numpy.fft._pocketfft

module documentation

(source)

Discrete Fourier Transforms

Routines in this module:

fft(a, n=None, axis=-1, norm="backward") ifft(a, n=None, axis=-1, norm="backward") rfft(a, n=None, axis=-1, norm="backward") irfft(a, n=None, axis=-1, norm="backward") hfft(a, n=None, axis=-1, norm="backward") ihfft(a, n=None, axis=-1, norm="backward") fftn(a, s=None, axes=None, norm="backward") ifftn(a, s=None, axes=None, norm="backward") rfftn(a, s=None, axes=None, norm="backward") irfftn(a, s=None, axes=None, norm="backward") fft2(a, s=None, axes=(-2,-1), norm="backward") ifft2(a, s=None, axes=(-2, -1), norm="backward") rfft2(a, s=None, axes=(-2,-1), norm="backward") irfft2(a, s=None, axes=(-2, -1), norm="backward")

i = inverse transform r = transform of purely real data h = Hermite transform n = n-dimensional transform 2 = 2-dimensional transform (Note: 2D routines are just nD routines with different default behavior.)

Function	`fft`	Compute the one-dimensional discrete Fourier Transform.
Function	`fft2`	Compute the 2-dimensional discrete Fourier Transform.
Function	`fftn`	Compute the N-dimensional discrete Fourier Transform.
Function	`hfft`	Compute the FFT of a signal that has Hermitian symmetry, i.e., a real spectrum.
Function	`ifft`	Compute the one-dimensional inverse discrete Fourier Transform.
Function	`ifft2`	Compute the 2-dimensional inverse discrete Fourier Transform.
Function	`ifftn`	Compute the N-dimensional inverse discrete Fourier Transform.
Function	`ihfft`	Compute the inverse FFT of a signal that has Hermitian symmetry.
Function	`irfft`	Computes the inverse of `rfft`.
Function	`irfft2`	Computes the inverse of `rfft2`.
Function	`irfftn`	Computes the inverse of `rfftn`.
Function	`rfft`	Compute the one-dimensional discrete Fourier Transform for real input.
Function	`rfft2`	Compute the 2-dimensional FFT of a real array.
Function	`rfftn`	Compute the N-dimensional discrete Fourier Transform for real input.
Variable	`array_function_dispatch`	Undocumented
Function	`_cook_nd_args`	Undocumented
Function	`_fft_dispatcher`	Undocumented
Function	`_fftn_dispatcher`	Undocumented
Function	`_get_backward_norm`	Undocumented
Function	`_get_forward_norm`	Undocumented
Function	`_raw_fft`	Undocumented
Function	`_raw_fftnd`	Undocumented
Function	`_swap_direction`	Undocumented
Constant	`_SWAP_DIRECTION_MAP`	Undocumented

@array_function_dispatch(_fft_dispatcher)
def fft(a, n=None, axis=-1, norm=None): (source) ¶

Compute the one-dimensional discrete Fourier Transform.

This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm [CT].

See Also

numpy.fft: for definition of the DFT and conventions used.
ifft: The inverse of fft.
fft2: The two-dimensional FFT.
fftn: The n-dimensional FFT.
rfftn: The n-dimensional FFT of real input.
fftfreq: Frequency bins for given FFT parameters.

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.

The DFT is defined, with the conventions used in this implementation, in the documentation for the numpy.fft module.

References

[CT]	Cooley, James W., and John W. Tukey, 1965, "An algorithm for the machine calculation of complex Fourier series," Math. Comput. 19: 297-301.

Examples

>>> np.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, as described in the numpy.fft documentation:

>>> import matplotlib.pyplot as plt
>>> t = np.arange(256)
>>> sp = np.fft.fft(np.sin(t))
>>> freq = np.fft.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()

Parameters
a:`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:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - 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 `axis` is not a valid axis of `a`.

@array_function_dispatch(_fftn_dispatcher)
def fft2(a, s=None, axes=(-2, -1), norm=None): (source) ¶

Compute the 2-dimensional discrete Fourier Transform.

