package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Return the Discrete Cosine Transform of arbitrary type sequence x.

Parameters ---------- x : array_like The input array. type :

, 2, 3, 4

, optional Type of the DCT (see Notes). Default type is 2. n : int, optional Length of the transform. If ``n < x.shapeaxis``, `x` is truncated. If ``n > x.shapeaxis``, `x` is zero-padded. The default results in ``n = x.shapeaxis``. axis : int, optional Axis along which the dct is computed; the default is over the last axis (i.e., ``axis=-1``). norm : None, 'ortho', optional Normalization mode (see Notes). Default is None. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. 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 ------- y : ndarray of real The transformed input array.

See Also -------- idct : Inverse DCT

Notes ----- For a single dimension array ``x``, ``dct(x, norm='ortho')`` is equal to MATLAB ``dct(x)``.

For ``norm=None``, there is no scaling on `dct` and the `idct` is scaled by ``1/N`` where ``N`` is the 'logical' size of the DCT. For ``norm='ortho'`` both directions are scaled by the same factor ``1/sqrt(N)``.

There are, theoretically, 8 types of the DCT, only the first 4 types are implemented in SciPy.'The' DCT generally refers to DCT type 2, and 'the' Inverse DCT generally refers to DCT type 3.

**Type I**

There are several definitions of the DCT-I; we use the following (for ``norm=None``)

.. math::

y_k = x_0 + (-1)^k x_N-1 + 2 \sum_n=1^N-2 x_n \cos\left( \frac\pi k nN-1 \right)

If ``norm='ortho'``, ``x0`` and ``xN-1`` are multiplied by a scaling factor of :math:`\sqrt

`, and ``yk`` is multiplied by a scaling factor ``f``

.. math::

f = \begincases \frac

\sqrt\frac{1N-1

}

& \textfk=0\text or N-1, \\ \frac

\sqrt\frac{2N-1

}

& \textotherwise \endcases

.. note:: The DCT-I is only supported for input size > 1.

**Type II**

There are several definitions of the DCT-II; we use the following (for ``norm=None``)

.. math::

y_k = 2 \sum_n=0^N-1 x_n \cos\left(\frac\pi k(2n+1)

N

\right)

If ``norm='ortho'``, ``yk`` is multiplied by a scaling factor ``f``

.. math:: f = \begincases \sqrt\frac{1

N

}

& \textfk=0, \\ \sqrt\frac{1

N

}

& \textotherwise \endcases

which makes the corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``).

**Type III**

There are several definitions, we use the following (for ``norm=None``)

.. math::

y_k = x_0 + 2 \sum_n=1^N-1 x_n \cos\left(\frac\pi(2k+1)n

N

\right)

or, for ``norm='ortho'``

.. math::

y_k = \fracx_0\sqrt{N

}

1. \sqrt\frac{2N

}

\sum_n=1^N-1 x_n \cos\left(\frac\pi(2k+1)n

N

\right)

The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up to a factor `2N`. The orthonormalized DCT-III is exactly the inverse of the orthonormalized DCT-II.

**Type IV**

There are several definitions of the DCT-IV; we use the following (for ``norm=None``)

.. math::

y_k = 2 \sum_n=0^N-1 x_n \cos\left(\frac\pi(2k+1)(2n+1)

N

\right)

If ``norm='ortho'``, ``yk`` is multiplied by a scaling factor ``f``

.. math::

f = \frac

\sqrt{2N

}

References ---------- .. 1 'A Fast Cosine Transform in One and Two Dimensions', by J. Makhoul, `IEEE Transactions on acoustics, speech and signal processing` vol. 28(1), pp. 27-34, :doi:`10.1109/TASSP.1980.1163351` (1980). .. 2 Wikipedia, 'Discrete cosine transform', https://en.wikipedia.org/wiki/Discrete_cosine_transform

Examples -------- The Type 1 DCT is equivalent to the FFT (though faster) for real, even-symmetrical inputs. The output is also real and even-symmetrical. Half of the FFT input is used to generate half of the FFT output:

>>> from scipy.fft import fft, dct >>> fft(np.array(4., 3., 5., 10., 5., 3.)).real array( 30., -8., 6., -2., 6., -8.) >>> dct(np.array(4., 3., 5., 10.), 1) array( 30., -8., 6., -2.)

Return multidimensional Discrete Cosine Transform along the specified axes.

