# scipy.fftpack.dct

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

  Return the Discrete Cosine Transform of arbitrary type sequence x.

  Parameters:
    x array_like

      The input array.

    type{1, 2, 3, 4}, optional

      Type of the DCT (see Notes). Default type is 2.

    n int, 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]`.

    axis int, 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_x bool, optional

      If True, the contents of x can be destroyed; the default is False.

  Returns:
    y ndarray 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)`.

  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`

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

  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 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`

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

  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.])