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

See Also

numpy.fft: Overall view of discrete Fourier transforms, with definitions and conventions used.
ifft2: The inverse two-dimensional FFT.
fft: The one-dimensional FFT.
fftn: The n-dimensional FFT.
fftshift: Shifts zero-frequency terms to the center of the array. For two-dimensional input, swaps first and third quadrants, and second and fourth quadrants.

Notes

fft2 is just fftn with a different default for axes.

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

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

Examples

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

Parameters
a:`array_like`	Input array, can be complex
s:`sequence` of `ints`, optional	Shape (length of each transformed axis) of the output (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). This corresponds to `n` for `fft(x, n)`. Along each axis, if the given shape is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros. if `s` is not given, the shape of the input along the axes specified by `axes` is used.
axes:`sequence` of `ints`, optional	Axes over which to compute the FFT. If not given, the last two axes are used. A repeated index in `axes` means the transform over that axis is performed multiple times. A one-element sequence means that a one-dimensional FFT is performed.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - The truncated or zero-padded input, transformed along the axes indicated by `axes`, or the last two axes if `axes` is not given.
Raises
`ValueError`	If `s` and `axes` have different length, or `axes` not given and `len(s) != 2`.
`IndexError`	If an element of `axes` is larger than than the number of axes of `a`.

@array_function_dispatch(_fftn_dispatcher)
def fftn(a, s=None, axes=None, norm=None): (source) ¶

Compute the N-dimensional discrete Fourier Transform.

This function computes the N-dimensional discrete Fourier Transform over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT).

See Also

numpy.fft: Overall view of discrete Fourier transforms, with definitions and conventions used.
ifftn: The inverse of fftn, the inverse n-dimensional FFT.
fft: The one-dimensional FFT, with definitions and conventions used.
rfftn: The n-dimensional FFT of real input.
fft2: The two-dimensional FFT.
fftshift: Shifts zero-frequency terms to centre of array

Notes

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

See numpy.fft for details, definitions and conventions used.

Examples

>>> a = np.mgrid[:3, :3, :3][0]
>>> np.fft.fftn(a, axes=(1, 2))
array([[[ 0.+0.j,   0.+0.j,   0.+0.j], # may vary
        [ 0.+0.j,   0.+0.j,   0.+0.j],
        [ 0.+0.j,   0.+0.j,   0.+0.j]],
       [[ 9.+0.j,   0.+0.j,   0.+0.j],
        [ 0.+0.j,   0.+0.j,   0.+0.j],
        [ 0.+0.j,   0.+0.j,   0.+0.j]],
       [[18.+0.j,   0.+0.j,   0.+0.j],
        [ 0.+0.j,   0.+0.j,   0.+0.j],
        [ 0.+0.j,   0.+0.j,   0.+0.j]]])
>>> np.fft.fftn(a, (2, 2), axes=(0, 1))
array([[[ 2.+0.j,  2.+0.j,  2.+0.j], # may vary
        [ 0.+0.j,  0.+0.j,  0.+0.j]],
       [[-2.+0.j, -2.+0.j, -2.+0.j],
        [ 0.+0.j,  0.+0.j,  0.+0.j]]])

>>> import matplotlib.pyplot as plt
>>> [X, Y] = np.meshgrid(2 * np.pi * np.arange(200) / 12,
...                      2 * np.pi * np.arange(200) / 34)
>>> S = np.sin(X) + np.cos(Y) + np.random.uniform(0, 1, X.shape)
>>> FS = np.fft.fftn(S)
>>> plt.imshow(np.log(np.abs(np.fft.fftshift(FS))**2))
<matplotlib.image.AxesImage object at 0x...>
>>> plt.show()

Parameters
a:`array_like`	Input array, can be complex.
s:`sequence` of `ints`, optional	Shape (length of each transformed axis) of the output (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). This corresponds to `n` for `fft(x, n)`. Along any axis, if the given shape is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros. if `s` is not given, the shape of the input along the axes specified by `axes` is used.
axes:`sequence` of `ints`, optional	Axes over which to compute the FFT. If not given, the last `len(s)` axes are used, or all axes if `s` is also not specified. Repeated indices in `axes` means that the transform over that axis is performed multiple times.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - The truncated or zero-padded input, transformed along the axes indicated by `axes`, or by a combination of `s` and `a`, as explained in the parameters section above.
Raises
`ValueError`	If `s` and `axes` have different length.
`IndexError`	If an element of `axes` is larger than than the number of axes of `a`.

