""" Real spectrum tranforms (DCT, DST, MDCT) """ from __future__ import division, print_function, absolute_import __all__ = ['dct', 'idct', 'dst', 'idst'] import numpy as np from scipy.fftpack import _fftpack from scipy.fftpack.basic import _datacopied, _fix_shape, _asfarray import atexit atexit.register(_fftpack.destroy_ddct1_cache) atexit.register(_fftpack.destroy_ddct2_cache) atexit.register(_fftpack.destroy_dct1_cache) atexit.register(_fftpack.destroy_dct2_cache) atexit.register(_fftpack.destroy_ddst1_cache) atexit.register(_fftpack.destroy_ddst2_cache) atexit.register(_fftpack.destroy_dst1_cache) atexit.register(_fftpack.destroy_dst2_cache) def 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 ---------- .. [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 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.]) """ if type == 1 and norm is not None: raise NotImplementedError( "Orthonormalization not yet supported for DCT-I") return _dct(x, type, n, axis, normalize=norm, overwrite_x=overwrite_x) def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False): """ Return the Inverse Discrete Cosine Transform of an arbitrary type sequence. 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 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. Returns ------- idct : ndarray of real The transformed input array. See Also -------- dct : Forward DCT Notes ----- 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`. Examples -------- 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.]) """ if type == 1 and norm is not None: raise NotImplementedError( "Orthonormalization not yet supported for IDCT-I") # Inverse/forward type table _TP = {1:1, 2:3, 3:2} return _dct(x, _TP[type], n, axis, normalize=norm, overwrite_x=overwrite_x) def _get_dct_fun(type, dtype): try: name = {'float64':'ddct%d', 'float32':'dct%d'}[dtype.name] except KeyError: raise ValueError("dtype %s not supported" % dtype) try: f = getattr(_fftpack, name % type) except AttributeError as e: raise ValueError(str(e) + ". Type %d not understood" % type) return f def _get_norm_mode(normalize): try: nm = {None:0, 'ortho':1}[normalize] except KeyError: raise ValueError("Unknown normalize mode %s" % normalize) return nm def __fix_shape(x, n, axis, dct_or_dst): tmp = _asfarray(x) copy_made = _datacopied(tmp, x) if n is None: n = tmp.shape[axis] elif n != tmp.shape[axis]: tmp, copy_made2 = _fix_shape(tmp, n, axis) copy_made = copy_made or copy_made2 if n < 1: raise ValueError("Invalid number of %s data points " "(%d) specified." % (dct_or_dst, n)) return tmp, n, copy_made def _raw_dct(x0, type, n, axis, nm, overwrite_x): f = _get_dct_fun(type, x0.dtype) return _eval_fun(f, x0, n, axis, nm, overwrite_x) def _raw_dst(x0, type, n, axis, nm, overwrite_x): f = _get_dst_fun(type, x0.dtype) return _eval_fun(f, x0, n, axis, nm, overwrite_x) def _eval_fun(f, tmp, n, axis, nm, overwrite_x): if axis == -1 or axis == len(tmp.shape) - 1: return f(tmp, n, nm, overwrite_x) tmp = np.swapaxes(tmp, axis, -1) tmp = f(tmp, n, nm, overwrite_x) return np.swapaxes(tmp, axis, -1) def _dct(x, type, n=None, axis=-1, overwrite_x=False, normalize=None): """ Return Discrete Cosine Transform of arbitrary type sequence x. Parameters ---------- 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. Returns ------- z : ndarray """ x0, n, copy_made = __fix_shape(x, n, axis, 'DCT') if type == 1 and n < 2: raise ValueError("DCT-I is not defined for size < 2") overwrite_x = overwrite_x or copy_made nm = _get_norm_mode(normalize) if np.iscomplexobj(x0): return (_raw_dct(x0.real, type, n, axis, nm, overwrite_x) + 1j * _raw_dct(x0.imag, type, n, axis, nm, overwrite_x)) else: return _raw_dct(x0, type, n, axis, nm, overwrite_x) def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False): """ Return the Discrete Sine Transform of arbitrary type sequence x. Parameters ---------- 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. Returns ------- dst : ndarray of reals The transformed input array. See Also -------- idst : Inverse DST Notes ----- 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. .. versionadded:: 0.11.0 References ---------- .. [1] Wikipedia, "Discrete sine transform", http://en.wikipedia.org/wiki/Discrete_sine_transform """ if type == 1 and norm is not None: raise NotImplementedError( "Orthonormalization not yet supported for IDCT-I") return _dst(x, type, n, axis, normalize=norm, overwrite_x=overwrite_x) def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False): """ Return the Inverse Discrete Sine Transform of an arbitrary type sequence. Parameters ---------- 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. Returns ------- idst : ndarray of real The transformed input array. See Also -------- dst : Forward DST Notes ----- '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`. .. versionadded:: 0.11.0 """ if type == 1 and norm is not None: raise NotImplementedError( "Orthonormalization not yet supported for IDCT-I") # Inverse/forward type table _TP = {1:1, 2:3, 3:2} return _dst(x, _TP[type], n, axis, normalize=norm, overwrite_x=overwrite_x) def _get_dst_fun(type, dtype): try: name = {'float64':'ddst%d', 'float32':'dst%d'}[dtype.name] except KeyError: raise ValueError("dtype %s not supported" % dtype) try: f = getattr(_fftpack, name % type) except AttributeError as e: raise ValueError(str(e) + ". Type %d not understood" % type) return f def _dst(x, type, n=None, axis=-1, overwrite_x=False, normalize=None): """ Return Discrete Sine Transform of arbitrary type sequence x. Parameters ---------- 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) Returns ------- z : real ndarray """ x0, n, copy_made = __fix_shape(x, n, axis, 'DST') if type == 1 and n < 2: raise ValueError("DST-I is not defined for size < 2") overwrite_x = overwrite_x or copy_made nm = _get_norm_mode(normalize) if np.iscomplexobj(x0): return (_raw_dst(x0.real, type, n, axis, nm, overwrite_x) + 1j * _raw_dst(x0.imag, type, n, axis, nm, overwrite_x)) else: return _raw_dst(x0, type, n, axis, nm, overwrite_x)