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Fft

dct

function dct
val dct :
  ?type_:[`Two | `Three | `Four | `One] ->
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

Return the Discrete Cosine Transform of arbitrary type sequence x.

Parameters

  • x : array_like The input array.

  • type : {1, 2, 3, 4}, optional Type of the DCT (see Notes). Default type is 2.

  • n : int, optional Length of the transform. If n < x.shape[axis], x is truncated. If n > x.shape[axis], x is zero-padded. The default results in n = x.shape[axis].

  • 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)

y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left( \frac{\pi k n}{N-1} \right)

If norm='ortho', x[0] and x[N-1] are multiplied by a scaling factor of :math:\sqrt{2}, and y[k] is multiplied by a scaling factor f

f = \begin{cases} \frac{1}{2}\sqrt{\frac{1}{N-1}} & \text{if }k=0\text{ or }N-1, \\ \frac{1}{2}\sqrt{\frac{2}{N-1}} & \text{otherwise} \end{cases}

.. 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)

y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right)

If norm='ortho', y[k] is multiplied by a scaling factor f

f = \begin{cases} \sqrt{\frac{1}{4N}} & \text{if }k=0, \\ \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}

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)

y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right)

or, for norm='ortho'

y_k = \frac{x_0}{\sqrt{N}} + \sqrt{\frac{2}{N}} \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\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)

y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)

If norm='ortho', y[k] is multiplied by a scaling factor f

f = \frac{1}{\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.])

dctn

function dctn
val dctn :
  ?type_:[`Two | `Three | `Four | `One] ->
  ?s:[`Array_like_of_ints of Py.Object.t | `I of int] ->
  ?axes:[`Array_like_of_ints of Py.Object.t | `I of int] ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

Return multidimensional Discrete Cosine Transform along the specified axes.

Parameters

  • x : array_like The input array.

  • type : {1, 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 s[i] > x.shape[i], the ith dimension is padded with zeros. If s[i] < x.shape[i], the ith dimension is truncated to length s[i]. 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

dst

function dst
val dst :
  ?type_:[`Two | `Three | `Four | `One] ->
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

Return the Discrete Sine Transform of arbitrary type sequence x.

Parameters

  • x : array_like The input array.

  • type : {1, 2, 3, 4}, optional Type of the DST (see Notes). Default type is 2.

  • n : int, optional Length of the transform. If n < x.shape[axis], x is truncated. If n > x.shape[axis], x is zero-padded. The default results in n = x.shape[axis].

  • 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.

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

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

if norm='ortho', y[k] is multiplied by a scaling factor f

f = \begin{cases} \sqrt{\frac{1}{4N}} & \text{if }k = 0, \\ \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}

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

y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left( \frac{\pi(2k+1)(n+1)}{2N}\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

y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\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

dstn

function dstn
val dstn :
  ?type_:[`Two | `Three | `Four | `One] ->
  ?s:[`Array_like_of_ints of Py.Object.t | `I of int] ->
  ?axes:[`Array_like_of_ints of Py.Object.t | `I of int] ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

Return multidimensional Discrete Sine Transform along the specified axes.

Parameters

  • x : array_like The input array.

  • type : {1, 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 s[i] > x.shape[i], the ith dimension is padded with zeros. If s[i] < x.shape[i], the ith dimension is truncated to length s[i]. 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

fft

function fft
val fft :
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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 ::

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

The frequency term f=k/n is found at y[k]. At y[n/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()

fft2

function fft2
val fft2 :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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 (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.

  • 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, :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        ]])

fftfreq

function fftfreq
val fftfreq :
  ?d:[`F of float | `I of int | `Bool of bool | `S of string] ->
  n:int ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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] / (dn) if n is even f = [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (dn) 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])

fftn

function fftn
val fftn :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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 (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.

  • 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, :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()

fftshift

function fftshift
val fftshift :
  ?axes:[`Shape_tuple of Py.Object.t | `I of int] ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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 y[0] 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.]])

get_workers

function get_workers
val get_workers :
  unit ->
  Py.Object.t

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

hfft

function hfft
val hfft :
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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.])

hfft2

function hfft2
val hfft2 :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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.

hfftn

function hfftn
val hfftn :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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 (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.

  • 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::

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 its 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.]]])

idct

function idct
val idct :
  ?type_:[`Two | `Three | `Four | `One] ->
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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

Parameters

  • x : array_like The input array.

  • type : {1, 2, 3, 4}, optional Type of the DCT (see Notes). Default type is 2.

  • n : int, optional Length of the transform. If n < x.shape[axis], x is truncated. If n > x.shape[axis], x is zero-padded. The default results in n = x.shape[axis].