Parameters ---------- x : array_like The input array. type :

, 2, 3, 4

, optional Type of the DCT (see Notes). Default type is 2. s : int or array_like of ints or None, optional The shape of the result. If both `s` and `axes` (see below) are None, `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is ``scipy.take(x.shape, axes, axis=0)``. If ``si > x.shapei``, the ith dimension is padded with zeros. If ``si < x.shapei``, the ith dimension is truncated to length ``si``. If any element of `s` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints or None, optional Axes over which the DCT is computed. 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 Notes). Default is None. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. 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 ------- y : ndarray of real The transformed input array.

See Also -------- idctn : Inverse multidimensional DCT

Notes ----- For full details of the DCT types and normalization modes, as well as references, see `dct`.

Examples -------- >>> from scipy.fft import dctn, idctn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idctn(dctn(y))) True

Return the Discrete Sine Transform of arbitrary type sequence x.

Parameters ---------- x : array_like The input array. type :

, 2, 3, 4

, optional Type of the DST (see Notes). Default type is 2. n : int, optional Length of the transform. If ``n < x.shapeaxis``, `x` is truncated. If ``n > x.shapeaxis``, `x` is zero-padded. The default results in ``n = x.shapeaxis``. axis : int, optional Axis along which the dst is computed; the default is over the last axis (i.e., ``axis=-1``). norm : None, 'ortho', optional Normalization mode (see Notes). Default is None. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. 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 ------- dst : ndarray of reals The transformed input array.

See Also -------- idst : Inverse DST

Notes ----- For a single dimension array ``x``.

For ``norm=None``, there is no scaling on the `dst` and the `idst` is scaled by ``1/N`` where ``N`` is the 'logical' size of the DST. For ``norm='ortho'`` both directions are scaled by the same factor ``1/sqrt(N)``.

There are, theoretically, 8 types of the DST for different combinations of even/odd boundary conditions and boundary off sets 1_, only the first 4 types are implemented in SciPy.

**Type I**

There are several definitions of the DST-I; we use the following for ``norm=None``. DST-I assumes the input is odd around `n=-1` and `n=N`.

.. math::

y_k = 2 \sum_n=0^N-1 x_n \sin\left(\frac\pi(k+1)(n+1)N+1\right)

Note that the DST-I is only supported for input size > 1. The (unnormalized) DST-I is its own inverse, up to a factor `2(N+1)`. The orthonormalized DST-I is exactly its own inverse.

**Type II**

There are several definitions of the DST-II; we use the following for ``norm=None``. DST-II assumes the input is odd around `n=-1/2` and `n=N-1/2`; the output is odd around :math:`k=-1` and even around `k=N-1`

.. math::

y_k = 2 \sum_n=0^N-1 x_n \sin\left(\frac\pi(k+1)(2n+1)

N

\right)

if ``norm='ortho'``, ``yk`` is multiplied by a scaling factor ``f``

.. math::

f = \begincases \sqrt\frac{1

N

}

& \textfk = 0, \\ \sqrt\frac{1

N

}

& \textotherwise \endcases

**Type III**

There are several definitions of the DST-III, we use the following (for ``norm=None``). DST-III assumes the input is odd around `n=-1` and even around `n=N-1`

.. math::

y_k = (-1)^k x_N-1 + 2 \sum_n=0^N-2 x_n \sin\left( \frac\pi(2k+1)(n+1)

N

\right)

The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up to a factor `2N`. The orthonormalized DST-III is exactly the inverse of the orthonormalized DST-II.

**Type IV**

There are several definitions of the DST-IV, we use the following (for ``norm=None``). DST-IV assumes the input is odd around `n=-0.5` and even around `n=N-0.5`

.. math::

y_k = 2 \sum_n=0^N-1 x_n \sin\left(\frac\pi(2k+1)(2n+1)

N

\right)

The (unnormalized) DST-IV is its own inverse, up to a factor `2N`. The orthonormalized DST-IV is exactly its own inverse.

References ---------- .. 1 Wikipedia, 'Discrete sine transform', https://en.wikipedia.org/wiki/Discrete_sine_transform

Return multidimensional Discrete Sine Transform along the specified axes.

Parameters ---------- x : array_like The input array. type :

, 2, 3, 4

, optional Type of the DST (see Notes). Default type is 2. s : int or array_like of ints or None, optional The shape of the result. If both `s` and `axes` (see below) are None, `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is ``scipy.take(x.shape, axes, axis=0)``. If ``si > x.shapei``, the ith dimension is padded with zeros. If ``si < x.shapei``, the ith dimension is truncated to length ``si``. If any element of `shape` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints or None, optional Axes over which the DST is computed. 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 Notes). Default is None. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. 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 ------- y : ndarray of real The transformed input array.