@array_function_dispatch(_fft_dispatcher)
def hfft(a, n=None, axis=-1, norm=None): (source) ¶

Compute the FFT of a signal that has Hermitian symmetry, i.e., a real spectrum.

See Also

rfft: Compute the one-dimensional FFT for real input.
ihfft: The inverse of hfft.

Notes

hfft/ihfft are a pair analogous to rfft/irfft, but for the opposite case: here the signal has Hermitian symmetry in the time domain and is real in the frequency domain. So here it's hfft for which you must supply the length of the result if it is to be odd.

even: ihfft(hfft(a, 2*len(a) - 2)) == a, within roundoff error,
odd: ihfft(hfft(a, 2*len(a) - 1)) == a, within roundoff error.

The correct interpretation of the hermitian input depends on the length of the original data, as given by n. This is because each input shape could correspond to either an odd or even length signal. By default, hfft assumes an even output length which puts the last entry at the Nyquist frequency; aliasing with its symmetric counterpart. By Hermitian symmetry, the value is thus treated as purely real. To avoid losing information, the shape of the full signal must be given.

Examples

>>> signal = np.array([1, 2, 3, 4, 3, 2])
>>> np.fft.fft(signal)
array([15.+0.j,  -4.+0.j,   0.+0.j,  -1.-0.j,   0.+0.j,  -4.+0.j]) # may vary
>>> np.fft.hfft(signal[:4]) # Input first half of signal
array([15.,  -4.,   0.,  -1.,   0.,  -4.])
>>> np.fft.hfft(signal, 6)  # Input entire signal and truncate
array([15.,  -4.,   0.,  -1.,   0.,  -4.])

>>> signal = np.array([[1, 1.j], [-1.j, 2]])
>>> np.conj(signal.T) - signal   # check Hermitian symmetry
array([[ 0.-0.j,  -0.+0.j], # may vary
       [ 0.+0.j,  0.-0.j]])
>>> freq_spectrum = np.fft.hfft(signal)
>>> freq_spectrum
array([[ 1.,  1.],
       [ 2., -2.]])

Parameters
a:`array_like`	The input array.
n:`int`, optional	Length of the transformed axis of the output. For `n` output points, `n//2 + 1` input points are necessary. If the input is longer than this, it is cropped. If it is shorter than this, it is padded with zeros. If `n` is not given, it is taken to be `2*(m-1)` where `m` is the length of the input along the axis specified by `axis`.
axis:`int`, optional	Axis over which to compute the FFT. If not given, the last axis is used.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
`ndarray`	out - 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, `2m - 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 `2m - 1` in the typical case,
Raises
`IndexError`	If `axis` is not a valid axis of `a`.

@array_function_dispatch(_fft_dispatcher)
def ifft(a, n=None, axis=-1, norm=None): (source) ¶

Compute the one-dimensional inverse discrete Fourier Transform.

This function computes the inverse of the one-dimensional n-point discrete Fourier transform computed by fft. In other words, ifft(fft(a)) == a to within numerical accuracy. For a general description of the algorithm and definitions, see numpy.fft.

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

a[0] should contain the zero frequency term,
a[1:n//2] should contain the positive-frequency terms,
a[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, A[n//2] represents the sum of the values at the positive and negative Nyquist frequencies, as the two are aliased together. See numpy.fft for details.

See Also

numpy.fft: An introduction, with definitions and general explanations.
fft: The one-dimensional (forward) FFT, of which ifft is the inverse
ifft2: The two-dimensional inverse FFT.
ifftn: The n-dimensional 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.

