number of axes in an M-dimensional array by
|
||
|
|
means of the Fast Fourier Transform (FFT). In other words,
|
||
|
|
``ifftn(fftn(a)) == a`` to within numerical accuracy.
|
||
|
|
For a description of the definitions and conventions used, see `numpy.fft`.
|
||
|
|
|
||
|
|
The input, analogously to `ifft`, should be ordered in the same way as is
|
||
|
|
returned by `fftn`, i.e. it should have the term for zero frequency
|
||
|
|
in all axes in the low-order corner, the positive frequency terms in the
|
||
|
|
first half of all axes, the term for the Nyquist frequency in the middle
|
||
|
|
of all axes and the negative frequency terms in the second half of all
|
||
|
|
axes, in order of decreasingly negative frequency.
|
||
|
|
|
||
|
|
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 ``ifft(x, n)``.
|
||
|
|
Along any 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.
|
||
|
|
if `s` is not given, the shape of the input along the axes specified
|
||
|
|
by `axes` is used. See notes for issue on `ifft` zero padding.
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
Axes over which to compute the IFFT. If not given, the last ``len(s)``
|
||
|
|
axes are used, or all axes if `s` is also not specified.
|
||
|
|
Repeated indices in `axes` means that the inverse transform over that
|
||
|
|
axis is performed multiple times.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : complex ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` or `a`,
|
||
|
|
as explained in the parameters section above.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
ValueError
|
||
|
|
If `s` and `axes` have different length.
|
||
|
|
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.
|
||
|
|
fftn : The forward *n*-dimensional FFT, of which `ifftn` is the inverse.
|
||
|
|
ifft : The one-dimensional inverse FFT.
|
||
|
|
ifft2 : The two-dimensional inverse FFT.
|
||
|
|
ifftshift : Undoes `fftshift`, shifts zero-frequency terms to beginning
|
||
|
|
of array.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
See `numpy.fft` for definitions and conventions used.
|
||
|
|
|
||
|
|
Zero-padding, analogously with `ifft`, is performed by appending zeros to
|
||
|
|
the input along the specified dimension. Although this is the common
|
||
|
|
approach, it might lead to surprising results. If another form of zero
|
||
|
|
padding is desired, it must be performed before `ifftn` is called.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> a = np.eye(4)
|
||
|
|
>>> np.fft.ifftn(np.fft.fftn(a, axes=(0,)), axes=(1,))
|
||
|
|
array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], # may vary
|
||
|
|
[0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],
|
||
|
|
[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],
|
||
|
|
[0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])
|
||
|
|
|
||
|
|
|
||
|
|
Create and plot an image with band-limited frequency content:
|
||
|
|
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> n = np.zeros((200,200), dtype=complex)
|
||
|
|
>>> n[60:80, 20:40] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20, 20)))
|
||
|
|
>>> im = np.fft.ifftn(n).real
|
||
|
|
>>> plt.imshow(im)
|
||
|
|
<matplotlib.image.AxesImage object at 0x...>
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
return _raw_fftnd(a, s, axes, ifft, norm)
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fftn_dispatcher)
|
||
|
|
def fft2(a, s=None, axes=(-2, -1), norm=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.
|
||
|
|
if `s` is not given, the shape of the input along the axes specified
|
||
|
|
by `axes` is used.
|
||
|
|
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.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
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
|
||
|
|
--------
|
||
|
|
>>> 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 ]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
return _raw_fftnd(a, s, axes, fft, norm)
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fftn_dispatcher)
|
||
|
|
def ifft2(a, s=None, axes=(-2, -1), norm=None):
|
||
|
|
"""
|
||
|
|
Compute the 2-dimensional inverse discrete Fourier Transform.
|
||
|
|
|
||
|
|
This function computes the inverse of the 2-dimensional discrete Fourier
|
||
|
|
Transform over any number of axes in an M-dimensional array by means of
|
||
|
|
the Fast Fourier Transform (FFT). In other words, ``ifft2(fft2(a)) == a``
|
||
|
|
to within numerical accuracy. By default, the inverse transform is
|
||
|
|
computed over the last two axes of the input array.