See Also -------- idstn : Inverse multidimensional DST

Notes ----- For full details of the DST types and normalization modes, as well as references, see `dst`.

Examples -------- >>> from scipy.fft import dstn, idstn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idstn(dstn(y))) True

Compute the 1-D discrete Fourier Transform.

This function computes the 1-D *n*-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm 1_.

Parameters ---------- x : array_like Input array, can be complex. n : int, optional Length of the transformed axis of the output. If `n` is smaller than the length of the input, the input is cropped. If it is larger, the input is padded with zeros. If `n` is not given, the length of the input along the axis specified by `axis` is used. axis : int, optional Axis over which to compute the FFT. If not given, the last axis is used. norm : None, 'ortho', optional Normalization mode. Default is None, meaning no normalization on the forward transforms and scaling by ``1/n`` on the `ifft`. For ``norm='ortho'``, both directions are scaled by ``1/sqrt(n)``. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. See the notes below 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 below for more details. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

Returns ------- out : complex ndarray The truncated or zero-padded input, transformed along the axis indicated by `axis`, or the last one if `axis` is not specified.

Raises ------ IndexError if `axes` is larger than the last axis of `x`.

See Also -------- ifft : The inverse of `fft`. fft2 : The 2-D FFT. fftn : The N-D FFT. rfftn : The N-D FFT of real input. fftfreq : Frequency bins for given FFT parameters. next_fast_len : Size to pad input to for most efficient transforms

Notes -----

FFT (Fast Fourier Transform) refers to a way the discrete Fourier Transform (DFT) can be calculated efficiently, by using symmetries in the calculated terms. The symmetry is highest when `n` is a power of 2, and the transform is therefore most efficient for these sizes. For poorly factorizable sizes, `scipy.fft` uses Bluestein's algorithm 2_ and so is never worse than O(`n` log `n`). Further performance improvements may be seen by zero-padding the input using `next_fast_len`.

If ``x`` is a 1d array, then the `fft` is equivalent to ::

yk = np.sum(x * np.exp(-2j * np.pi * k * np.arange(n)/n))

The frequency term ``f=k/n`` is found at ``yk``. At ``yn/2`` we reach the Nyquist frequency and wrap around to the negative-frequency terms. So, for an 8-point transform, the frequencies of the result are 0, 1, 2, 3, -4, -3, -2, -1. To rearrange the fft output so that the zero-frequency component is centered, like -4, -3, -2, -1, 0, 1, 2, 3, use `fftshift`.

Transforms can be done in single, double, or extended precision (long double) floating point. Half precision inputs will be converted to single precision and non-floating-point inputs will be converted to double precision.

If the data type of ``x`` is real, a 'real FFT' algorithm is automatically used, which roughly halves the computation time. To increase efficiency a little further, use `rfft`, which does the same calculation, but only outputs half of the symmetrical spectrum. If the data are both real and symmetrical, the `dct` can again double the efficiency, by generating half of the spectrum from half of the signal.

When ``overwrite_x=True`` is specified, the memory referenced by ``x`` may be used by the implementation in any way. This may include reusing the memory for the result, but this is in no way guaranteed. You should not rely on the contents of ``x`` after the transform as this may change in future without warning.

The ``workers`` argument specifies the maximum number of parallel jobs to split the FFT computation into. This will execute independent 1-D FFTs within ``x``. So, ``x`` must be at least 2-D and the non-transformed axes must be large enough to split into chunks. If ``x`` is too small, fewer jobs may be used than requested.

References ---------- .. 1 Cooley, James W., and John W. Tukey, 1965, 'An algorithm for the machine calculation of complex Fourier series,' *Math. Comput.* 19: 297-301. .. 2 Bluestein, L., 1970, 'A linear filtering approach to the computation of discrete Fourier transform'. *IEEE Transactions on Audio and Electroacoustics.* 18 (4): 451-455.

Examples -------- >>> import scipy.fft >>> scipy.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8)) array(-2.33486982e-16+1.14423775e-17j, 8.00000000e+00-1.25557246e-15j, 2.33486982e-16+2.33486982e-16j, 0.00000000e+00+1.22464680e-16j, -1.14423775e-17+2.33486982e-16j, 0.00000000e+00+5.20784380e-16j, 1.14423775e-17+1.14423775e-17j, 0.00000000e+00+1.22464680e-16j)

In this example, real input has an FFT which is Hermitian, i.e., symmetric in the real part and anti-symmetric in the imaginary part:

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

Compute the 2-D discrete Fourier Transform

This function computes the N-D discrete Fourier Transform over any axes in an M-D 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 (``s0`` refers to axis 0, ``s1`` 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

Returns ------- out : complex ndarray The truncated or zero-padded input, transformed along the axes indicated by `axes`, or the last two axes if `axes` is not given.

