File size: 25,386 Bytes
7885a28 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 |
from ._basic import _dispatch
from scipy._lib.uarray import Dispatchable
import numpy as np
__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn']
@_dispatch
def dctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False,
workers=None, *, orthogonalize=None):
"""
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.
s : int or array_like of ints or None, optional
The shape of the result. If both `s` and `axes` (see below) are None,
`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is
``numpy.take(x.shape, axes, axis=0)``.
If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros.
If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length
``s[i]``.
If any element of `s` is -1, the size of the corresponding dimension of
`x` is used.
axes : int or array_like of ints or None, optional
Axes over which the DCT is computed. If not given, the last ``len(s)``
axes are used, or all axes if `s` is also not specified.
norm : {"backward", "ortho", "forward"}, optional
Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
workers : int, optional
Maximum number of workers to use for parallel computation. If negative,
the value wraps around from ``os.cpu_count()``.
See :func:`~scipy.fft.fft` for more details.
orthogonalize : bool, optional
Whether to use the orthogonalized DCT variant (see Notes).
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
.. versionadded:: 1.8.0
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
--------
>>> import numpy as np
>>> from scipy.fft import dctn, idctn
>>> rng = np.random.default_rng()
>>> y = rng.standard_normal((16, 16))
>>> np.allclose(y, idctn(dctn(y)))
True
"""
return (Dispatchable(x, np.ndarray),)
@_dispatch
def idctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False,
workers=None, orthogonalize=None):
"""
Return multidimensional Inverse 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.
s : int or array_like of ints or None, optional
The shape of the result. If both `s` and `axes` (see below) are
None, `s` is ``x.shape``; if `s` is None but `axes` is
not None, then `s` is ``numpy.take(x.shape, axes, axis=0)``.
If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros.
If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length
``s[i]``.
If any element of `s` is -1, the size of the corresponding dimension of
`x` is used.
axes : int or array_like of ints or None, optional
Axes over which the IDCT is computed. If not given, the last ``len(s)``
axes are used, or all axes if `s` is also not specified.
norm : {"backward", "ortho", "forward"}, optional
Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
workers : int, optional
Maximum number of workers to use for parallel computation. If negative,
the value wraps around from ``os.cpu_count()``.
See :func:`~scipy.fft.fft` for more details.
orthogonalize : bool, optional
Whether to use the orthogonalized IDCT variant (see Notes).
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
.. versionadded:: 1.8.0
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
--------
>>> import numpy as np
>>> from scipy.fft import dctn, idctn
>>> rng = np.random.default_rng()
>>> y = rng.standard_normal((16, 16))
>>> np.allclose(y, idctn(dctn(y)))
True
"""
return (Dispatchable(x, np.ndarray),)
@_dispatch
def dstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False,
workers=None, orthogonalize=None):
"""
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 DST (see Notes). Default type is 2.
s : int or array_like of ints or None, optional
The shape of the result. If both `s` and `axes` (see below) are None,
`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is
``numpy.take(x.shape, axes, axis=0)``.
If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros.
If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length
``s[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 over which the DST is computed. If not given, the last ``len(s)``
axes are used, or all axes if `s` is also not specified.
norm : {"backward", "ortho", "forward"}, optional
Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
workers : int, optional
Maximum number of workers to use for parallel computation. If negative,
the value wraps around from ``os.cpu_count()``.
See :func:`~scipy.fft.fft` for more details.
orthogonalize : bool, optional
Whether to use the orthogonalized DST variant (see Notes).
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
.. versionadded:: 1.8.0
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
--------
>>> import numpy as np
>>> from scipy.fft import dstn, idstn
>>> rng = np.random.default_rng()
>>> y = rng.standard_normal((16, 16))
>>> np.allclose(y, idstn(dstn(y)))
True
"""
return (Dispatchable(x, np.ndarray),)
@_dispatch
def idstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False,
workers=None, orthogonalize=None):
"""
Return multidimensional Inverse Discrete Sine Transform along the specified axes.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3, 4}, optional
Type of the DST (see Notes). Default type is 2.
s : int or array_like of ints or None, optional
The shape of the result. If both `s` and `axes` (see below) are None,
`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is
``numpy.take(x.shape, axes, axis=0)``.
If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros.
If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length
``s[i]``.
If any element of `s` is -1, the size of the corresponding dimension of
`x` is used.
axes : int or array_like of ints or None, optional
Axes over which the IDST is computed. If not given, the last ``len(s)``
axes are used, or all axes if `s` is also not specified.
norm : {"backward", "ortho", "forward"}, optional
Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
workers : int, optional
Maximum number of workers to use for parallel computation. If negative,
the value wraps around from ``os.cpu_count()``.