|
||
|
|
|
||
|
|
The input, analogously to `ifft`, should be ordered in the same way as is
|
||
|
|
returned by `fft2`, i.e. it should have the term for zero frequency
|
||
|
|
in the low-order corner of the two 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
|
||
|
|
both axes, in order of decreasingly negative frequency.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : array_like
|
||
|
|
Input array, can be complex.
|
||
|
|
s : sequence of ints, optional
|
||
|
|
Shape (length of each axis) of the output (``s[0]`` refers to axis 0,
|
||
|
|
``s[1]`` to axis 1, etc.). This corresponds to `n` for ``ifft(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.
|
||
|
|
if `s` is not given, the shape of the input along the axes specified
|
||
|
|
by `axes` is used. See notes for issue on `ifft` zero padding.
|
||
|
|
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.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
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.
|
||
|
|
fft2 : The forward 2-dimensional FFT, of which `ifft2` is the inverse.
|
||
|
|
ifftn : The inverse of the *n*-dimensional FFT.
|
||
|
|
fft : The one-dimensional FFT.
|
||
|
|
ifft : The one-dimensional inverse FFT.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
`ifft2` is just `ifftn` with a different default for `axes`.
|
||
|
|
|
||
|
|
See `ifftn` for details and a plotting example, and `numpy.fft` for
|
||
|
|
definition and conventions used.
|
||
|
|
|
||
|
|
Zero-padding, analogously with `ifft`, is performed by appending zeros to
|
||
|
|
the input along the specified dimension. Although this is the common
|
||
|
|
approach, it might lead to surprising results. If another form of zero
|
||
|
|
padding is desired, it must be performed before `ifft2` is called.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> a = 4 * np.eye(4)
|
||
|
|
>>> np.fft.ifft2(a)
|
||
|
|
array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], # may vary
|
||
|
|
[0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],
|
||
|
|
[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],
|
||
|
|
[0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
return _raw_fftnd(a, s, axes, ifft, norm)
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fftn_dispatcher)
|
||
|
|
def rfftn(a, s=None, axes=None, norm=None):
|
||
|
|
"""
|
||
|
|
Compute the N-dimensional discrete Fourier Transform for real input.
|
||
|
|
|
||
|
|
This function computes the N-dimensional discrete Fourier Transform over
|
||
|
|
any number of axes in an M-dimensional real array by means of the Fast
|
||
|
|
Fourier Transform (FFT). By default, all axes are transformed, with the
|
||
|
|
real transform performed over the last axis, while the remaining
|
||
|
|
transforms are complex.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : array_like
|
||
|
|
Input array, taken to be real.
|
||
|
|
s : sequence of ints, optional
|
||
|
|
Shape (length along each transformed axis) to use from the input.
|
||
|
|
(``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.).
|
||
|
|
The final element of `s` corresponds to `n` for ``rfft(x, n)``, while
|
||
|
|
for the remaining axes, it corresponds to `n` for ``fft(x, n)``.
|
||
|
|
Along any 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.
|
||
|
|
if `s` is not given, the shape of the input along the axes specified
|
||
|
|
by `axes` is used.
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
Axes over which to compute the FFT. If not given, the last ``len(s)``
|
||
|
|
axes are used, or all axes if `s` is also not specified.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : complex ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` and `a`,
|
||
|
|
as explained in the parameters section above.
|
||
|
|
The length of the last axis transformed will be ``s[-1]//2+1``,
|
||
|
|
while the remaining transformed axes will have lengths according to
|
||
|
|
`s`, or unchanged from the input.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
ValueError
|
||
|
|
If `s` and `axes` have different length.
|
||
|
|
IndexError
|
||
|
|
If an element of `axes` is larger than than the number of axes of `a`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
irfftn : The inverse of `rfftn`, i.e. the inverse of the n-dimensional FFT
|
||
|
|
of real input.
|
||
|
|
fft : The one-dimensional FFT, with definitions and conventions used.
|
||
|
|
rfft : The one-dimensional FFT of real input.
|
||
|
|
fftn : The n-dimensional FFT.
|
||
|
|
rfft2 : The two-dimensional FFT of real input.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The transform for real input is performed over the last transformation
|
||
|
|
axis, as by `rfft`, then the transform over the remaining axes is
|
||
|
|
performed as by `fftn`. The order of the output is as for `rfft` for the
|
||
|
|
final transformation axis, and as for `fftn` for the remaining
|
||
|
|
transformation axes.