Raises ------ ValueError If `s` and `axes` have different length, or `axes` not given and ``len(s) != 2``. IndexError If an element of `axes` is larger than than the number of axes of `x`.

See Also -------- ifft2 : The inverse 2-D FFT. fft : The 1-D FFT. fftn : The N-D FFT. fftshift : Shifts zero-frequency terms to the center of the array. For 2-D 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, :50 >>> 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 the Discrete Fourier Transform sample frequencies.

The returned float array `f` contains the frequency bin centers in cycles per unit of the sample spacing (with zero at the start). For instance, if the sample spacing is in seconds, then the frequency unit is cycles/second.

Given a window length `n` and a sample spacing `d`::

f = 0, 1, ..., n/2-1, -n/2, ..., -1 / (d*n) if n is even f = 0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1 / (d*n) if n is odd

Parameters ---------- n : int Window length. d : scalar, optional Sample spacing (inverse of the sampling rate). Defaults to 1.

Returns ------- f : ndarray Array of length `n` containing the sample frequencies.

Examples -------- >>> signal = np.array(-2, 8, 6, 4, 1, 0, 3, 5, dtype=float) >>> fourier = np.fft.fft(signal) >>> n = signal.size >>> timestep = 0.1 >>> freq = np.fft.fftfreq(n, d=timestep) >>> freq array( 0. , 1.25, 2.5 , ..., -3.75, -2.5 , -1.25)

Compute the N-D discrete Fourier Transform.

This function computes the N-D discrete Fourier Transform over any number of axes in an M-D 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 (``s0`` refers to axis 0, ``s1`` 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

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-D FFT. fft : The 1-D FFT, with definitions and conventions used. rfftn : The N-D FFT of real input. fft2 : The 2-D 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, :30 >>> 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()

Shift the zero-frequency component to the center of the spectrum.

This function swaps half-spaces for all axes listed (defaults to all). Note that ``y0`` is the Nyquist component only if ``len(x)`` is even.

Parameters ---------- x : array_like Input array. axes : int or shape tuple, optional Axes over which to shift. Default is None, which shifts all axes.

Returns ------- y : ndarray The shifted array.

See Also -------- ifftshift : The inverse of `fftshift`.

Examples -------- >>> freqs = np.fft.fftfreq(10, 0.1) >>> freqs array( 0., 1., 2., ..., -3., -2., -1.) >>> np.fft.fftshift(freqs) array(-5., -4., -3., -2., -1., 0., 1., 2., 3., 4.)

Shift the zero-frequency component only along the second axis:

>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3) >>> freqs array([ 0., 1., 2.], [ 3., 4., -4.], [-3., -2., -1.]) >>> np.fft.fftshift(freqs, axes=(1,)) array([ 2., 0., 1.], [-4., 3., 4.], [-1., -3., -2.])

Returns the default number of workers within the current context

Examples -------- >>> from scipy import fft >>> fft.get_workers() 1 >>> with fft.set_workers(4): ... fft.get_workers() 4

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

.. versionadded:: 1.5.0

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 1-D FFT for real input. ihfft : The inverse of `hfft`. hfftn : Compute the N-D 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.)

Compute the 2-D 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

Returns ------- out : ndarray The real result of the 2-D Hermitian complex real FFT.

See Also -------- hfftn : Compute the N-D discrete Fourier Transform for Hermitian complex input.

Notes ----- This is really just `hfftn` with different default behavior. For more details see `hfftn`.

Compute the N-D FFT of Hermitian symmetric complex input, i.e., a signal with a real spectrum.

This function computes the N-D discrete Fourier Transform for a Hermitian symmetric complex input over any number of axes in an M-D 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 (``s0`` refers to axis 0, ``s1`` 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

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-D FFT with real spectrum. Inverse of `hfftn`. fft : The 1-D FFT, with definitions and conventions used. rfft : Forward FFT of real input.

Notes -----

For a 1-D signal ``x`` to have a real spectrum, it must satisfy the Hermitian property::

xi == np.conj(x-i) for all i

This generalizes into higher dimensions by reflecting over each axis in turn::

xi, 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 its own conjugate::

xi, j == np.conj(xj, 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 the Inverse Discrete Cosine Transform of an arbitrary type sequence.

