scipy.fftpack.dct¶
- scipy.fftpack.dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False)[source]¶
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.
axis : int, optional
Axis over which to compute the transform.
norm : {None, ‘ortho’}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=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
[R33] ‘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). [R34] 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:
>>> fft(array([4., 3., 5., 10., 5., 3.])).real array([ 30., -8., 6., -2., 6., -8.]) >>> dct(array([4., 3., 5., 10.]), 1) array([ 30., -8., 6., -2.])