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# 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}, 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 3 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`):

                                         N-2
      y[k] = x[0] + (-1)**k x[N-1] + 2 * sum x[n]*cos(pi*k*n/(N-1))
                                         n=1

  Only None is supported as normalization mode for DCT-I. Note also that 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`):

                N-1
      y[k] = 2* sum x[n]*cos(pi*k*(2n+1)/(2*N)), 0 <= k < N.
                n=0

  If `norm='ortho'`, `y[k]` is multiplied by a scaling factor f:

      f = sqrt(1/(4*N)) if k = 0,
      f = sqrt(1/(2*N)) otherwise.

  Which makes the corresponding matrix of coefficients orthonormal (`OO' = Id`).

  Type III

  There are several definitions, we use the following (for `norm=None`):

                        N-1
      y[k] = x[0] + 2 * sum x[n]*cos(pi*(k+0.5)*n/N), 0 <= k < N.
                        n=1

  or, for `norm='ortho'` and 0 <= k < N:

                                          N-1
      y[k] = x[0] / sqrt(N) + sqrt(2/N) * sum x[n]*cos(pi*(k+0.5)*n/N)
                                          n=1

  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.

  References

  [R42] | ‘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, http://dx.doi.org/10.1109/TASSP.1980.1163351 (1980).

  [R43] | Wikipedia, “Discrete cosine transform”, http://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
      >>> 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.])