Parameters ---------- x : array_like The input array. type :

, 2, 3, 4

, optional Type of the DCT (see Notes). Default type is 2. n : int, optional Length of the transform. If ``n < x.shapeaxis``, `x` is truncated. If ``n > x.shapeaxis``, `x` is zero-padded. The default results in ``n = x.shapeaxis``. axis : int, optional Axis along which the idct is computed; the default is over the last axis (i.e., ``axis=-1``). norm : None, 'ortho', optional Normalization mode (see Notes). Default is None. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. 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 ------- idct : ndarray of real The transformed input array.

See Also -------- dct : Forward DCT

Notes ----- For a single dimension array `x`, ``idct(x, norm='ortho')`` is equal to MATLAB ``idct(x)``.

'The' IDCT is the IDCT-II, which is the same as the normalized DCT-III.

The IDCT is equivalent to a normal DCT except for the normalization and type. DCT type 1 and 4 are their own inverse and DCTs 2 and 3 are each other's inverses.

Examples -------- The Type 1 DCT is equivalent to the DFT for real, even-symmetrical inputs. The output is also real and even-symmetrical. Half of the IFFT input is used to generate half of the IFFT output:

>>> from scipy.fft import ifft, idct >>> ifft(np.array( 30., -8., 6., -2., 6., -8.)).real array( 4., 3., 5., 10., 5., 3.) >>> idct(np.array( 30., -8., 6., -2.), 1) array( 4., 3., 5., 10.)

Return multidimensional Discrete Cosine Transform along the specified axes.

Parameters ---------- x : array_like The input array. type :

, 2, 3, 4

, optional Type of the DCT (see Notes). Default type is 2. s : int or array_like of ints or None, optional The shape of the result. If both `s` and `axes` (see below) are None, `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is ``scipy.take(x.shape, axes, axis=0)``. If ``si > x.shapei``, the ith dimension is padded with zeros. If ``si < x.shapei``, the ith dimension is truncated to length ``si``. If any element of `s` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints or None, optional Axes over which the IDCT is computed. 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 Notes). Default is None. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. 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 ------- y : ndarray of real The transformed input array.

See Also -------- dctn : multidimensional DCT

Notes ----- For full details of the IDCT types and normalization modes, as well as references, see `idct`.

Examples -------- >>> from scipy.fft import dctn, idctn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idctn(dctn(y))) True

Return the Inverse Discrete Sine Transform of an arbitrary type sequence.

Parameters ---------- x : array_like The input array. type :

, 2, 3, 4

, optional Type of the DST (see Notes). Default type is 2. n : int, optional Length of the transform. If ``n < x.shapeaxis``, `x` is truncated. If ``n > x.shapeaxis``, `x` is zero-padded. The default results in ``n = x.shapeaxis``. axis : int, optional Axis along which the idst is computed; the default is over the last axis (i.e., ``axis=-1``). norm : None, 'ortho', optional Normalization mode (see Notes). Default is None. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. 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 ------- idst : ndarray of real The transformed input array.

See Also -------- dst : Forward DST

Notes -----

'The' IDST is the IDST-II, which is the same as the normalized DST-III.

The IDST is equivalent to a normal DST except for the normalization and type. DST type 1 and 4 are their own inverse and DSTs 2 and 3 are each other's inverses.

Return multidimensional Discrete Sine Transform along the specified axes.

Parameters ---------- x : array_like The input array. type :

, 2, 3, 4

, optional Type of the DST (see Notes). Default type is 2. s : int or array_like of ints or None, optional The shape of the result. If both `s` and `axes` (see below) are None, `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is ``scipy.take(x.shape, axes, axis=0)``. If ``si > x.shapei``, the ith dimension is padded with zeros. If ``si < x.shapei``, the ith dimension is truncated to length ``si``. If any element of `s` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints or None, optional Axes over which the IDST is computed. 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 Notes). Default is None. overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. 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 ------- y : ndarray of real The transformed input array.

See Also -------- dstn : multidimensional DST

Notes ----- For full details of the IDST types and normalization modes, as well as references, see `idst`.

Examples -------- >>> from scipy.fft import dstn, idstn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idstn(dstn(y))) True

Compute the 1-D inverse discrete Fourier Transform.

This function computes the inverse of the 1-D *n*-point discrete Fourier transform computed by `fft`. In other words, ``ifft(fft(x)) == x`` to within numerical accuracy.

