xrscipy.fftpack.dct

xrscipy.fftpack.dct(x, coord, type=2, n=None, norm=None)

dct(x, coord, type=2, n=None, norm=None)

Parameters:
  • obj (xarray object) – The input array.

  • coord (string) – Coordinate along which the dct is computed. The coordinate must be evenly spaced.

  • 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].

  • norm ({None, 'ortho'}, optional) – Normalization mode (see Notes). Default is None.

Returns:

y – The transformed input array.

Return type:

ndarray of real

See also

idct

Inverse DCT

numpy.fftpack.dct

scipy.fft.dct : Original scipy implementation

Notes

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

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 \(\sqrt{2}\), and y[k] is multiplied by a scaling factor f

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

New in version 1.2.0: Orthonormalization in DCT-I.

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

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

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

\[f = \frac{1}{\sqrt{2N}}\]

New in version 1.2.0: Support for DCT-IV.

References

Examples

None 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:

Examples

>>> from scipy.fftpack import fft, dct
>>> import numpy as np
>>> fft(np.array([4., 3., 5., 10., 5., 3.])).real
array([ 30.,  -8.,   6.,  -2.,   6.,  -8.])

Examples

>>> dct(np.array([4., 3., 5., 10.]), 1)
array([ 30.,  -8.,   6.,  -2.])