Examples

>>> np.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 = np.fft.ifft(n)
>>> plt.plot(t, s.real, label='real')
[<matplotlib.lines.Line2D object at ...>]
>>> plt.plot(t, s.imag, '--', label='imaginary')
[<matplotlib.lines.Line2D object at ...>]
>>> plt.legend()
<matplotlib.legend.Legend object at ...>
>>> plt.show()

Parameters
a:`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:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - 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 `axis` is not a valid axis of `a`.

@array_function_dispatch(_fftn_dispatcher)
def ifft2(a, s=None, axes=(-2, -1), norm=None): (source) ¶

Compute the 2-dimensional inverse discrete Fourier Transform.

This function computes the inverse of the 2-dimensional discrete Fourier Transform over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). In other words, ifft2(fft2(a)) == a to within numerical accuracy. By default, the inverse transform is computed over the last two axes of the input array.

The input, analogously to ifft, should be ordered in the same way as is returned by fft2, i.e. it should have the term for zero frequency in the low-order corner of the two axes, the positive frequency terms in the first half of these axes, the term for the Nyquist frequency in the middle of the axes and the negative frequency terms in the second half of both axes, in order of decreasingly negative frequency.

See Also

numpy.fft: Overall view of discrete Fourier transforms, with definitions and conventions used.
fft2: The forward 2-dimensional FFT, of which ifft2 is the inverse.
ifftn: The inverse of the n-dimensional FFT.
fft: The one-dimensional FFT.
ifft: The one-dimensional inverse FFT.

Notes

ifft2 is just ifftn with a different default for axes.

See ifftn for details and a plotting example, and numpy.fft for definition and conventions used.

Zero-padding, analogously with ifft, is performed by appending zeros to the input along the specified dimension. Although this is the common approach, it might lead to surprising results. If another form of zero padding is desired, it must be performed before ifft2 is called.

Examples

>>> a = 4 * np.eye(4)
>>> np.fft.ifft2(a)
array([[1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j], # may vary
       [0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j],
       [0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
       [0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]])

Parameters
a:`array_like`	Input array, can be complex.
s:`sequence` of `ints`, optional	Shape (length of each axis) of the output (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). This corresponds to `n` for `ifft(x, n)`. Along each axis, if the given shape is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros. if `s` is not given, the shape of the input along the axes specified by `axes` is used. See notes for issue on `ifft` zero padding.
axes:`sequence` of `ints`, optional	Axes over which to compute the FFT. If not given, the last two axes are used. A repeated index in `axes` means the transform over that axis is performed multiple times. A one-element sequence means that a one-dimensional FFT is performed.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - The truncated or zero-padded input, transformed along the axes indicated by `axes`, or the last two axes if `axes` is not given.
Raises
`ValueError`	If `s` and `axes` have different length, or `axes` not given and `len(s) != 2`.
`IndexError`	If an element of `axes` is larger than than the number of axes of `a`.

@array_function_dispatch(_fftn_dispatcher)
def ifftn(a, s=None, axes=None, norm=None): (source) ¶

Compute the N-dimensional inverse discrete Fourier Transform.

This function computes the inverse of the N-dimensional discrete Fourier Transform over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). In other words, ifftn(fftn(a)) == a to within numerical accuracy. For a description of the definitions and conventions used, see numpy.fft.

The input, analogously to ifft, should be ordered in the same way as is returned by fftn, i.e. it should have the term for zero frequency in all axes in the low-order corner, the positive frequency terms in the first half of all axes, the term for the Nyquist frequency in the middle of all axes and the negative frequency terms in the second half of all axes, in order of decreasingly negative frequency.

See Also

numpy.fft: Overall view of discrete Fourier transforms, with definitions and conventions used.
fftn: The forward n-dimensional FFT, of which ifftn is the inverse.
ifft: The one-dimensional inverse FFT.
ifft2: The two-dimensional inverse FFT.
ifftshift: Undoes fftshift, shifts zero-frequency terms to beginning of array.

Notes

See numpy.fft for definitions and conventions used.