The input should be ordered in the same way as is returned by `fft`, i.e.,

* ``x0`` should contain the zero frequency term, * ``x1:n//2`` should contain the positive-frequency terms, * ``xn//2 + 1:`` should contain the negative-frequency terms, in increasing order starting from the most negative frequency.

For an even number of input points, ``xn//2`` represents the sum of the values at the positive and negative Nyquist frequencies, as the two are aliased together. See `fft` for details.

Parameters ---------- x : array_like Input array, can be complex. n : int, optional Length of the transformed axis of the output. If `n` is smaller than the length of the input, the input is cropped. If it is larger, the input is padded with zeros. If `n` is not given, the length of the input along the axis specified by `axis` is used. See notes about padding issues. axis : int, optional Axis over which to compute the inverse DFT. 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

Returns ------- out : complex ndarray The truncated or zero-padded input, transformed along the axis indicated by `axis`, or the last one if `axis` is not specified.

Raises ------ IndexError If `axes` is larger than the last axis of `x`.

See Also -------- fft : The 1-D (forward) FFT, of which `ifft` is the inverse. ifft2 : The 2-D inverse FFT. ifftn : The N-D inverse FFT.

Notes ----- If the input parameter `n` is larger than the size of the input, the input is padded by appending zeros at the end. Even though this is the common approach, it might lead to surprising results. If a different padding is desired, it must be performed before calling `ifft`.

If ``x`` is a 1-D array, then the `ifft` is equivalent to ::

yk = np.sum(x * np.exp(2j * np.pi * k * np.arange(n)/n)) / len(x)

As with `fft`, `ifft` has support for all floating point types and is optimized for real input.

Examples -------- >>> import scipy.fft >>> scipy.fft.ifft(0, 4, 0, 0) array( 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j) # may vary

Create and plot a band-limited signal with random phases:

>>> import matplotlib.pyplot as plt >>> t = np.arange(400) >>> n = np.zeros((400,), dtype=complex) >>> n40:60 = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,))) >>> s = scipy.fft.ifft(n) >>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--') <matplotlib.lines.Line2D object at ...>, <matplotlib.lines.Line2D object at ...> >>> plt.legend(('real', 'imaginary')) <matplotlib.legend.Legend object at ...> >>> plt.show()

Compute the 2-D inverse discrete Fourier Transform.

This function computes the inverse of the 2-D discrete Fourier Transform over any number of axes in an M-D 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 (``s0`` refers to axis 0, ``s1`` 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

Returns ------- out : complex ndarray The truncated or zero-padded input, transformed along the axes indicated by `axes`, or the last two axes if `axes` is not given.

Raises ------ ValueError If `s` and `axes` have different length, or `axes` not given and ``len(s) != 2``. IndexError If an element of `axes` is larger than than the number of axes of `x`.

See Also -------- fft2 : The forward 2-D FFT, of which `ifft2` is the inverse. ifftn : The inverse of the N-D FFT. fft : The 1-D FFT. ifft : The 1-D 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])

Compute the N-D inverse discrete Fourier Transform.

This function computes the inverse of the N-D discrete Fourier Transform over any number of axes in an M-D 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 (``s0`` refers to axis 0, ``s1`` 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

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-D FFT, of which `ifftn` is the inverse. ifft : The 1-D inverse FFT. ifft2 : The 2-D 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) >>> n60:80, 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()

The inverse of `fftshift`. Although identical for even-length `x`, the functions differ by one sample for odd-length `x`.

Parameters ---------- x : array_like Input array. axes : int or shape tuple, optional Axes over which to calculate. Defaults to None, which shifts all axes.

Returns ------- y : ndarray The shifted array.

See Also -------- fftshift : Shift zero-frequency component to the center of the spectrum.

Examples -------- >>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3) >>> freqs array([ 0., 1., 2.], [ 3., 4., -4.], [-3., -2., -1.]) >>> np.fft.ifftshift(np.fft.fftshift(freqs)) array([ 0., 1., 2.], [ 3., 4., -4.], [-3., -2., -1.])

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

.. versionadded:: 1.5.0

Returns ------- out : complex ndarray The truncated or zero-padded input, transformed along the axis indicated by `axis`, or the last one if `axis` is not specified. 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

Compute the 2-D 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

Returns ------- out : ndarray The result of the inverse real 2-D FFT.

See Also -------- ihfftn : Compute the inverse of the N-D FFT of Hermitian input.

Notes ----- This is really `ihfftn` with different defaults. For more details see `ihfftn`.

