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. Ifn > x.shape[axis]
, x is zero-padded. The default results inn = 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 MATLABdct(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]
andx[N-1]
are multiplied by a scaling factor of \(\sqrt{2}\), andy[k]
is multiplied by a scaling factorf
\[\begin{split}f = \begin{cases} \frac{1}{2}\sqrt{\frac{1}{N-1}} & \text{if }k=0\text{ or }N-1, \\ \frac{1}{2}\sqrt{\frac{2}{N-1}} & \text{otherwise} \end{cases}\end{split}\]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 factorf
\[\begin{split}f = \begin{cases} \sqrt{\frac{1}{4N}} & \text{if }k=0, \\ \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}\end{split}\]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 factorf
\[f = \frac{1}{\sqrt{2N}}\]New 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.])