Examples

>>> a = np.eye(4)
>>> np.fft.ifftn(np.fft.fftn(a, axes=(0,)), axes=(1,))
array([[1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j], # may vary
       [0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],
       [0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
       [0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j]])

Create and plot an image with band-limited frequency content:

>>> import matplotlib.pyplot as plt
>>> n = np.zeros((200,200), dtype=complex)
>>> n[60:80, 20:40] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20, 20)))
>>> im = np.fft.ifftn(n).real
>>> plt.imshow(im)
<matplotlib.image.AxesImage object at 0x...>
>>> plt.show()

Parameters
a:`array_like`	Input array, can be complex.
s:`sequence` of `ints`, optional	Shape (length of each transformed axis) of the output (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). This corresponds to `n` for `ifft(x, n)`. Along any axis, if the given shape is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros. if `s` is not given, the shape of the input along the axes specified by `axes` is used. See notes for issue on `ifft` zero padding.
axes:`sequence` of `ints`, optional	Axes over which to compute the IFFT. If not given, the last `len(s)` axes are used, or all axes if `s` is also not specified. Repeated indices in `axes` means that the inverse transform over that axis is performed multiple times.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - The truncated or zero-padded input, transformed along the axes indicated by `axes`, or by a combination of `s` or `a`, as explained in the parameters section above.
Raises
`ValueError`	If `s` and `axes` have different length.
`IndexError`	If an element of `axes` is larger than than the number of axes of `a`.

@array_function_dispatch(_fft_dispatcher)
def ihfft(a, n=None, axis=-1, norm=None): (source) ¶

Compute the inverse FFT of a signal that has Hermitian symmetry.

See Also

hfft, irfft

Notes

even: ihfft(hfft(a, 2*len(a) - 2)) == a, within roundoff error,
odd: ihfft(hfft(a, 2*len(a) - 1)) == a, within roundoff error.

Examples

>>> spectrum = np.array([ 15, -4, 0, -1, 0, -4])
>>> np.fft.ifft(spectrum)
array([1.+0.j,  2.+0.j,  3.+0.j,  4.+0.j,  3.+0.j,  2.+0.j]) # may vary
>>> np.fft.ihfft(spectrum)
array([ 1.-0.j,  2.-0.j,  3.-0.j,  4.-0.j]) # may vary

Parameters
a:`array_like`	Input array.
n:`int`, optional	Length of the inverse FFT, the number of points along transformation axis in the input to use. If `n` is smaller than the length of the input, the input is cropped. If it is larger, the input is padded with zeros. If `n` is not given, the length of the input along the axis specified by `axis` is used.
axis:`int`, optional	Axis over which to compute the inverse FFT. If not given, the last axis is used.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - 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`.

@array_function_dispatch(_fft_dispatcher)
def irfft(a, n=None, axis=-1, norm=None): (source) ¶

Computes the inverse of rfft.

This function computes the inverse of the one-dimensional n-point discrete Fourier Transform of real input computed by rfft. In other words, irfft(rfft(a), len(a)) == a to within numerical accuracy. (See Notes below for why len(a) is necessary here.)

The input is expected to be in the form returned by rfft, i.e. the real zero-frequency term followed by the complex positive frequency terms in order of increasing frequency. Since the discrete Fourier Transform of real input is Hermitian-symmetric, the negative frequency terms are taken to be the complex conjugates of the corresponding positive frequency terms.

See Also

numpy.fft: For definition of the DFT and conventions used.
rfft: The one-dimensional FFT of real input, of which irfft is inverse.
fft: The one-dimensional FFT.
irfft2: The inverse of the two-dimensional FFT of real input.
irfftn: The inverse of the n-dimensional FFT of real input.

Notes

Returns the real valued n-point inverse discrete Fourier transform of a, where a contains the non-negative frequency terms of a Hermitian-symmetric sequence. n is the length of the result, not the input.

If you specify an n such that a must be zero-padded or truncated, the extra/removed values will be added/removed at high frequencies. One can thus resample a series to m points via Fourier interpolation by: a_resamp = irfft(rfft(a), m).