Compute the N-D inverse discrete Fourier Transform for a real spectrum.

This function computes the N-D inverse discrete Fourier Transform over any number of axes in an M-D 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. (``s0`` refers to axis 0, ``s1`` to axis 1, etc.). 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

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-D FFT of Hermitian input. hfft : The 1-D FFT of Hermitian input. fft : The 1-D FFT, with definitions and conventions used. fftn : The N-D FFT. hfft2 : The 2-D 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]])

Computes the inverse of `rfft`.

This function computes the inverse of the 1-D *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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

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 1-D FFT of real input, of which `irfft` is inverse. fft : The 1-D FFT. irfft2 : The inverse of the 2-D FFT of real input. irfftn : The inverse of the N-D 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.

Computes the inverse of `rfft2`

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

.. versionadded:: 1.5.0

Returns ------- out : ndarray The result of the inverse real 2-D FFT.

See Also -------- rfft2 : The 2-D FFT of real input. irfft : The inverse of the 1-D FFT of real input. irfftn : The inverse of the N-D FFT of real input.

Notes ----- This is really `irfftn` with different defaults. For more details see `irfftn`.

Computes the inverse of `rfftn`

This function computes the inverse of the N-D discrete Fourier Transform for real input over any number of axes in an M-D 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 (``s0`` refers to axis 0, ``s1`` 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

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-D FFT of real input, of which `ifftn` is the inverse. fft : The 1-D FFT, with definitions and conventions used. irfft : The inverse of the 1-D FFT of real input. irfft2 : The inverse of the 2-D 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)) >>> x0, 0, 0 = 3 * 2 * 2 >>> scipy.fft.irfftn(x) array([[1., 1.], [1., 1.]], [[1., 1.], [1., 1.]], [[1., 1.], [1., 1.]])

Register a backend for permanent use.

Registered backends have the lowest priority and will be tried after the global backend.

Parameters ---------- backend: object, 'scipy' The backend to use. Can either be a ``str`` containing the name of a known backend 'scipy' or an object that implements the uarray protocol.

Raises ------ ValueError: If the backend does not implement ``numpy.scipy.fft``.

Examples -------- We can register a new fft backend:

>>> from scipy.fft import fft, register_backend, set_global_backend >>> class NoopBackend: # Define an invalid Backend ... __ua_domain__ = 'numpy.scipy.fft' ... def __ua_function__(self, func, args, kwargs): ... return NotImplemented >>> set_global_backend(NoopBackend()) # Set the invalid backend as global >>> register_backend('scipy') # Register a new backend >>> fft(1) # The registered backend is called because the global backend returns `NotImplemented` array(1.+0.j) >>> set_global_backend('scipy') # Restore global backend to default

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

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

Parameters ---------- 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 : 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

Returns ------- out : complex ndarray The truncated or zero-padded input, transformed along the axis indicated by `axis`, or the last one if `axis` is not specified. If `n` is even, the length of the transformed axis is ``(n/2)+1``. If `n` is odd, the length is ``(n+1)/2``.

Raises ------ IndexError If `axis` is larger than the last axis of `a`.

See Also -------- irfft : The inverse of `rfft`. fft : The 1-D FFT of general (complex) input. fftn : The N-D FFT. rfft2 : The 2-D FFT of real input. rfftn : The N-D FFT of real input.

Notes ----- When the DFT is computed for purely real input, the output is Hermitian-symmetric, i.e., the negative frequency terms are just the complex conjugates of the corresponding positive-frequency terms, and the negative-frequency terms are therefore redundant. This function does not compute the negative frequency terms, and the length of the transformed axis of the output is therefore ``n//2 + 1``.

When ``X = rfft(x)`` and fs is the sampling frequency, ``X0`` 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 -------- >>> import scipy.fft >>> scipy.fft.fft(0, 1, 0, 0) array( 1.+0.j, 0.-1.j, -1.+0.j, 0.+1.j) # may vary >>> scipy.fft.rfft(0, 1, 0, 0) array( 1.+0.j, 0.-1.j, -1.+0.j) # may vary

Notice how the final element of the `fft` output is the complex conjugate of the second element, for real input. For `rfft`, this symmetry is exploited to compute only the non-negative frequency terms.

Compute the 2-D 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

Returns ------- out : ndarray The result of the real 2-D FFT.

See Also -------- irfft2 : The inverse of the 2-D FFT of real input. rfft : The 1-D FFT of real input. rfftn : Compute the N-D discrete Fourier Transform for real input.

