pyffs.czt¶

Methods for computing the chirp Z-transform.

czt(x, A, W, M, axis=-1)[source]¶

Chirp Z-Transform.

This implementation follows the semantics defined in The Chirp Z-Transform.

Parameters

x (ndarray) – (…, N, …) input array.
A (float or complex) – Circular offset from the positive real-axis.
W (float or complex) – Circular spacing between transform points.
M (int) – Length of the transform.
axis (int) – Dimension of x along which the samples are stored.

Returns

X – (…, M, …) transformed input along the axis indicated by axis.

Return type

ndarray

Notes

Due to numerical instability when using large M, this implementation only supports transforms where A and W have unit norm.

Examples

Implementation of the DFT:

>>> N = M = 10
>>> x = np.random.randn(N, 3) + 1j * np.random.randn(N, 3)  # multi-dim

>>> dft_x = np.fft.fft(x, axis=0)
>>> czt_x = czt(x, A=1, W=np.exp(-1j * 2 * np.pi / N), M=M, axis=0)

>>> np.allclose(dft_x, czt_x)
True

Implementation of the iDFT:

>>> N = M = 10
>>> x = np.random.randn(N) + 1j * np.random.randn(N)

>>> idft_x = np.fft.ifft(x)
>>> czt_x = czt(x, A=1, W=np.exp(1j * 2 * np.pi / N), M=M)

>>> np.allclose(idft_x, czt_x / N)  # czt() does not do the scaling.
True

cztn(x, A, W, M, axes=None)[source]¶

Multi-dimensional Chirp Z-transform.

Parameters

x (ndarray) – (…, N_1, N_2, …, N_D, …) input values.
A (list(float or complex)) – Circular offset from the positive real-axis, for each dimension.
W (list(float or complex)) – Circular spacing between transform points, for each dimension.
M (list(int)) – Length of transform for each dimension.
axes (tuple) – Dimensions of x along which transform should be applied.

Returns

x_czt – (…, M_1, M_2, …, M_D, …) transformed input along the axes indicated by axes.

Return type

ndarray

Notes

Due to numerical instability when using large M, this implementation only supports transforms where each element of A and W has unit norm.

Examples

Implementation of N-dimensional DFT:

>>> N = M = 10
>>> W = np.exp(-1j * 2 * np.pi / N)
>>> x = np.random.randn(N, N, N) + 1j * np.random.randn(N, N, N)  # extra dimension

>>> dft_x = np.fft.fftn(x, axes=(1, 2))
>>> czt_x = cztn(x, A=[1, 1], W=[W, W], M=[M, M], axes=(1, 2))

>>> np.allclose(dft_x, czt_x)
True