The correct interpretation of the hermitian input depends on the length of the original data, as given by n. This is because each input shape could correspond to either an odd or even length signal. By default, irfft assumes an even output length which puts the last entry at the Nyquist frequency; aliasing with its symmetric counterpart. By Hermitian symmetry, the value is thus treated as purely real. To avoid losing information, the correct length of the real input must be given.

Examples

>>> np.fft.ifft([1, -1j, -1, 1j])
array([0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]) # may vary
>>> np.fft.irfft([1, -1j, -1])
array([0.,  1.,  0.,  0.])

Notice how the last term in the input to the ordinary ifft is the complex conjugate of the second term, and the output has zero imaginary part everywhere. When calling irfft, the negative frequencies are not specified, and the output array is purely real.

Parameters
a:`array_like`	The input array.
n:`int`, optional	Length of the transformed axis of the output. For `n` output points, `n//2+1` input points are necessary. If the input is longer than this, it is cropped. If it is shorter than this, it is padded with zeros. If `n` is not given, it is taken to be `2*(m-1)` where `m` is the length of the input along the axis specified by `axis`.
axis:`int`, optional	Axis over which to compute the inverse FFT. If not given, the last axis is used.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
`ndarray`	out - The truncated or zero-padded input, transformed along the axis indicated by `axis`, or the last one if `axis` is not specified. The length of the transformed axis is `n`, or, if `n` is not given, `2*(m-1)` where `m` is the length of the transformed axis of the input. To get an odd number of output points, `n` must be specified.
Raises
`IndexError`	If `axis` is not a valid axis of `a`.

@array_function_dispatch(_fftn_dispatcher)
def irfft2(a, s=None, axes=(-2, -1), norm=None): (source) ¶

Computes the inverse of rfft2.

See Also

rfft2: The forward two-dimensional FFT of real input, of which irfft2 is the inverse.
rfft: The one-dimensional FFT for real input.
irfft: The inverse of the one-dimensional FFT of real input.
irfftn: Compute the inverse of the N-dimensional FFT of real input.

Notes

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

Examples

>>> a = np.mgrid[:5, :5][0]
>>> A = np.fft.rfft2(a)
>>> np.fft.irfft2(A, s=a.shape)
array([[0., 0., 0., 0., 0.],
       [1., 1., 1., 1., 1.],
       [2., 2., 2., 2., 2.],
       [3., 3., 3., 3., 3.],
       [4., 4., 4., 4., 4.]])

Parameters
a:`array_like`	The input array
s:`sequence` of `ints`, optional	Shape of the real output to the inverse FFT.
axes:`sequence` of `ints`, optional	The axes over which to compute the inverse fft. Default is the last two axes.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
`ndarray`	out - The result of the inverse real 2-D FFT.

@array_function_dispatch(_fftn_dispatcher)
def irfftn(a, s=None, axes=None, norm=None): (source) ¶

Computes the inverse of rfftn.

This function computes the inverse of the N-dimensional discrete Fourier Transform for real input over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). In other words, irfftn(rfftn(a), a.shape) == a to within numerical accuracy. (The a.shape is necessary like len(a) is for irfft, and for the same reason.)

The input should be ordered in the same way as is returned by rfftn, i.e. as for irfft for the final transformation axis, and as for ifftn along all the other axes.

See Also

rfftn: The forward n-dimensional FFT of real input, of which ifftn is the inverse.
fft: The one-dimensional FFT, with definitions and conventions used.
irfft: The inverse of the one-dimensional FFT of real input.
irfft2: The inverse of the two-dimensional FFT of real input.

Notes

See fft for definitions and conventions used.

See rfft for definitions and conventions used for real input.

The correct interpretation of the hermitian input depends on the shape of the original data, as given by s. This is because each input shape could correspond to either an odd or even length signal. By default, irfftn assumes an even output length which puts the last entry at the Nyquist frequency; aliasing with its symmetric counterpart. When performing the final complex to real transform, the last value is thus treated as purely real. To avoid losing information, the correct shape of the real input must be given.

