Return the Discrete Cosine Transform of arbitrary type sequence x.
Parameters ---------- x : array_like The input array. type :
, 2, 3, 4
, optional Type of the DCT (see Notes). Default type is 2. n : int, optional Length of the transform. If ``n < x.shapeaxis``, `x` is truncated. If ``n > x.shapeaxis``, `x` is zero-padded. The default results in ``n = x.shapeaxis``. 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``)
.. math::
y_k = x_0 + (-1)^k x_N-1 + 2 \sum_n=1^N-2 x_n \cos\left( \frac\pi k nN-1 \right)
If ``norm='ortho'``, ``x0`` and ``xN-1`` are multiplied by a scaling factor of :math:`\sqrt
`, and ``yk`` is multiplied by a scaling factor ``f``
.. math::
f = \begincases \frac
\sqrt\frac{1N-1
}
& \textfk=0\text or N-1, \\ \frac
\sqrt\frac{2N-1
}
& \textotherwise \endcases
.. versionadded:: 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``)
.. math::
y_k = 2 \sum_n=0^N-1 x_n \cos\left(\frac\pi k(2n+1)
N
\right)
If ``norm='ortho'``, ``yk`` is multiplied by a scaling factor ``f``
.. math:: f = \begincases \sqrt\frac{1
N
}
& \textfk=0, \\ \sqrt\frac{1
N
}
& \textotherwise \endcases
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``)
.. math::
y_k = x_0 + 2 \sum_n=1^N-1 x_n \cos\left(\frac\pi(2k+1)n
N
\right)
or, for ``norm='ortho'``
.. math::
y_k = \fracx_0\sqrt{N
}
- \sqrt
\frac{2N
}
\sum_n=1^N-1 x_n \cos\left(\frac\pi(2k+1)n
N
\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``)
.. math::
y_k = 2 \sum_n=0^N-1 x_n \cos\left(\frac\pi(2k+1)(2n+1)
N
\right)
If ``norm='ortho'``, ``yk`` is multiplied by a scaling factor ``f``
.. math::
f = \frac
\sqrt{2N
}
.. versionadded:: 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 >>> 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.)