The LIB module variable

The LIB variable specifies the default library to use by the module. The Numpy library is used by default with LIB sets to "numpy". Possible values are

• "numpy" to use Numpy, or

• "fftw" to use pyFFTW, if installed.

The variable can be changed globally

udft.LIB = "numpy"

In addition, each function has a parameter to change the library used at call time. A ValueError is raided if the value is not recognized.

Discrete Fourier Transform

dftn

udft.dftn(inarray, ndim=None, lib=None)

ND unitary discrete Fourier transform.

Parameters

• inarray (array-like) – The array to transform.

• ndim (int, optional) – The ndim last axes along which to compute the transform. All axes by default.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The DFT of inarray with same shape.

Return type

array-like

idftn

udft.idftn(inarray, ndim=None, lib=None)

ND unitary inverse discrete Fourier transform.

Parameters

• inarray (array-like) – The array to transform.

• ndim (int, optional) – The ndim last axes along wich to compute the transform. All axes by default.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The IDFT of inarray with same shape.

Return type

array-like

dft

udft.dft(inarray, lib=None)

1D unitary discrete Fourier transform.

Compute the unitary DFT on the last axis.

Parameters

• inarray (array-like) – The array to transform.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The DFT of inarray with same shape.

Return type

array-like

idft

udft.idft(inarray, lib=None)

1D unitary inverse discrete Fourier transform.

Compute the unitary inverse DFT transform on the last axis.

Parameters

• inarray (array-like) – The array to transform.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The DFT of inarray with same shape.

Return type

array-like

dft2

udft.dft2(inarray, lib=None)

2D unitary discrete Fourier transform.

Compute the unitary DFT on the last 2 axes.

Parameters

• inarray (array-like) – The array to transform.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The DFT of inarray with same shape.

Return type

array-like

idft2

udft.idft2(inarray, lib=None)

2D unitary inverse discrete Fourier transform.

Compute the unitary IDFT on the last 2 axes.

Parameters

• inarray (array-like) – The array to transform.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The IDFT of inarray with same shape.

Return type

array-like

Real Discrete Fourier Transform

The transforms here suppose input of real values. In direct transform, the last transformed axis has length N // 2 + 1.

Note

The exact output shape for real transform can’t be guessed from input shape. Therefor, only one inverse transform irdftn() is provided and that function ask for the output shape. The dimension ndim corresponds to the length of shape.

rdftn

udft.rdftn(inarray, ndim=None, lib=None)

ND real unitary discrete Fourier transform.

Consider the Hermitian property of output with input having real values.

Parameters

• inarray (array-like) – The array of real values to transform.

• ndim (int, optional) – The ndim last axes along which to compute the transform. All axes by default.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The real DFT of inarray (the last axe as N // 2 + 1 length).

Return type

array-like

irdftn

udft.irdftn(inarray, shape, lib=None)

ND real unitary inverse discrete Fourier transform.

Consider the Hermitian property of input and return real values.

Parameters

• inarray (array-like) – The array of complex values to transform.

• shape (tuple of int) – The output shape of the len(shape) last axes. The transform is applied on the n=len(shape) axes.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The real IDFT of inarray.

Return type

array-like

rdft

udft.rdft(inarray, lib=None)

1D real unitary discrete Fourier transform.

Compute the unitary real DFT on the last axis. Consider the Hermitian property of output with input having real values.

Parameters

• inarray (array-like) – The array to transform.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The real DFT of inarray, where the last dim has length N//2+1.

Return type

array-like

rdft2

udft.rdft2(inarray, lib=None)

2D real unitary discrete Fourier transform.

Compute the unitary real DFT on the last 2 axes. Consider the Hermitian property of output when input has real values.

Parameters

• inarray (array-like) – The array to transform.

• lib (str, optional) – Specify the library to compute the Fourier transform. See set_lib get_lib functions for the default.

Returns

outarray – The real DFT of inarray, where the last dim has length N//2+1.

Return type

array-like

Convolution related

ir2fr

udft.ir2fr(imp_resp, shape, origin=None, real=True)

Compute the frequency response from impulse responses.

This function makes the necessary correct zero-padding, zero convention, correct DFT etc.

The DFT is performed on the last len(shape) dimensions for efficiency (C-order array). Use numpy implementation.

Parameters

• imp_resp (array-like) – The impulse responses.

• shape (tuple of int) – A tuple of integer corresponding to the target shape of the frequency responses, without hermitian property. The DFT is performed on the len(shape) last axes of ndarray.

• origin (tuple of int, optional) – The index of the origin (0 coordinate) of the impulse response. The center of the array by default (shape[i] // 2).

• real (boolean, optional) – If True, imp_resp is supposed real, and real DFT is used.

Returns

out – The frequency responses of shape shape on the last len(shape) dimensions. If real is True, the last dimension as lenght N//2+1.

Return type

array-like

Notes

• The output is returned as C-contiguous array.

• For convolution, the result must be used with unitary discrete Fourier transform for the signal (udftn or equivalent).

fr2ir

udft.fr2ir(freq_resp, shape, origin=None, real=True)

Return the impulse responses from frequency responses.

This function makes the necessary correct zero-padding, zero convention, correct DFT etc. to compute the impulse responses from frequency responses.

The IR array is supposed to have the origin in the middle of the array.

The Fourier transform is performed on the last len(shape) dimensions for efficiency (C-order array). Use np.fft.

Parameters

• freq_resp (array-like) – The frequency responses.

• shape (tuple of int) – Output shape of the impulse responses.

• origin (tuple of int, optional) – The index of the origin (0, 0) of output the impulse response. The center by default (shape[i] // 2).

• real (boolean, optional) – If True, imp_resp is supposed real, and real DFT is used.

Returns

out – The impulse responses of shape shape on the last len(shape) axes.

Return type

array-like

Notes

• The output is returned as C-contiguous array.

• For convolution, the result has to be used with unitary discrete Fourier transform for the signal (udftn or equivalent).

diff_ir

udft.diff_ir(ndim=1, axis=0, norm=False)

Return the impulse response of first order differences.

Parameters

• ndim (int, optional) – The number of dimensions of the array on which the diff will apply.

• axis (int, optional) – The axis (dimension) where the diff operates.

Returns

out – The impulse response

Return type

array_like

laplacian

udft.laplacian(ndim, norm=False)

Return the Laplacian impulse response.

The second-order difference in each axes.

Parameters

ndim (int) – The dimension of the Laplacian.

Returns

out – The impulse response

Return type

array_like

Other

hnorm

udft.hnorm(inarray, inshape)

Hermitian l2-norm of array in discrete Fourier space.

Compute the l2-norm of complex array

\[\|x\|_2 = \sqrt{\sum_{n=1}^{N} |x_n|^2}\]

considering the Hermitian property. Must be used with rdftn. Equivalent of np.linalg.norm for array applied on full Fourier space array (those obtained with dftn).

Parameters

• inarray (array-like of shape (.. + inshape)) – The input array with half of the Fourier plan.

• inshape (tuple of int) – The shape of the original array oarr where inarray=rdft(oarr).

Returns

norm

Return type

float

crandn

Warning

At that time, this function does not respect the hermitian property, this is planned.

udft.crandn(shape)

Draw from white complex Normal.

Draw unitary DFT of real white Gaussian field of zero mean and unit variance. Does not consider hermitian property, shape is supposed to consider half of the frequency plane already.

Return type

ndarray