See :func:`~scipy.fft.fft` for more details.
orthogonalize : bool, optional
Whether to use the orthogonalized IDST variant (see Notes).
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
.. versionadded:: 1.8.0
Returns
-------
y : ndarray of real
The transformed input array.
See Also
--------
dstn : multidimensional DST
Notes
-----
For full details of the IDST types and normalization modes, as well as
references, see `idst`.
Examples
--------
>>> import numpy as np
>>> from scipy.fft import dstn, idstn
>>> rng = np.random.default_rng()
>>> y = rng.standard_normal((16, 16))
>>> np.allclose(y, idstn(dstn(y)))
True
"""
return (Dispatchable(x, np.ndarray),)
@_dispatch
def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None,
orthogonalize=None):
r"""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 : {"backward", "ortho", "forward"}, optional
Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
workers : int, optional
Maximum number of workers to use for parallel computation. If negative,
the value wraps around from ``os.cpu_count()``.
See :func:`~scipy.fft.fft` for more details.
orthogonalize : bool, optional
Whether to use the orthogonalized DCT variant (see Notes).
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
.. versionadded:: 1.8.0
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)``.
.. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct
correspondence with the direct Fourier transform. To recover
it you must specify ``orthogonalize=False``.
For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same
overall factor in both directions. By default, the transform is also
orthogonalized which for types 1, 2 and 3 means the transform definition is
modified to give orthogonality of the DCT matrix (see below).
For ``norm="backward"``, there is no scaling on `dct` and the `idct` is
scaled by ``1/N`` where ``N`` is the "logical" size of the DCT. For
``norm="forward"`` the ``1/N`` normalization is applied to the forward
`dct` instead and the `idct` is unnormalized.
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="backward"``)
.. math::
y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left(
\frac{\pi k n}{N-1} \right)
If ``orthogonalize=True``, ``x[0]`` and ``x[N-1]`` are multiplied by a
scaling factor of :math:`\sqrt{2}`, and ``y[0]`` and ``y[N-1]`` are divided
by :math:`\sqrt{2}`. When combined with ``norm="ortho"``, this makes the
corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``).
.. 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="backward"``)
.. math::
y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right)
If ``orthogonalize=True``, ``y[0]`` is divided by :math:`\sqrt{2}` which,
when combined with ``norm="ortho"``, makes the corresponding matrix of
coefficients orthonormal (``O @ O.T = np.eye(N)``).
**Type III**
There are several definitions, we use the following (for
``norm="backward"``)
.. math::
y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right)
If ``orthogonalize=True``, ``x[0]`` terms are multiplied by
:math:`\sqrt{2}` which, when combined with ``norm="ortho"``, makes the
corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(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.
**Type IV**
There are several definitions of the DCT-IV; we use the following
(for ``norm="backward"``)
.. math::
y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)
``orthogonalize`` has no effect here, as the DCT-IV matrix is already
orthogonal up to a scale factor of ``2N``.
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.fft import fft, dct
>>> import numpy as np
>>> 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.])
"""
return (Dispatchable(x, np.ndarray),)
@_dispatch
def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False,
workers=None, orthogonalize=None):
"""
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 : {"backward", "ortho", "forward"}, optional
Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
workers : int, optional
Maximum number of workers to use for parallel computation. If negative,
the value wraps around from ``os.cpu_count()``.
See :func:`~scipy.fft.fft` for more details.
orthogonalize : bool, optional
Whether to use the orthogonalized IDCT variant (see Notes).
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
.. versionadded:: 1.8.0
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)``.
.. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct
correspondence with the inverse direct Fourier transform. To
recover it you must specify ``orthogonalize=False``.
For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same
overall factor in both directions. By default, the transform is also
orthogonalized which for types 1, 2 and 3 means the transform definition is
modified to give orthogonality of the IDCT matrix (see `dct` for the full
definitions).
'The' IDCT is the IDCT-II, which is the same as the normalized DCT-III.
The IDCT is equivalent to a normal DCT except for the normalization and
type. DCT type 1 and 4 are their own inverse and DCTs 2 and 3 are each
other's inverses.
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.fft import ifft, idct
>>> import numpy as np
>>> 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)
array([ 4., 3., 5., 10.])
"""
return (Dispatchable(x, np.ndarray),)
@_dispatch
def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None,
orthogonalize=None):
r"""
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 : {"backward", "ortho", "forward"}, optional
Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
workers : int, optional
Maximum number of workers to use for parallel computation. If negative,
the value wraps around from ``os.cpu_count()``.
See :func:`~scipy.fft.fft` for more details.
orthogonalize : bool, optional
Whether to use the orthogonalized DST variant (see Notes).
