scipy.fft.

dct#

scipy.fft.dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, orthogonalize=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{“backward”, “ortho”, “forward”}, optional: Normalization mode (see Notes). Default is “backward”.
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.
orthogonalizebool, optional: Whether to use the orthogonalized DCT variant (see Notes). Defaults to True when norm="ortho" and False otherwise.

Added in version 1.8.0.

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).

Warning

For type in {1, 2, 3}, norm="ortho" breaks the direct correspondence with the direct Fourier transform. To recover it you must specify orthogonalize=False.

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")

\[y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left( \frac{\pi k n}{N-1} \right)\]

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

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="backward")

\[y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right)\]

If orthogonalize=True, y[0] is divided by \(\sqrt{2}\) 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")

\[y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right)\]

If orthogonalize=True, x[0] terms are multiplied by \(\sqrt{2}\) 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")

\[y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)\]

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

Array API Standard Support

dct has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	⛔	⛔
Dask	⚠️ computes graph	n/a

See Support for the array API standard for more information.

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
>>> 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.])