  • 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.])

idctn

function idctn
val idctn :
  ?type_:[`Two | `Three | `Four | `One] ->
  ?s:[`Array_like_of_ints of Py.Object.t | `I of int] ->
  ?axes:[`Array_like_of_ints of Py.Object.t | `I of int] ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

Return multidimensional Discrete Cosine Transform along the specified axes.

Parameters

  • x : array_like The input array.

  • type : {1, 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 s[i] > x.shape[i], the ith dimension is padded with zeros. If s[i] < x.shape[i], the ith dimension is truncated to length s[i]. 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

idst

function idst
val idst :
  ?type_:[`Two | `Three | `Four | `One] ->
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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

Parameters

  • x : array_like The input array.

  • type : {1, 2, 3, 4}, optional Type of the DST (see Notes). Default type is 2.

  • n : int, optional Length of the transform. If n < x.shape[axis], x is truncated. If n > x.shape[axis], x is zero-padded. The default results in n = x.shape[axis].

  • 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.

idstn

function idstn
val idstn :
  ?type_:[`Two | `Three | `Four | `One] ->
  ?s:[`Array_like_of_ints of Py.Object.t | `I of int] ->
  ?axes:[`Array_like_of_ints of Py.Object.t | `I of int] ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

Return multidimensional Discrete Sine Transform along the specified axes.

Parameters

  • x : array_like The input array.

  • type : {1, 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 s[i] > x.shape[i], the ith dimension is padded with zeros. If s[i] < x.shape[i], the ith dimension is truncated to length s[i]. 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

ifft

function ifft
val ifft :
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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.,

  • x[0] should contain the zero frequency term,
  • x[1:n//2] should contain the positive-frequency terms,
  • x[n//2 + 1:] should contain the negative-frequency terms, in increasing order starting from the most negative frequency.

For an even number of input points, x[n//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 ::

y[k] = 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)
>>> n[40:60] = 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()

ifft2

function ifft2
val ifft2 :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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 (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.

  • 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]])

ifftn

function ifftn
val ifftn :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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 (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.

  • 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)
>>> 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()

ifftshift

function ifftshift
val ifftshift :
  ?axes:[`Shape_tuple of Py.Object.t | `I of int] ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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.]])

ihfft

function ihfft
val ihfft :
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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

ihfft2

function ihfft2
val ihfft2 :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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.

ihfftn

function ihfftn
val ihfftn :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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. (s[0] refers to axis 0, s[1] 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]]])

irfft

function irfft
val irfft :
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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.

irfft2

function irfft2
val irfft2 :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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.

irfftn

function irfftn
val irfftn :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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 (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.

  • 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))
>>> x[0, 0, 0] = 3 * 2 * 2
>>> scipy.fft.irfftn(x)
array([[[1.,  1.],
        [1.,  1.]],
       [[1.,  1.],
        [1.,  1.]],
       [[1.,  1.],
        [1.,  1.]]])

register_backend

function register_backend
val register_backend :
  [`PyObject of Py.Object.t | `Scipy] ->
  Py.Object.t

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

rfft

function rfft
val rfft :
  ?n:int ->
  ?axis:int ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:Py.Object.t ->
  unit ->
  Py.Object.t

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, X[0] contains the zero-frequency term 0*fs, which is real due to Hermitian symmetry.

If n is even, A[-1] contains the term representing both positive and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely real. If n is odd, there is no term at fs/2; A[-1] contains the largest positive frequency (fs/2*(n-1)/n), and is complex in the general case.

If the input a contains an imaginary part, it is silently discarded.

Examples

>>> 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.

rfft2

function rfft2
val rfft2 :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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.

rfftfreq

function rfftfreq
val rfftfreq :
  ?d:[`F of float | `I of int | `Bool of bool | `S of string] ->
  n:int ->
  unit ->
  [`ArrayLike|`Ndarray|`Object] Np.Obj.t

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] / (dn) if n is even f = [0, 1, ..., (n-1)/2-1, (n-1)/2] / (dn) 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.])

rfftn

function rfftn
val rfftn :
  ?s:int list ->
  ?axes:int list ->
  ?norm:string ->
  ?overwrite_x:bool ->
  ?workers:int ->
  ?plan:Py.Object.t ->
  x:[>`Ndarray] Np.Obj.t ->
  unit ->
  Py.Object.t

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. (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.

  • 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]]])

set_backend

function set_backend
val set_backend :
  ?coerce:bool ->
  ?only:bool ->
  backend:[`PyObject of Py.Object.t | `Scipy] ->
  unit ->
  Py.Object.t

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])

set_global_backend

function set_global_backend
val set_global_backend :
  [`PyObject of Py.Object.t | `Scipy] ->
  Py.Object.t

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])

set_workers

function set_workers
val set_workers :
  int ->
  Py.Object.t

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)

skip_backend

function skip_backend
val skip_backend :
  [`PyObject of Py.Object.t | `Scipy] ->
  Py.Object.t

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 ...