B &]\\]@sdZddlmZmZmZddddddd d gZdd lZdd lm Z dd l m Z m Z m Z ddlmZdd lZee jee jee jee jee jee jee jee jee jee jd,ddZd-ddZd.dd Zd/dd Zd0ddZ d1ddZ!ddZ"ddZ#ddZ$ddZ%d d!Z&d"d#Z'd2d$d%Z(d3d&dZ)d4d'dZ*d(d)Z+d5d*d+Z,d S)6z+ Real spectrum transforms (DCT, DST, MDCT) )divisionprint_functionabsolute_importdctidctdstidstdctnidctndstnidstnN)_fftpack) _datacopied _fix_shape _asfarray)_init_nd_shape_and_axesFc CsJt|}t|||\}}x*t||D]\}}t||||||d}q&W|S)aR Return multidimensional Discrete Cosine Transform along the specified axes. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, optional Type of the DCT (see Notes). Default type is 2. shape : int or array_like of ints or None, 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]``. If any element of `shape` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints 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. Returns ------- y : ndarray of real The transformed input array. See Also -------- idctn : Inverse multidimensional DCT Notes ----- For full details of the DCT types and normalization modes, as well as references, see `dct`. Examples -------- >>> from scipy.fftpack import dctn, idctn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) True )typenaxisnorm overwrite_x)np asanyarrayrzipr)xrshapeaxesrrraxr;lib/python3.7/site-packages/scipy/fftpack/realtransforms.pyr s 1 c CsJt|}t|||\}}x*t||D]\}}t||||||d}q&W|S)aL Return multidimensional Discrete Cosine Transform along the specified axes. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, optional Type of the DCT (see Notes). Default type is 2. shape : int or array_like of ints or None, 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]``. If any element of `shape` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints 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. Returns ------- y : ndarray of real The transformed input array. See Also -------- dctn : multidimensional DCT Notes ----- For full details of the IDCT types and normalization modes, as well as references, see `idct`. Examples -------- >>> from scipy.fftpack import dctn, idctn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) True )rrrrr)rrrrr)rrrrrrrrrrr r Ts 1   c CsJt|}t|||\}}x*t||D]\}}t||||||d}q&W|S)aP Return multidimensional Discrete Sine Transform along the specified axes. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, optional Type of the DCT (see Notes). Default type is 2. shape : int or array_like of ints or None, 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]``. If any element of `shape` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints 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. Returns ------- y : ndarray of real The transformed input array. See Also -------- idstn : Inverse multidimensional DST Notes ----- For full details of the DST types and normalization modes, as well as references, see `dst`. Examples -------- >>> from scipy.fftpack import dstn, idstn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) True )rrrrr)rrrrr)rrrrrrrrrrr r s 1 c CsJt|}t|||\}}x*t||D]\}}t||||||d}q&W|S)aJ Return multidimensional Discrete Sine Transform along the specified axes. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, optional Type of the DCT (see Notes). Default type is 2. shape : int or array_like of ints or None, 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]``. If any element of `shape` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints 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. Returns ------- y : ndarray of real The transformed input array. See Also -------- dctn : multidimensional DST Notes ----- For full details of the IDST types and normalization modes, as well as references, see `idst`. Examples -------- >>> from scipy.fftpack import dstn, idstn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) True )rrrrr)rrrrr)rrrrrrrrrrr r s 1   cCst||||||dS)at Return the Discrete Cosine Transform of arbitrary type sequence x. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, 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 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``):: N-2 y[k] = x[0] + (-1)**k x[N-1] + 2 * sum x[n]*cos(pi*k*n/(N-1)) n=1 If ``norm='ortho'``, ``x[0]`` and ``x[N-1]`` are multiplied by a scaling factor of ``sqrt(2)``, and ``y[k]`` is multiplied by a scaling factor `f`:: f = 0.5*sqrt(1/(N-1)) if k = 0 or N-1, f = 0.5*sqrt(2/(N-1)) otherwise. .. 