fftpack.realtransforms

Real spectrum tranforms (DCT, DST, MDCT)

Module Contents

Functions

_init_nd_shape_and_axes(x,shape,axes) Handle shape and axes arguments for dctn, idctn, dstn, idstn.
dctn(x,type=2,shape=None,axes=None,norm=None,overwrite_x=False) Return multidimensional Discrete Cosine Transform along the specified axes.
idctn(x,type=2,shape=None,axes=None,norm=None,overwrite_x=False) Return multidimensional Discrete Cosine Transform along the specified axes.
dstn(x,type=2,shape=None,axes=None,norm=None,overwrite_x=False) Return multidimensional Discrete Sine Transform along the specified axes.
idstn(x,type=2,shape=None,axes=None,norm=None,overwrite_x=False) Return multidimensional Discrete Sine Transform along the specified axes.
dct(x,type=2,n=None,axis=None,norm=None,overwrite_x=False) Return the Discrete Cosine Transform of arbitrary type sequence x.
idct(x,type=2,n=None,axis=None,norm=None,overwrite_x=False) Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.
_get_dct_fun(type,dtype)
_get_norm_mode(normalize)
__fix_shape(x,n,axis,dct_or_dst)
_raw_dct(x0,type,n,axis,nm,overwrite_x)
_raw_dst(x0,type,n,axis,nm,overwrite_x)
_eval_fun(f,tmp,n,axis,nm,overwrite_x)
_dct(x,type,n=None,axis=None,overwrite_x=False,normalize=None) Return Discrete Cosine Transform of arbitrary type sequence x.
dst(x,type=2,n=None,axis=None,norm=None,overwrite_x=False) Return the Discrete Sine Transform of arbitrary type sequence x.
idst(x,type=2,n=None,axis=None,norm=None,overwrite_x=False) Return the Inverse Discrete Sine Transform of an arbitrary type sequence.
_get_dst_fun(type,dtype)
_dst(x,type,n=None,axis=None,overwrite_x=False,normalize=None) Return Discrete Sine Transform of arbitrary type sequence x.
_init_nd_shape_and_axes(x, shape, axes)

Handle shape and axes arguments for dctn, idctn, dstn, idstn.

dctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False)

Return multidimensional Discrete Cosine Transform along the specified axes.

x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
shape : tuple of ints, optional
The shape of the result. If both shape and axes (see below) are None, shape is x.shape; if shape is None but axes is not None, then shape is scipy.take(x.shape, axes, axis=0). If shape[i] > x.shape[i], the i-th dimension is padded with zeros. If shape[i] < x.shape[i], the i-th dimension is truncated to length shape[i].
axes : tuple or None, optional
Axes along which the DCT is computed; the default is over all axes.
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.

idctn : Inverse multidimensional DCT

For full details of the DCT types and normalization modes, as well as references, see dct.

>>> from scipy.fftpack import dctn, idctn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho'))
True
idctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False)

Return multidimensional Discrete Cosine Transform along the specified axes.

x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
shape : tuple of ints, optional
The shape of the result. If both shape and axes (see below) are None, shape is x.shape; if shape is None but axes is not None, then shape is scipy.take(x.shape, axes, axis=0). If shape[i] > x.shape[i], the i-th dimension is padded with zeros. If shape[i] < x.shape[i], the i-th dimension is truncated to length shape[i].
axes : tuple or None, optional
Axes along which the IDCT is computed; the default is over all axes.
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.

dctn : multidimensional DCT

For full details of the IDCT types and normalization modes, as well as references, see idct.

>>> from scipy.fftpack import dctn, idctn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho'))
True
dstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False)

Return multidimensional Discrete Sine Transform along the specified axes.

x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
shape : tuple of ints, optional
The shape of the result. If both shape and axes (see below) are None, shape is x.shape; if shape is None but axes is not None, then shape is scipy.take(x.shape, axes, axis=0). If shape[i] > x.shape[i], the i-th dimension is padded with zeros. If shape[i] < x.shape[i], the i-th dimension is truncated to length shape[i].
axes : tuple or None, optional
Axes along which the DCT is computed; the default is over all axes.
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.

idstn : Inverse multidimensional DST

For full details of the DST types and normalization modes, as well as references, see dst.

>>> from scipy.fftpack import dstn, idstn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho'))
True
idstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False)

Return multidimensional Discrete Sine Transform along the specified axes.

x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
shape : tuple of ints, optional
The shape of the result. If both shape and axes (see below) are None, shape is x.shape; if shape is None but axes is not None, then shape is scipy.take(x.shape, axes, axis=0). If shape[i] > x.shape[i], the i-th dimension is padded with zeros. If shape[i] < x.shape[i], the i-th dimension is truncated to length shape[i].
axes : tuple or None, optional
Axes along which the IDCT is computed; the default is over all axes.
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.

dctn : multidimensional DST

For full details of the IDST types and normalization modes, as well as references, see idst.

