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.

New 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.

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.])