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- fftfor more details.
- orthogonalizebool, optional
- Whether to use the orthogonalized DCT variant (see Notes). Defaults to - Truewhen- norm="ortho"and- Falseotherwise.- 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- dctand- idctare 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- dctand the- idctis scaled by- 1/Nwhere- Nis the “logical” size of the DCT. For- norm="forward"the- 1/Nnormalization is applied to the forward- dctinstead and the- idctis 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")\[y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)\]- orthogonalizehas 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 >>> 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.])