Examples

>>> a = np.zeros((3, 2, 2))
>>> a[0, 0, 0] = 3 * 2 * 2
>>> np.fft.irfftn(a)
array([[[1.,  1.],
        [1.,  1.]],
       [[1.,  1.],
        [1.,  1.]],
       [[1.,  1.],
        [1.,  1.]]])

Parameters
a:`array_like`	Input array.
s:`sequence` of `ints`, optional	Shape (length of each transformed axis) of the output (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). `s` is also the number of input points used along this axis, except for the last axis, where `s[-1]//2+1` points of the input are used. Along any axis, if the shape indicated by `s` is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros. If `s` is not given, the shape of the input along the axes specified by axes is used. Except for the last axis which is taken to be `2*(m-1)` where `m` is the length of the input along that axis.
axes:`sequence` of `ints`, optional	Axes over which to compute the inverse FFT. If not given, the last `len(s)` axes are used, or all axes if `s` is also not specified. Repeated indices in `axes` means that the inverse transform over that axis is performed multiple times.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
`ndarray`	out - The truncated or zero-padded input, transformed along the axes indicated by `axes`, or by a combination of `s` or `a`, as explained in the parameters section above. The length of each transformed axis is as given by the corresponding element of `s`, or the length of the input in every axis except for the last one if `s` is not given. In the final transformed axis the length of the output when `s` is not given is `2*(m-1)` where `m` is the length of the final transformed axis of the input. To get an odd number of output points in the final axis, `s` must be specified.
Raises
`ValueError`	If `s` and `axes` have different length.
`IndexError`	If an element of `axes` is larger than than the number of axes of `a`.

@array_function_dispatch(_fft_dispatcher)
def rfft(a, n=None, axis=-1, norm=None): (source) ¶

Compute the one-dimensional discrete Fourier Transform for real input.

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

See Also

numpy.fft: For definition of the DFT and conventions used.
irfft: The inverse of rfft.
fft: The one-dimensional FFT of general (complex) input.
fftn: The n-dimensional FFT.
rfftn: The n-dimensional FFT of real input.

Notes

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

When A = rfft(a) and fs is the sampling frequency, A[0] contains the zero-frequency term 0*fs, which is real due to Hermitian symmetry.

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

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

Examples

>>> np.fft.fft([0, 1, 0, 0])
array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j]) # may vary
>>> np.fft.rfft([0, 1, 0, 0])
array([ 1.+0.j,  0.-1.j, -1.+0.j]) # may vary

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

Parameters
a:`array_like`	Input array
n:`int`, optional	Number of points along transformation axis in the input to use. If `n` is smaller than the length of the input, the input is cropped. If it is larger, the input is padded with zeros. If `n` is not given, the length of the input along the axis specified by `axis` is used.
axis:`int`, optional	Axis over which to compute the FFT. If not given, the last axis is used.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - The truncated or zero-padded input, transformed along the axis indicated by `axis`, or the last one if `axis` is not specified. If `n` is even, the length of the transformed axis is `(n/2)+1`. If `n` is odd, the length is `(n+1)/2`.
Raises
`IndexError`	If `axis` is not a valid axis of `a`.

@array_function_dispatch(_fftn_dispatcher)
def rfft2(a, s=None, axes=(-2, -1), norm=None): (source) ¶

Compute the 2-dimensional FFT of a real array.

See Also

rfftn: Compute the N-dimensional discrete Fourier Transform for real input.

Notes

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

Examples

>>> a = np.mgrid[:5, :5][0]
>>> np.fft.rfft2(a)
array([[ 50.  +0.j        ,   0.  +0.j        ,   0.  +0.j        ],
       [-12.5+17.20477401j,   0.  +0.j        ,   0.  +0.j        ],
       [-12.5 +4.0614962j ,   0.  +0.j        ,   0.  +0.j        ],
       [-12.5 -4.0614962j ,   0.  +0.j        ,   0.  +0.j        ],
       [-12.5-17.20477401j,   0.  +0.j        ,   0.  +0.j        ]])

Parameters
a:`array`	Input array, taken to be real.
s:`sequence` of `ints`, optional	Shape of the FFT.
axes:`sequence` of `ints`, optional	Axes over which to compute the FFT.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
`ndarray`	out - The result of the real 2-D FFT.

