`scipy.fft._realtransforms:dct`

source: build-install/usr/lib/python3.14/site-packages/scipy/fft/_realtransforms.py:273

Signature

def dct(x, type = 2, n = None, axis = -1, norm = None, overwrite_x = False, workers = None, orthogonalize = None)

Summary

Return the Discrete Cosine Transform of arbitrary type sequence x.

Parameters

x : array_like: The input array.
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].
axis : int, optional: Axis along which the dct is computed; the default is over the last axis (i.e., axis=-1).
norm : {"backward", "ortho", "forward"}, optional: Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional: If True, the contents of x can be destroyed; the default is False.
workers : int, optional: Maximum number of workers to use for parallel computation. If negative, the value wraps around from os.cpu_count(). See fft for more details.
orthogonalize : bool, optional: Whether to use the orthogonalized DCT variant (see Notes). Defaults to True when norm="ortho" and False otherwise.
versionadded 1.8.0

Returns

y : ndarray of real: The transformed input array.

Notes

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

For norm="ortho" both the dct and idct are scaled by the same overall factor in both directions. By default, the transform is also orthogonalized which for types 1, 2 and 3 means the transform definition is modified to give orthogonality of the DCT matrix (see below).

For norm="backward", there is no scaling on dct and the idct is scaled by 1/N where N is the "logical" size of the DCT. For norm="forward" the 1/N normalization is applied to the forward dct instead and the idct is unnormalized.

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="backward")

y_{k} = x_{0} + (- 1)^{k} x_{N - 1} + 2 n = 1 \sum N - 2 x_{n} cos (\frac{π k n}{N - 1})

If orthogonalize=True, x[0] and x[N-1] are multiplied by a scaling factor of $2$ , and y[0] and y[N-1] are divided by $2$ . When combined with norm="ortho", this makes the corresponding matrix of coefficients orthonormal (O @ O.T = np.eye(N)).

Type II

There are several definitions of the DCT-II; we use the following (for norm="backward")

y_{k} = 2 n = 0 \sum N - 1 x_{n} cos (\frac{π k ( 2 n + 1 )}{2 N})

If orthogonalize=True, y[0] is divided by $2$ which, when combined with norm="ortho", 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="backward")

y_{k} = x_{0} + 2 n = 1 \sum N - 1 x_{n} cos (\frac{π ( 2 k + 1 ) n}{2 N})

If orthogonalize=True, x[0] terms are multiplied by $2$ which, when combined with norm="ortho", makes the corresponding matrix of coefficients orthonormal (O @ O.T = np.eye(N)).

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="backward")

y_{k} = 2 n = 0 \sum N - 1 x_{n} cos (\frac{π ( 2 k + 1 ) ( 2 n + 1 )}{4 N})

orthogonalize has no effect here, as the DCT-IV matrix is already orthogonal up to a scale factor of 2N.

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.fft import fft, dct
import numpy as np

✓

fft(np.array([4., 3., 5., 10., 5., 3.])).real
dct(np.array([4., 3., 5., 10.]), 1)

✗

array([30., -8.,  6., -2.])

Aliases

scipy.fft.dct

Signature

Summary

Parameters

Returns

Notes

Examples

See also

Aliases