`scipy.fftpack._realtransforms:dct`

source: /scipy/fftpack/_realtransforms.py :235

Signature

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

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 : {None, 'ortho'}, optional: Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional: If True, the contents of x can be destroyed; the default is False.

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).

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

If norm='ortho', x[0] and x[N-1] are multiplied by a scaling factor of $2$ , and y[k] is multiplied by a scaling factor f

f = ⎩ ⎨ ⎧ \frac{1}{2} \frac{1}{N - 1} \frac{1}{2} \frac{2}{N - 1} if k = 0 or N - 1, otherwise

Type II

There are several definitions of the DCT-II; we use the following (for norm=None)

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

If norm='ortho', y[k] is multiplied by a scaling factor f

f = ⎩ ⎨ ⎧ \frac{1}{4 N} \frac{1}{2 N} if k = 0, otherwise

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

or, for norm='ortho'

y_{k} = \frac{x _{0}}{N} + \frac{2}{N} n = 1 \sum N - 1 x_{n} cos (\frac{π ( 2 k + 1 ) n}{2 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=None)

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

If norm='ortho', y[k] is multiplied by a scaling factor f

f = \frac{1}{2 N}

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

✗

Aliases

scipy.fftpack.dct

Signature

Summary

Parameters

Returns

Notes

Examples

See also

Aliases