scipy.fftpack.dct#

scipy.fftpack.dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False)[source]#

Return the Discrete Cosine Transform of arbitrary type sequence x.

Parameters:

xarray_like: The input array.
type{1, 2, 3, 4}, optional: Type of the DCT (see Notes). Default type is 2.
nint, 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].
axisint, 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_xbool, optional: If True, the contents of x can be destroyed; the default is False.

Returns:

yndarray 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).

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

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

Added 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

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

Added in version 1.2.0: Support for DCT-IV.

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