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``):: 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. **Type IV** There are several definitions of the DCT-IV; we use the following (for ``norm=None``):: N-1 y[k] = 2* sum x[n]*cos(pi*(2k+1)*(2n+1)/(4*N)), 0 <= k < N. n=0 If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor `f`:: f = 0.5*sqrt(2/N) .. 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.]) ) normalizer)_dct)rrrrrrrrr rs cCs&ddddd}t|||||||dS)a" Return the Inverse Discrete Cosine Transform of an arbitrary type sequence. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, 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. IDCT of type 4 is the DCT of type 4. 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.]) r)r$rr%r&)r"r)r#)rrrrrr_TPrrr rs8c Csyddd|j}Wn tk r4td|YnXytt||}Wn6tk r~}ztt|d|Wdd}~XYnX|S)Nzddct%dzdct%d)float64float32zdtype %s not supportedz. Type %d not understood)nameKeyError ValueErrorgetattrr AttributeErrorstr)rdtyper*ferrr _get_dct_funs&r3cCs8yddd|}Wn tk r2td|YnX|S)Nrr$)NZorthozUnknown normalize mode %s)r+r,)r"nmrrr _get_norm_modes r5cCsnt|}t||}|dkr&|j|}n&||j|krLt|||\}}|pJ|}|dkrdtd||f|||fS)Nr$z0Invalid number of %s data points (%d) specified.)rrrrr,)rrrZ dct_or_dsttmp copy_madeZ copy_made2rrr __fix_shapes   r8cCst||j}t||||||S)N)r3r0 _eval_fun)x0rrrr4rr1rrr _raw_dcts r;cCst||j}t||||||S)N) _get_dst_funr0r9)r:rrrr4rr1rrr _raw_dsts r=cCsR|dks|t|jdkr(|||||St||d}|||||}t||dS)Nr!r$)lenrrZswapaxes)r1r6rrr4rrrr r9s r9c Cst|||d\}}}|dkr,|dkr,td|p2|}t|}t|rrt|j|||||dt|j|||||St||||||SdS)a~ 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 ZDCTr$rz!DCT-I is not defined for size < 2y?N)r8r,r5r iscomplexobjr;realimag) rrrrrr"r:r7r4rrr r#s r#cCst||||||dS)a Return the Discrete Sine Transform of arbitrary type sequence x. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, 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 Note that the DST-I is only supported for input size > 1 The (unnormalized) DST-I is its own inverse, up to a factor `2(N+1)`. The orthonormalized DST-I is exactly its own inverse. **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) DST-III is the inverse of the (unnormalized) DST-II, up to a factor `2N`. The orthonormalized DST-III is exactly the inverse of the orthonormalized DST-II. .. versionadded:: 0.11.0 **Type IV** There are several definitions of the DST-IV, we use the following (for ``norm=None``). DST-IV assumes the input is odd around n=-0.5 and even around n=N-0.5 :: N-1 y[k] = 2* sum x[n]*sin(pi*(k+0.5)*(n+0.5)/N), 0 <= k < N. n=0 The (unnormalized) DST-IV is its own inverse, up to a factor `2N`. The orthonormalized DST-IV is exactly its own inverse. .. versionadded:: 1.2.0 Support for DST-IV. References ---------- .. [1] Wikipedia, "Discrete sine transform", https://en.wikipedia.org/wiki/Discrete_sine_transform )r"r)_dst)rrrrrrrrr rsicCs&ddddd}t|||||||dS)a Return the Inverse Discrete Sine Transform of an arbitrary type sequence. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, 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 r$r%rr&)r$rr%r&)r"r)rB)rrrrrrr'rrr rs+c Csyddd|j}Wn tk r4td|YnXytt||}Wn6tk r~}ztt|d|Wdd}~XYnX|S)Nzddst%dzdst%d)r(r)zdtype %s not supportedz. Type %d not understood)r*r+r,r-r r.r/)rr0r*r1r2rrr r<s&r<c Cst|||d\}}}|dkr,|dkr,td|p2|}t|}t|rrt|j|||||dt|j|||||St||||||SdS)a Return Discrete Sine Transform of arbitrary type sequence x. Parameters ---------- x : array_like input array. n : int, optional Length of the transform. axis : int, optional Axis along which the dst is computed. (default=-1) overwrite_x : bool, optional If True the contents of x can be destroyed. (default=False) Returns ------- z : real ndarray ZDSTr$rz!DST-I is not defined for size < 2y?N)r8r,r5rr?r=r@rA) rrrrrr"r:r7r4rrr rBs rB)rNNNF)rNNNF)rNNNF)rNNNF)rNr!NF)rNr!NF)Nr!FN)rNr!NF)rNr!NF)Nr!FN)-__doc__Z __future__rrr__all__ZnumpyrZ scipy.fftpackr Zscipy.fftpack.basicrrrZscipy.fftpack.helperratexitregisterZdestroy_ddct1_cacheZdestroy_ddct2_cacheZdestroy_ddct4_cacheZdestroy_dct1_cacheZdestroy_dct2_cacheZdestroy_dct4_cacheZdestroy_ddst1_cacheZdestroy_ddst2_cacheZdestroy_dst1_cacheZdestroy_dst2_cacher r r r rrr3r5r8r;r=r9r#rrr<rBrrrr sF             8 9 8 9  <  # l /