|
||
|
|
|
||
|
|
See `fft` for details, definitions and conventions used.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> a = np.ones((2, 2, 2))
|
||
|
|
>>> np.fft.rfftn(a)
|
||
|
|
array([[[8.+0.j, 0.+0.j], # may vary
|
||
|
|
[0.+0.j, 0.+0.j]],
|
||
|
|
[[0.+0.j, 0.+0.j],
|
||
|
|
[0.+0.j, 0.+0.j]]])
|
||
|
|
|
||
|
|
>>> np.fft.rfftn(a, axes=(2, 0))
|
||
|
|
array([[[4.+0.j, 0.+0.j], # may vary
|
||
|
|
[4.+0.j, 0.+0.j]],
|
||
|
|
[[0.+0.j, 0.+0.j],
|
||
|
|
[0.+0.j, 0.+0.j]]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
a = asarray(a)
|
||
|
|
s, axes = _cook_nd_args(a, s, axes)
|
||
|
|
a = rfft(a, s[-1], axes[-1], norm)
|
||
|
|
for ii in range(len(axes)-1):
|
||
|
|
a = fft(a, s[ii], axes[ii], norm)
|
||
|
|
return a
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fftn_dispatcher)
|
||
|
|
def rfft2(a, s=None, axes=(-2, -1), norm=None):
|
||
|
|
"""
|
||
|
|
Compute the 2-dimensional FFT of a real array.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : array
|
||
|
|
Input array, taken to be real.
|
||
|
|
s : sequence of ints, optional
|
||
|
|
Shape of the FFT.
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
Axes over which to compute the FFT.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The result of the real 2-D FFT.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
rfftn : Compute the N-dimensional discrete Fourier Transform for real
|
||
|
|
input.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is really just `rfftn` with different default behavior.
|
||
|
|
For more details see `rfftn`.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> a = np.mgrid[:5, :5][0]
|
||
|
|
>>> np.fft.rfft2(a)
|
||
|
|
array([[ 50. +0.j , 0. +0.j , 0. +0.j ],
|
||
|
|
[-12.5+17.20477401j, 0. +0.j , 0. +0.j ],
|
||
|
|
[-12.5 +4.0614962j , 0. +0.j , 0. +0.j ],
|
||
|
|
[-12.5 -4.0614962j , 0. +0.j , 0. +0.j ],
|
||
|
|
[-12.5-17.20477401j, 0. +0.j , 0. +0.j ]])
|
||
|
|
"""
|
||
|
|
return rfftn(a, s, axes, norm)
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fftn_dispatcher)
|
||
|
|
def irfftn(a, s=None, axes=None, norm=None):
|
||
|
|
"""
|
||
|
|
Computes the inverse of `rfftn`.
|
||
|
|
|
||
|
|
This function computes the inverse of the N-dimensional discrete
|
||
|
|
Fourier Transform for real input over any number of axes in an
|
||
|
|
M-dimensional array by means of the Fast Fourier Transform (FFT). In
|
||
|
|
other words, ``irfftn(rfftn(a), a.shape) == a`` to within numerical
|
||
|
|
accuracy. (The ``a.shape`` is necessary like ``len(a)`` is for `irfft`,
|
||
|
|
and for the same reason.)
|
||
|
|
|
||
|
|
The input should be ordered in the same way as is returned by `rfftn`,
|
||
|
|
i.e. as for `irfft` for the final transformation axis, and as for `ifftn`
|
||
|
|
along all the other axes.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : array_like
|
||
|
|
Input array.
|
||
|
|
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.). `s` is also the
|
||
|
|
number of input points used along this axis, except for the last axis,
|
||
|
|
where ``s[-1]//2+1`` points of the input are used.
|
||
|
|
Along any axis, if the shape indicated by `s` is smaller than that of
|
||
|
|
the input, the input is cropped. If it is larger, the input is padded
|
||
|
|
with zeros. If `s` is not given, the shape of the input along the axes
|
||
|
|
specified by axes is used. Except for the last axis which is taken to
|
||
|
|
be ``2*(m-1)`` where ``m`` is the length of the input along that axis.
