SciPy

scipy.fft.dct

scipy.fft.dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None)[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.

workersint, 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.

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).

For norm=None, 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='ortho' both directions are scaled by the same factor 1/sqrt(N).

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

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

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

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

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.fft import fft, dct
>>> 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.])

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