1507 lines
57 KiB
Python
1507 lines
57 KiB
Python
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from scipy._lib.uarray import generate_multimethod, Dispatchable
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import numpy as np
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def _x_replacer(args, kwargs, dispatchables):
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"""
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uarray argument replacer to replace the transform input array (``x``)
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"""
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if len(args) > 0:
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return (dispatchables[0],) + args[1:], kwargs
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kw = kwargs.copy()
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kw['x'] = dispatchables[0]
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return args, kw
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def _dispatch(func):
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"""
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Function annotation that creates a uarray multimethod from the function
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"""
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return generate_multimethod(func, _x_replacer, domain="numpy.scipy.fft")
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@_dispatch
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def fft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=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 [1]_.
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Parameters
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----------
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x : 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 : {None, "ortho"}, optional
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Normalization mode. Default is None, meaning no normalization on the
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forward transforms and scaling by ``1/n`` on the `ifft`.
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For ``norm="ortho"``, both directions are scaled by ``1/sqrt(n)``.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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See the notes below for more details.
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workers : int, optional
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Maximum number of workers to use for parallel computation. If negative,
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the value wraps around from ``os.cpu_count()``. See below for more
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details.
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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 `axes` is larger than the last axis of `x`.
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See Also
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--------
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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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next_fast_len : Size to pad input to for most efficient transforms
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Notes
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-----
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FFT (Fast Fourier Transform) refers to a way the discrete Fourier Transform
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(DFT) can be calculated efficiently, by using symmetries in the calculated
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terms. The symmetry is highest when `n` is a power of 2, and the transform
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is therefore most efficient for these sizes. For poorly factorizable sizes,
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`scipy.fft` uses Bluestein's algorithm [2]_ and so is never worse than
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O(`n` log `n`). Further performance improvements may be seen by zero-padding
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the input using `next_fast_len`.
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If ``x`` is a 1d array, then the `fft` is equivalent to ::
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y[k] = np.sum(x * np.exp(-2j * np.pi * k * np.arange(n)/n))
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The frequency term ``f=k/n`` is found at ``y[k]``. At ``y[n/2]`` we reach
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the Nyquist frequency and wrap around to the negative-frequency terms. So,
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for an 8-point transform, the frequencies of the result are
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[0, 1, 2, 3, -4, -3, -2, -1]. To rearrange the fft output so that the
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zero-frequency component is centered, like [-4, -3, -2, -1, 0, 1, 2, 3],
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use `fftshift`.
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Transforms can be done in single, double or extended precision (long
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double) floating point. Half precision inputs will be converted to single
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precision and non floating-point inputs will be converted to double
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precision.
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If the data type of ``x`` is real, a "real FFT" algorithm is automatically
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used, which roughly halves the computation time. To increase efficiency
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a little further, use `rfft`, which does the same calculation, but only
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outputs half of the symmetrical spectrum. If the data is both real and
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symmetrical, the `dct` can again double the efficiency, by generating
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half of the spectrum from half of the signal.
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When ``overwrite_x=True`` is specified, the memory referenced by ``x`` may
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be used by the implementation in any way. This may include reusing the
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memory for the result, but this is in no way guaranteed. You should not
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rely on the contents of ``x`` after the transform as this may change in
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future without warning.
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The ``workers`` argument specifies the maximum number of parallel jobs to
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split the FFT computation into. This will execute independent 1-dimensional
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FFTs within ``x``. So, ``x`` must be at least 2-dimensional and the
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non-transformed axes must be large enough to split into chunks. If ``x`` is
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too small, fewer jobs may be used than requested.
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References
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----------
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.. [1] 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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.. [2] Bluestein, L., 1970, "A linear filtering approach to the
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computation of discrete Fourier transform". *IEEE Transactions on
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Audio and Electroacoustics.* 18 (4): 451-455.
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Examples
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--------
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>>> import scipy.fft
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>>> scipy.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:
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>>> from scipy.fft import fft, fftfreq, fftshift
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>>> import matplotlib.pyplot as plt
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>>> t = np.arange(256)
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>>> sp = fftshift(fft(np.sin(t)))
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>>> freq = fftshift(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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return (Dispatchable(x, np.ndarray),)
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@_dispatch
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def ifft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=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(x)) == x`` to within numerical accuracy.
