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.

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.

y : ndarray of real

The transformed input array.

See also

Inverse DCT


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):

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

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

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):

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

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

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


[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”,


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.])

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