@array_function_dispatch(_fftn_dispatcher)
def rfftn(a, s=None, axes=None, norm=None): (source) ¶

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

This function computes the N-dimensional discrete Fourier Transform over any number of axes in an M-dimensional real array by means of the Fast Fourier Transform (FFT). By default, all axes are transformed, with the real transform performed over the last axis, while the remaining transforms are complex.

See Also

irfftn: The inverse of rfftn, i.e. the inverse of the n-dimensional FFT of real input.
fft: The one-dimensional FFT, with definitions and conventions used.
rfft: The one-dimensional FFT of real input.
fftn: The n-dimensional FFT.
rfft2: The two-dimensional FFT of real input.

Notes

The transform for real input is performed over the last transformation axis, as by rfft, then the transform over the remaining axes is performed as by fftn. The order of the output is as for rfft for the final transformation axis, and as for fftn for the remaining transformation axes.

See fft for details, definitions and conventions used.

Examples

>>> a = np.ones((2, 2, 2))
>>> np.fft.rfftn(a)
array([[[8.+0.j,  0.+0.j], # may vary
        [0.+0.j,  0.+0.j]],
       [[0.+0.j,  0.+0.j],
        [0.+0.j,  0.+0.j]]])

>>> np.fft.rfftn(a, axes=(2, 0))
array([[[4.+0.j,  0.+0.j], # may vary
        [4.+0.j,  0.+0.j]],
       [[0.+0.j,  0.+0.j],
        [0.+0.j,  0.+0.j]]])

Parameters
a:`array_like`	Input array, taken to be real.
s:`sequence` of `ints`, optional	Shape (length along each transformed axis) to use from the input. (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.). The final element of `s` corresponds to `n` for `rfft(x, n)`, while for the remaining axes, it corresponds to `n` for `fft(x, n)`. Along any axis, if the given shape is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros. if `s` is not given, the shape of the input along the axes specified by `axes` is used.
axes:`sequence` of `ints`, optional	Axes over which to compute the FFT. If not given, the last `len(s)` axes are used, or all axes if `s` is also not specified.
norm:{"backward", "ortho", "forward"}, optional	New in version 1.10.0. Normalization mode (see `numpy.fft`). Default is "backward". Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor. New in version 1.20.0: The "backward", "forward" values were added.
Returns
complex ndarray	out - The truncated or zero-padded input, transformed along the axes indicated by `axes`, or by a combination of `s` and `a`, as explained in the parameters section above. The length of the last axis transformed will be `s[-1]//2+1`, while the remaining transformed axes will have lengths according to `s`, or unchanged from the input.
Raises
`ValueError`	If `s` and `axes` have different length.
`IndexError`	If an element of `axes` is larger than than the number of axes of `a`.

array_function_dispatch = (source) ¶

Undocumented

def _cook_nd_args(a, s=None, axes=None, invreal=0): (source) ¶

Undocumented

def _fft_dispatcher(a, n=None, axis=None, norm=None): (source) ¶

Undocumented

def _fftn_dispatcher(a, s=None, axes=None, norm=None): (source) ¶

Undocumented

def _get_backward_norm(n, norm): (source) ¶

Undocumented

def _get_forward_norm(n, norm): (source) ¶

Undocumented

def _raw_fft(a, n, axis, is_real, is_forward, inv_norm): (source) ¶

Undocumented

def _raw_fftnd(a, s=None, axes=None, function=fft, norm=None): (source) ¶

Undocumented

def _swap_direction(norm): (source) ¶

Undocumented

_SWAP_DIRECTION_MAP: dict = (source) ¶

Undocumented

Value

{'backward': 'forward',
 None: 'forward',
 'ortho': 'ortho',
 'forward': 'backward'}