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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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* ``x[0]`` should contain the zero frequency term,
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* ``x[1:n//2]`` should contain the positive-frequency terms,
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* ``x[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, ``x[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 `fft` for details.
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Parameters
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----------
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x : 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 : {None, "ortho"}, optional
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Normalization mode (see `fft`). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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See :func:`fft` for more details.
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workers : int, optional
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Maximum number of workers to use for parallel computation. If negative,
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the value wraps around from ``os.cpu_count()``.
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See :func:`~scipy.fft.fft` for more details.
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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 `axes` is larger than the last axis of `x`.
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See Also
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--------
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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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Notes
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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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is padded by appending zeros at the end. Even though this is the common
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approach, it might lead to surprising results. If a different padding is
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desired, it must be performed before calling `ifft`.
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If ``x`` is a 1d array, then the `ifft` is equivalent to ::
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y[k] = np.sum(x * np.exp(2j * np.pi * k * np.arange(n)/n)) / len(x)
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As with `fft`, `ifft` has support for all floating point types and is
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optimized for real input.
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Examples
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--------
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>>> import scipy.fft
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>>> scipy.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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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 = scipy.fft.ifft(n)
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>>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
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[<matplotlib.lines.Line2D object at ...>, <matplotlib.lines.Line2D object at ...>]
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>>> plt.legend(('real', 'imaginary'))
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<matplotlib.legend.Legend object at ...>
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>>> plt.show()
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"""
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return (Dispatchable(x, np.ndarray),)
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@_dispatch
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def rfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None):
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"""
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Compute the one-dimensional discrete Fourier Transform for real input.
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This function computes the one-dimensional *n*-point discrete Fourier
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Transform (DFT) of a real-valued array by means of an efficient algorithm
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called the Fast Fourier Transform (FFT).
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Parameters
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----------
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a : array_like
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Input array
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n : int, optional
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Number of points along transformation axis in the input to use.
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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 : {None, "ortho"}, optional
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Normalization mode (see `fft`). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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See :func:`fft` for more details.
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workers : int, optional
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Maximum number of workers to use for parallel computation. If negative,
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the value wraps around from ``os.cpu_count()``.
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See :func:`~scipy.fft.fft` for more details.
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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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If `n` is even, the length of the transformed axis is ``(n/2)+1``.
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If `n` is odd, the length is ``(n+1)/2``.
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Raises
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------
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IndexError
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If `axis` is larger than the last axis of `a`.
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See Also
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--------
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irfft : The inverse of `rfft`.
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fft : The one-dimensional FFT of general (complex) input.
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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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Notes
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-----
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When the DFT is computed for purely real input, the output is
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Hermitian-symmetric, i.e. the negative frequency terms are just the complex
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conjugates of the corresponding positive-frequency terms, and the
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negative-frequency terms are therefore redundant. This function does not
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compute the negative frequency terms, and the length of the transformed
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axis of the output is therefore ``n//2 + 1``.
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When ``X = rfft(x)`` and fs is the sampling frequency, ``X[0]`` contains
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the zero-frequency term 0*fs, which is real due to Hermitian symmetry.
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If `n` is even, ``A[-1]`` contains the term representing both positive
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and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely
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real. If `n` is odd, there is no term at fs/2; ``A[-1]`` contains
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the largest positive frequency (fs/2*(n-1)/n), and is complex in the
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general case.
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If the input `a` contains an imaginary part, it is silently discarded.
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Examples
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--------
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>>> import scipy.fft
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>>> scipy.fft.fft([0, 1, 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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>>> scipy.fft.rfft([0, 1, 0, 0])
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array([ 1.+0.j, 0.-1.j, -1.+0.j]) # may vary
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Notice how the final element of the `fft` output is the complex conjugate
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of the second element, for real input. For `rfft`, this symmetry is
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exploited to compute only the non-negative frequency terms.