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
.. versionadded:: 1.8.0
Returns
-------
dst : ndarray of reals
The transformed input array.
See Also
--------
idst : Inverse DST
Notes
-----
.. warning:: For ``type in {2, 3}``, ``norm="ortho"`` breaks the direct
correspondence with the direct Fourier transform. To recover
it you must specify ``orthogonalize=False``.
For ``norm="ortho"`` both the `dst` and `idst` are scaled by the same
overall factor in both directions. By default, the transform is also
orthogonalized which for types 2 and 3 means the transform definition is
modified to give orthogonality of the DST matrix (see below).
For ``norm="backward"``, there is no scaling on the `dst` and the `idst` is
scaled by ``1/N`` where ``N`` is the "logical" size of the DST.
There are, theoretically, 8 types of the DST for different combinations of
even/odd boundary conditions and boundary off sets [1]_, only the first
4 types are implemented in SciPy.
**Type I**
There are several definitions of the DST-I; we use the following for
``norm="backward"``. DST-I assumes the input is odd around :math:`n=-1` and
:math:`n=N`.
.. math::
y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(n+1)}{N+1}\right)
Note that the DST-I is only supported for input size > 1.
The (unnormalized) DST-I is its own inverse, up to a factor :math:`2(N+1)`.
The orthonormalized DST-I is exactly its own inverse.
``orthogonalize`` has no effect here, as the DST-I matrix is already
orthogonal up to a scale factor of ``2N``.
**Type II**
There are several definitions of the DST-II; we use the following for
``norm="backward"``. DST-II assumes the input is odd around :math:`n=-1/2` and
:math:`n=N-1/2`; the output is odd around :math:`k=-1` and even around :math:`k=N-1`
.. math::
y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(2n+1)}{2N}\right)
If ``orthogonalize=True``, ``y[-1]`` is divided :math:`\sqrt{2}` which, when
combined with ``norm="ortho"``, makes the corresponding matrix of
coefficients orthonormal (``O @ O.T = np.eye(N)``).
**Type III**
There are several definitions of the DST-III, we use the following (for
``norm="backward"``). DST-III assumes the input is odd around :math:`n=-1` and
even around :math:`n=N-1`
.. math::
y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left(
\frac{\pi(2k+1)(n+1)}{2N}\right)
If ``orthogonalize=True``, ``x[-1]`` is multiplied by :math:`\sqrt{2}`
which, when combined with ``norm="ortho"``, makes the corresponding matrix
of coefficients orthonormal (``O @ O.T = np.eye(N)``).
The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up
to a factor :math:`2N`. The orthonormalized DST-III is exactly the inverse of the
orthonormalized DST-II.
**Type IV**
There are several definitions of the DST-IV, we use the following (for
``norm="backward"``). DST-IV assumes the input is odd around :math:`n=-0.5` and
even around :math:`n=N-0.5`
.. math::
y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right)
``orthogonalize`` has no effect here, as the DST-IV matrix is already
orthogonal up to a scale factor of ``2N``.
The (unnormalized) DST-IV is its own inverse, up to a factor :math:`2N`. The
orthonormalized DST-IV is exactly its own inverse.
References
----------
.. [1] Wikipedia, "Discrete sine transform",
https://en.wikipedia.org/wiki/Discrete_sine_transform
"""
return (Dispatchable(x, np.ndarray),)
@_dispatch
def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False,
workers=None, orthogonalize=None):
"""
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 : {"backward", "ortho", "forward"}, optional
Normalization mode (see Notes). Default is "backward".
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
workers : int, optional
Maximum number of workers to use for parallel computation. If negative,
the value wraps around from ``os.cpu_count()``.
See :func:`~scipy.fft.fft` for more details.
orthogonalize : bool, optional
Whether to use the orthogonalized IDST variant (see Notes).
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
.. versionadded:: 1.8.0
Returns
-------
idst : ndarray of real
The transformed input array.
See Also
--------
dst : Forward DST
Notes
-----
.. warning:: For ``type in {2, 3}``, ``norm="ortho"`` breaks the direct
correspondence with the inverse direct Fourier transform.
For ``norm="ortho"`` both the `dst` and `idst` are scaled by the same
overall factor in both directions. By default, the transform is also
orthogonalized which for types 2 and 3 means the transform definition is
modified to give orthogonality of the DST matrix (see `dst` for the full
definitions).
'The' IDST is the IDST-II, which is the same as the normalized DST-III.
The IDST is equivalent to a normal DST except for the normalization and
type. DST type 1 and 4 are their own inverse and DSTs 2 and 3 are each
other's inverses.
"""
return (Dispatchable(x, np.ndarray),)
|