Notes ----- This is really just `rfftn` with different default behavior. For more details see `rfftn`.

Return the Discrete Fourier Transform sample frequencies (for usage with rfft, irfft).

The returned float array `f` contains the frequency bin centers in cycles per unit of the sample spacing (with zero at the start). For instance, if the sample spacing is in seconds, then the frequency unit is cycles/second.

Given a window length `n` and a sample spacing `d`::

f = 0, 1, ..., n/2-1, n/2 / (d*n) if n is even f = 0, 1, ..., (n-1)/2-1, (n-1)/2 / (d*n) if n is odd

Unlike `fftfreq` (but like `scipy.fftpack.rfftfreq`) the Nyquist frequency component is considered to be positive.

Parameters ---------- n : int Window length. d : scalar, optional Sample spacing (inverse of the sampling rate). Defaults to 1.

Returns ------- f : ndarray Array of length ``n//2 + 1`` containing the sample frequencies.

Examples -------- >>> signal = np.array(-2, 8, 6, 4, 1, 0, 3, 5, -3, 4, dtype=float) >>> fourier = np.fft.rfft(signal) >>> n = signal.size >>> sample_rate = 100 >>> freq = np.fft.fftfreq(n, d=1./sample_rate) >>> freq array( 0., 10., 20., ..., -30., -20., -10.) >>> freq = np.fft.rfftfreq(n, d=1./sample_rate) >>> freq array( 0., 10., 20., 30., 40., 50.)

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

This function computes the N-D discrete Fourier Transform over any number of axes in an M-D 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. (``s0`` refers to axis 0, ``s1`` 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. plan: object, optional This argument is reserved for passing in a precomputed plan provided by downstream FFT vendors. It is currently not used in SciPy.

.. versionadded:: 1.5.0

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-D FFT of real input. fft : The 1-D FFT, with definitions and conventions used. rfft : The 1-D FFT of real input. fftn : The N-D FFT. rfft2 : The 2-D 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]])

Context manager to set the backend within a fixed scope.

Upon entering the ``with`` statement, the given backend will be added to the list of available backends with the highest priority. Upon exit, the backend is reset to the state before entering the scope.

Parameters ---------- backend: object, 'scipy' The backend to use. Can either be a ``str`` containing the name of a known backend 'scipy' or an object that implements the uarray protocol. coerce: bool, optional Whether to allow expensive conversions for the ``x`` parameter. e.g., copying a NumPy array to the GPU for a CuPy backend. Implies ``only``. only: bool, optional If only is ``True`` and this backend returns ``NotImplemented``, then a BackendNotImplemented error will be raised immediately. Ignoring any lower priority backends.

Examples -------- >>> import scipy.fft as fft >>> with fft.set_backend('scipy', only=True): ... fft.fft(1) # Always calls the scipy implementation array(1.+0.j)

Sets the global fft backend

The global backend has higher priority than registered backends, but lower priority than context-specific backends set with `set_backend`.

Parameters ---------- backend: object, 'scipy' The backend to use. Can either be a ``str`` containing the name of a known backend 'scipy' or an object that implements the uarray protocol.

Raises ------ ValueError: If the backend does not implement ``numpy.scipy.fft``.

Notes ----- This will overwrite the previously set global backend, which, by default, is the SciPy implementation.

Examples -------- We can set the global fft backend:

>>> from scipy.fft import fft, set_global_backend >>> set_global_backend('scipy') # Sets global backend. 'scipy' is the default backend. >>> fft(1) # Calls the global backend array(1.+0.j)

Context manager for the default number of workers used in `scipy.fft`

Parameters ---------- workers : int The default number of workers to use

Examples -------- >>> from scipy import fft, signal >>> x = np.random.randn(128, 64) >>> with fft.set_workers(4): ... y = signal.fftconvolve(x, x)

Context manager to skip a backend within a fixed scope.

Within the context of a ``with`` statement, the given backend will not be called. This covers backends registered both locally and globally. Upon exit, the backend will again be considered.

Parameters ---------- backend: object, 'scipy' The backend to skip. Can either be a ``str`` containing the name of a known backend 'scipy' or an object that implements the uarray protocol.

Examples -------- >>> import scipy.fft as fft >>> fft.fft(1) # Calls default SciPy backend array(1.+0.j) >>> with fft.skip_backend('scipy'): # We explicitly skip the SciPy backend ... fft.fft(1) # leaving no implementation available Traceback (most recent call last): ... BackendNotImplementedError: No selected backends had an implementation ...