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"""
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return (Dispatchable(x, np.ndarray),)
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@_dispatch
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def irfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None):
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"""
|
||
|
|
Compute the inverse of the n-point DFT for real input.
|
||
|
|
|
||
|
|
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(x), len(x)) == x`` 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
|
||
|
|
----------
|
||
|
|
x : 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 : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
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 larger than the last axis of `x`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
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 `x`, where `x` 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 default value of `n` assumes an even output length. By the Hermitian
|
||
|
|
symmetry, the last imaginary component must be 0 and so is ignored. To
|
||
|
|
avoid losing information, the correct length of the real input *must* be
|
||
|
|
given.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> scipy.fft.ifft([1, -1j, -1, 1j])
|
||
|
|
array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]) # may vary
|
||
|
|
>>> scipy.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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def hfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None):
|
||
|
|
"""
|
||
|
|
Compute the FFT of a signal that has Hermitian symmetry, i.e., a real
|
||
|
|
spectrum.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : 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 : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
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 larger than the last axis of `a`.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
rfft : Compute the one-dimensional FFT for real input.
|
||
|
|
ihfft : The inverse of `hfft`.
|
||
|
|
hfftn : Compute the n-dimensional FFT of a Hermitian signal.
|
||
|
|
|
||
|
|
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
|
||
|
|
--------
|
||
|
|
>>> from scipy.fft import fft, hfft
|
||
|
|
>>> a = 2 * np.pi * np.arange(10) / 10
|
||
|
|
>>> signal = np.cos(a) + 3j * np.sin(3 * a)
|
||
|
|
>>> fft(signal).round(10)
|
||
|
|
array([ -0.+0.j, 5.+0.j, -0.+0.j, 15.-0.j, 0.+0.j, 0.+0.j,
|
||
|
|
-0.+0.j, -15.-0.j, 0.+0.j, 5.+0.j])
|
||
|
|
>>> hfft(signal[:6]).round(10) # Input first half of signal
|
||
|
|
array([ 0., 5., 0., 15., -0., 0., 0., -15., -0., 5.])
|
||
|
|
>>> hfft(signal, 10) # Input entire signal and truncate
|
||
|
|
array([ 0., 5., 0., 15., -0., 0., 0., -15., -0., 5.])
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def ihfft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None):
|
||
|
|
"""
|
||
|
|
Compute the inverse FFT of a signal that has Hermitian symmetry.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : 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 : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
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
|
||
|
|
--------
|
||
|
|
>>> from scipy.fft import ifft, ihfft
|
||
|
|
>>> spectrum = np.array([ 15, -4, 0, -1, 0, -4])
|
||
|
|
>>> ifft(spectrum)
|
||
|
|
array([1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.+0.j]) # may vary
|
||
|
|
>>> ihfft(spectrum)
|
||
|
|
array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j]) # may vary
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def fftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=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
|
||
|
|
----------
|
||
|
|
x : 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.
|
||
|
|
norm : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : complex ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` and `x`,
|
||
|
|
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 `x`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
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.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> x = np.mgrid[:3, :3, :3][0]
|
||
|
|
>>> scipy.fft.fftn(x, 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]]])
|
||
|
|
>>> scipy.fft.fftn(x, (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 = scipy.fft.fftn(S)
|
||
|
|
>>> plt.imshow(np.log(np.abs(scipy.fft.fftshift(FS))**2))
|
||
|
|
<matplotlib.image.AxesImage object at 0x...>
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def ifftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=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(x)) == x`` to within numerical accuracy.
|
||
|
|
|
||
|
|
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
|
||
|
|
----------
|
||
|
|
x : 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.
|
||
|
|
norm : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : complex ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` or `x`,
|
||
|
|
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 `x`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
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
|
||
|
|
-----
|
||
|
|
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
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> x = np.eye(4)
|
||
|
|
>>> scipy.fft.ifftn(scipy.fft.fftn(x, 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 = scipy.fft.ifftn(n).real
|
||
|
|
>>> plt.imshow(im)
|
||
|
|
<matplotlib.image.AxesImage object at 0x...>
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def fft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=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
|
||
|
|
----------
|
||
|
|
x : 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.
|
||
|
|
norm : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
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 `x`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
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 `fft` for
|
||
|
|
definitions and conventions used.
|
||
|
|
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> x = np.mgrid[:5, :5][0]
|
||
|
|
>>> scipy.fft.fft2(x)
|
||
|
|
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 (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def ifft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=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(x)) == x``
|
||
|
|
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
|
||
|
|
----------
|
||
|
|
x : 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.
|
||
|
|
norm : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
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 `x`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
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 `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
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> x = 4 * np.eye(4)
|
||
|
|
>>> scipy.fft.ifft2(x)
|
||
|
|
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 (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def rfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=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
|
||
|
|
----------
|
||
|
|
x : 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 : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : complex ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` and `x`,
|
||
|
|
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 `x`.