>>> from scipy.fftpack import dstn, idstn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho'))
True
dct(x, type=2, n=None, axis=None, norm=None, overwrite_x=False)

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.

idct : 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):

                                   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.

[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, http://dx.doi.org/10.1109/TASSP.1980.1163351 (1980).
[2]Wikipedia, “Discrete cosine transform”, http://en.wikipedia.org/wiki/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.])
idct(x, type=2, n=None, axis=None, norm=None, overwrite_x=False)

Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.

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 idct 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.
idct : ndarray of real
The transformed input array.

dct : Forward DCT

For a single dimension array x, idct(x, norm='ortho') is equal to MATLAB idct(x).

‘The’ IDCT is the IDCT of type 2, which is the same as DCT of type 3.

IDCT of type 1 is the DCT of type 1, IDCT of type 2 is the DCT of type 3, and IDCT of type 3 is the DCT of type 2. For the definition of these types, see dct.

The Type 1 DCT is equivalent to the DFT for real, even-symmetrical inputs. The output is also real and even-symmetrical. Half of the IFFT input is used to generate half of the IFFT output:

>>> from scipy.fftpack import ifft, idct
>>> ifft(np.array([ 30.,  -8.,   6.,  -2.,   6.,  -8.])).real
array([  4.,   3.,   5.,  10.,   5.,   3.])
>>> idct(np.array([ 30.,  -8.,   6.,  -2.]), 1) / 6
array([  4.,   3.,   5.,  10.])
_get_dct_fun(type, dtype)
_get_norm_mode(normalize)
__fix_shape(x, n, axis, dct_or_dst)
_raw_dct(x0, type, n, axis, nm, overwrite_x)
_raw_dst(x0, type, n, axis, nm, overwrite_x)
_eval_fun(f, tmp, n, axis, nm, overwrite_x)
_dct(x, type, n=None, axis=None, overwrite_x=False, normalize=None)

Return Discrete Cosine Transform of arbitrary type sequence x.

x : array_like
input array.
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).
overwrite_x : bool, optional
If True, the contents of x can be destroyed; the default is False.

z : ndarray

dst(x, type=2, n=None, axis=None, norm=None, overwrite_x=False)

Return the Discrete Sine Transform of arbitrary type sequence x.

x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DST (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 dst 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.
dst : ndarray of reals
The transformed input array.

idst : Inverse DST

For a single dimension array x.

There are theoretically 8 types of the DST for different combinations of even/odd boundary conditions and boundary off sets [1]_, only the first 3 types are implemented in scipy.

Type I

There are several definitions of the DST-I; we use the following for norm=None. DST-I assumes the input is odd around n=-1 and n=N.

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

Only None is supported as normalization mode for DCT-I. Note also that the DCT-I is only supported for input size > 1 The (unnormalized) DCT-I is its own inverse, up to a factor 2(N+1).

Type II

There are several definitions of the DST-II; we use the following for norm=None. DST-II assumes the input is odd around n=-1/2 and n=N-1/2; the output is odd around k=-1 and even around k=N-1

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

Type III

There are several definitions of the DST-III, we use the following (for norm=None). DST-III assumes the input is odd around n=-1 and even around n=N-1

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

The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up to a factor 2N. The orthonormalized DST-III is exactly the inverse of the orthonormalized DST-II.

New in version 0.11.0.

[1]Wikipedia, “Discrete sine transform”, http://en.wikipedia.org/wiki/Discrete_sine_transform
idst(x, type=2, n=None, axis=None, norm=None, overwrite_x=False)

Return the Inverse Discrete Sine Transform of an arbitrary type sequence.

x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DST (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 idst 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.
idst : ndarray of real
The transformed input array.

dst : Forward DST

‘The’ IDST is the IDST of type 2, which is the same as DST of type 3.

IDST of type 1 is the DST of type 1, IDST of type 2 is the DST of type 3, and IDST of type 3 is the DST of type 2. For the definition of these types, see dst.

New in version 0.11.0.

_get_dst_fun(type, dtype)
_dst(x, type, n=None, axis=None, overwrite_x=False, normalize=None)

Return Discrete Sine Transform of arbitrary type sequence x.

x : array_like
input array.
n : int, optional
Length of the transform.
axis : int, optional
Axis along which the dst is computed. (default=-1)
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)

z : real ndarray