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
Axes over which to compute the inverse FFT. If not given, the last
|
||
|
|
`len(s)` axes are used, or all axes if `s` is also not specified.
|
||
|
|
Repeated indices in `axes` means that the inverse transform over that
|
||
|
|
axis is performed multiple times.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` or `a`,
|
||
|
|
as explained in the parameters section above.
|
||
|
|
The length of each transformed axis is as given by the corresponding
|
||
|
|
element of `s`, or the length of the input in every axis except for the
|
||
|
|
last one if `s` is not given. In the final transformed axis the length
|
||
|
|
of the output when `s` is not given is ``2*(m-1)`` where ``m`` is the
|
||
|
|
length of the final transformed axis of the input. To get an odd
|
||
|
|
number of output points in the final axis, `s` must be specified.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
ValueError
|
||
|
|
If `s` and `axes` have different length.
|
||
|
|
IndexError
|
||
|
|
If an element of `axes` is larger than than the number of axes of `a`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
rfftn : The forward n-dimensional FFT of real input,
|
||
|
|
of which `ifftn` is the inverse.
|
||
|
|
fft : The one-dimensional FFT, with definitions and conventions used.
|
||
|
|
irfft : The inverse of the one-dimensional FFT of real input.
|
||
|
|
irfft2 : The inverse of the two-dimensional FFT of real input.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
See `fft` for definitions and conventions used.
|
||
|
|
|
||
|
|
See `rfft` for definitions and conventions used for real input.
|
||
|
|
|
||
|
|
The correct interpretation of the hermitian input depends on the shape of
|
||
|
|
the original data, as given by `s`. This is because each input shape could
|
||
|
|
correspond to either an odd or even length signal. By default, `irfftn`
|
||
|
|
assumes an even output length which puts the last entry at the Nyquist
|
||
|
|
frequency; aliasing with its symmetric counterpart. When performing the
|
||
|
|
final complex to real transform, the last value is thus treated as purely
|
||
|
|
real. To avoid losing information, the correct shape of the real input
|
||
|
|
**must** be given.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> a = np.zeros((3, 2, 2))
|
||
|
|
>>> a[0, 0, 0] = 3 * 2 * 2
|
||
|
|
>>> np.fft.irfftn(a)
|
||
|
|
array([[[1., 1.],
|
||
|
|
[1., 1.]],
|
||
|
|
[[1., 1.],
|
||
|
|
[1., 1.]],
|
||
|
|
[[1., 1.],
|
||
|
|
[1., 1.]]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
a = asarray(a)
|
||
|
|
s, axes = _cook_nd_args(a, s, axes, invreal=1)
|
||
|
|
for ii in range(len(axes)-1):
|
||
|
|
a = ifft(a, s[ii], axes[ii], norm)
|
||
|
|
a = irfft(a, s[-1], axes[-1], norm)
|
||
|
|
return a
|
||
|
|
|
||
|
|
|
||
|
|
@array_function_dispatch(_fftn_dispatcher)
|
||
|
|
def irfft2(a, s=None, axes=(-2, -1), norm=None):
|
||
|
|
"""
|
||
|
|
Computes the inverse of `rfft2`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : array_like
|
||
|
|
The input array
|
||
|
|
s : sequence of ints, optional
|
||
|
|
Shape of the real output to the inverse FFT.
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
The axes over which to compute the inverse fft.
|
||
|
|
Default is the last two axes.
|
||
|
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
|
.. versionadded:: 1.10.0
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 1.20.0
|
||
|
|
|
||
|
|
The "backward", "forward" values were added.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The result of the inverse real 2-D FFT.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
rfft2 : The forward two-dimensional FFT of real input,
|
||
|
|
of which `irfft2` is the inverse.
|
||
|
|
rfft : The one-dimensional FFT for real input.
|
||
|
|
irfft : The inverse of the one-dimensional FFT of real input.
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irfftn : Compute the inverse of the N-dimensional FFT of real input.
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Notes
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-----
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This is really `irfftn` with different defaults.
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For more details see `irfftn`.
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Examples
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--------
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>>> a = np.mgrid[:5, :5][0]
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>>> A = np.fft.rfft2(a)
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>>> np.fft.irfft2(A, s=a.shape)
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array([[0., 0., 0., 0., 0.],
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[1., 1., 1., 1., 1.],
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[2., 2., 2., 2., 2.],
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[3., 3., 3., 3., 3.],
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[4., 4., 4., 4., 4.]])
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"""
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return irfftn(a, s, axes, norm)
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