|
||
|
|
|
||
|
|
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
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> x = np.ones((2, 2, 2))
|
||
|
|
>>> scipy.fft.rfftn(x)
|
||
|
|
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]]])
|
||
|
|
|
||
|
|
>>> scipy.fft.rfftn(x, 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]]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def rfft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=None):
|
||
|
|
"""
|
||
|
|
Compute the 2-dimensional FFT of a real array.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : 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 : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
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`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def irfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None):
|
||
|
|
"""
|
||
|
|
Compute the inverse of the N-dimensional FFT of real input.
|
||
|
|
|
||
|
|
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(x), x.shape) == x`` 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
|
||
|
|
----------
|
||
|
|
x : 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.
|
||
|
|
norm : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` or `x`,
|
||
|
|
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 `x`.
|
||
|
|
|
||
|
|
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 default value of `s` assumes an even output length in the final
|
||
|
|
transformation axis. When performing the final complex to real
|
||
|
|
transformation, the Hermitian symmetry requires that the last imaginary
|
||
|
|
component along that axis must be 0 and so it is ignored. To avoid losing
|
||
|
|
information, the correct length of the real input *must* be given.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> x = np.zeros((3, 2, 2))
|
||
|
|
>>> x[0, 0, 0] = 3 * 2 * 2
|
||
|
|
>>> scipy.fft.irfftn(x)
|
||
|
|
array([[[1., 1.],
|
||
|
|
[1., 1.]],
|
||
|
|
[[1., 1.],
|
||
|
|
[1., 1.]],
|
||
|
|
[[1., 1.],
|
||
|
|
[1., 1.]]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def irfft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False,
|
||
|
|
workers=None):
|
||
|
|
"""
|
||
|
|
Compute the 2-dimensional inverse FFT of a real array.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : 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 : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The result of the inverse real 2-D FFT.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
irfftn : Compute the inverse of the N-dimensional FFT of real input.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is really `irfftn` with different defaults.
|
||
|
|
For more details see `irfftn`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def hfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None):
|
||
|
|
"""
|
||
|
|
Compute the N-dimensional FFT of Hermitian symmetric complex input, i.e. a
|
||
|
|
signal with a real spectrum.
|
||
|
|
|
||
|
|
This function computes the N-dimensional discrete Fourier Transform for a
|
||
|
|
Hermitian symmetric complex input over any number of axes in an
|
||
|
|
M-dimensional array by means of the Fast Fourier Transform (FFT). In other
|
||
|
|
words, ``ihfftn(hfftn(x, s)) == x`` to within numerical accuracy. (``s``
|
||
|
|
here is ``x.shape`` with ``s[-1] = x.shape[-1] * 2 - 1``, this is necessary
|
||
|
|
for the same reason ``x.shape`` would be necessary for `irfft`.)
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : 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.
|
||
|
|
norm : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` or `x`,
|
||
|
|
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 `x`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
ihfftn : The inverse n-dimensional FFT with real spectrum. Inverse of `hfftn`.
|
||
|
|
fft : The one-dimensional FFT, with definitions and conventions used.
|
||
|
|
rfft : Forward FFT of real input
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
|
||
|
|
For a 1 dimensional signal ``x`` to have a real spectrum, it must satisfy
|
||
|
|
the Hermitian property::
|
||
|
|
|
||
|
|
x[i] == np.conj(x[-i]) for all i
|
||
|
|
|
||
|
|
This generalizes into higher dimensions by reflecting over each axis in
|
||
|
|
turn::
|
||
|
|
|
||
|
|
x[i, j, k, ...] == np.conj(x[-i, -j, -k, ...]) for all i, j, k, ...
|
||
|
|
|
||
|
|
This should not be confused with a Hermitian matrix, for which the
|
||
|
|
transpose is it's own conjugate::
|
||
|
|
|
||
|
|
x[i, j] == np.conj(x[j, i]) for all i, j
|
||
|
|
|
||
|
|
|
||
|
|
The default value of `s` assumes an even output length in the final
|
||
|
|
transformation axis. When performing the final complex to real
|
||
|
|
transformation, the Hermitian symmetry requires that the last imaginary
|
||
|
|
component along that axis must be 0 and so it is ignored. To avoid losing
|
||
|
|
information, the correct length of the real input *must* be given.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> x = np.ones((3, 2, 2))
|
||
|
|
>>> scipy.fft.hfftn(x)
|
||
|
|
array([[[12., 0.],
|
||
|
|
[ 0., 0.]],
|
||
|
|
[[ 0., 0.],
|
||
|
|
[ 0., 0.]],
|
||
|
|
[[ 0., 0.],
|
||
|
|
[ 0., 0.]]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def hfft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False, workers=None):
|
||
|
|
"""
|
||
|
|
Compute the 2-dimensional FFT of a Hermitian complex array.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array
|
||
|
|
Input array, taken to be Hermitian complex.
|
||
|
|
s : sequence of ints, optional
|
||
|
|
Shape of the real output.
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
Axes over which to compute the FFT.
|
||
|
|
norm : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The real result of the 2D Hermitian complex real FFT.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
hfftn : Compute the N-dimensional discrete Fourier Transform for Hermitian
|
||
|
|
complex input.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is really just `hfftn` with different default behavior.
|
||
|
|
For more details see `hfftn`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def ihfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None):
|
||
|
|
"""
|
||
|
|
Compute the N-dimensional inverse discrete Fourier Transform for a real
|
||
|
|
spectrum.
|
||
|
|
|
||
|
|
This function computes the N-dimensional inverse discrete Fourier Transform
|
||
|
|
over any number of axes in an M-dimensional real array by means of the Fast
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|
|
Fourier Transform (FFT). By default, all axes are transformed, with the
|
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|
real transform performed over the last axis, while the remaining transforms
|
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|
|
are complex.
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|
|
|
||
|
|
Parameters
|
||
|
|
----------
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|
|
x : array_like
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|
|
Input array, taken to be real.
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|
|
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.).
|
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|
|
Along any axis, if the given shape is smaller than that of the input,
|
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|
|
the input is cropped. If it is larger, the input is padded with zeros.
|
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|
|
if `s` is not given, the shape of the input along the axes specified
|
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|
|
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 : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : complex ndarray
|
||
|
|
The truncated or zero-padded input, transformed along the axes
|
||
|
|
indicated by `axes`, or by a combination of `s` and `x`,
|
||
|
|
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 `x`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
hfftn : The forward n-dimensional FFT of Hermitian input.
|
||
|
|
hfft : The one-dimensional FFT of Hermitian input.
|
||
|
|
fft : The one-dimensional FFT, with definitions and conventions used.
|
||
|
|
fftn : The n-dimensional FFT.
|
||
|
|
hfft2 : The two-dimensional FFT of Hermitian input.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
|
||
|
|
The transform for real input is performed over the last transformation
|
||
|
|
axis, as by `ihfft`, then the transform over the remaining axes is
|
||
|
|
performed as by `ifftn`. The order of the output is the positive part of
|
||
|
|
the Hermitian output signal, in the same format as `rfft`.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import scipy.fft
|
||
|
|
>>> x = np.ones((2, 2, 2))
|
||
|
|
>>> scipy.fft.ihfftn(x)
|
||
|
|
array([[[1.+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]]])
|
||
|
|
>>> scipy.fft.ihfftn(x, axes=(2, 0))
|
||
|
|
array([[[1.+0.j, 0.+0.j], # may vary
|
||
|
|
[1.+0.j, 0.+0.j]],
|
||
|
|
[[0.+0.j, 0.+0.j],
|
||
|
|
[0.+0.j, 0.+0.j]]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
|
||
|
|
|
||
|
|
@_dispatch
|
||
|
|
def ihfft2(x, s=None, axes=(-2, -1), norm=None, overwrite_x=False,
|
||
|
|
workers=None):
|
||
|
|
"""
|
||
|
|
Compute the 2-dimensional inverse FFT of a real spectrum.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
The input array
|
||
|
|
s : sequence of ints, optional
|
||
|
|
Shape of the real input 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 : {None, "ortho"}, optional
|
||
|
|
Normalization mode (see `fft`). Default is None.
|
||
|
|
overwrite_x : bool, optional
|
||
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
|
See :func:`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 :func:`~scipy.fft.fft` for more details.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
out : ndarray
|
||
|
|
The result of the inverse real 2-D FFT.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
ihfftn : Compute the inverse of the N-dimensional FFT of Hermitian input.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is really `ihfftn` with different defaults.
|
||
|
|
For more details see `ihfftn`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
return (Dispatchable(x, np.ndarray),)
|