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Source code for torch.signal.windows.windows

# mypy: allow-untyped-decorators
# mypy: allow-untyped-defs
from typing import Optional, Iterable

import torch
from math import sqrt

from torch import Tensor
from torch._torch_docs import factory_common_args, parse_kwargs, merge_dicts

__all__ = [
    'bartlett',
    'blackman',
    'cosine',
    'exponential',
    'gaussian',
    'general_cosine',
    'general_hamming',
    'hamming',
    'hann',
    'kaiser',
    'nuttall',
]

window_common_args = merge_dicts(
    parse_kwargs(
        """
    M (int): the length of the window.
        In other words, the number of points of the returned window.
    sym (bool, optional): If `False`, returns a periodic window suitable for use in spectral analysis.
        If `True`, returns a symmetric window suitable for use in filter design. Default: `True`.
"""
    ),
    factory_common_args,
    {
        "normalization": "The window is normalized to 1 (maximum value is 1). However, the 1 doesn't appear if "
                         ":attr:`M` is even and :attr:`sym` is `True`.",
    }
)


def _add_docstr(*args):
    r"""Adds docstrings to a given decorated function.

    Specially useful when then docstrings needs string interpolation, e.g., with
    str.format().
    REMARK: Do not use this function if the docstring doesn't need string
    interpolation, just write a conventional docstring.

    Args:
        args (str):
    """

    def decorator(o):
        o.__doc__ = "".join(args)
        return o

    return decorator


def _window_function_checks(function_name: str, M: int, dtype: torch.dtype, layout: torch.layout) -> None:
    r"""Performs common checks for all the defined windows.
     This function should be called before computing any window.

     Args:
         function_name (str): name of the window function.
         M (int): length of the window.
         dtype (:class:`torch.dtype`): the desired data type of returned tensor.
         layout (:class:`torch.layout`): the desired layout of returned tensor.
     """
    if M < 0:
        raise ValueError(f'{function_name} requires non-negative window length, got M={M}')
    if layout is not torch.strided:
        raise ValueError(f'{function_name} is implemented for strided tensors only, got: {layout}')
    if dtype not in [torch.float32, torch.float64]:
        raise ValueError(f'{function_name} expects float32 or float64 dtypes, got: {dtype}')


[docs]@_add_docstr( r""" Computes a window with an exponential waveform. Also known as Poisson window. The exponential window is defined as follows: .. math:: w_n = \exp{\left(-\frac{|n - c|}{\tau}\right)} where `c` is the ``center`` of the window. """, r""" {normalization} Args: {M} Keyword args: center (float, optional): where the center of the window will be located. Default: `M / 2` if `sym` is `False`, else `(M - 1) / 2`. tau (float, optional): the decay value. Tau is generally associated with a percentage, that means, that the value should vary within the interval (0, 100]. If tau is 100, it is considered the uniform window. Default: 1.0. {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric exponential window of size 10 and with a decay value of 1.0. >>> # The center will be at (M - 1) / 2, where M is 10. >>> torch.signal.windows.exponential(10) tensor([0.0111, 0.0302, 0.0821, 0.2231, 0.6065, 0.6065, 0.2231, 0.0821, 0.0302, 0.0111]) >>> # Generates a periodic exponential window and decay factor equal to .5 >>> torch.signal.windows.exponential(10, sym=False,tau=.5) tensor([4.5400e-05, 3.3546e-04, 2.4788e-03, 1.8316e-02, 1.3534e-01, 1.0000e+00, 1.3534e-01, 1.8316e-02, 2.4788e-03, 3.3546e-04]) """.format( **window_common_args ), ) def exponential( M: int, *, center: Optional[float] = None, tau: float = 1.0, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False ) -> Tensor: if dtype is None: dtype = torch.get_default_dtype() _window_function_checks('exponential', M, dtype, layout) if tau <= 0: raise ValueError(f'Tau must be positive, got: {tau} instead.') if sym and center is not None: raise ValueError('Center must be None for symmetric windows') if M == 0: return torch.empty((0,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) if center is None: center = (M if not sym and M > 1 else M - 1) / 2.0 constant = 1 / tau k = torch.linspace(start=-center * constant, end=(-center + (M - 1)) * constant, steps=M, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) return torch.exp(-torch.abs(k))
[docs]@_add_docstr( r""" Computes a window with a simple cosine waveform, following the same implementation as SciPy. This window is also known as the sine window. The cosine window is defined as follows: .. math:: w_n = \sin\left(\frac{\pi (n + 0.5)}{M}\right) This formula differs from the typical cosine window formula by incorporating a 0.5 term in the numerator, which shifts the sample positions. This adjustment results in a window that starts and ends with non-zero values. """, r""" {normalization} Args: {M} Keyword args: {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric cosine window. >>> torch.signal.windows.cosine(10) tensor([0.1564, 0.4540, 0.7071, 0.8910, 0.9877, 0.9877, 0.8910, 0.7071, 0.4540, 0.1564]) >>> # Generates a periodic cosine window. >>> torch.signal.windows.cosine(10, sym=False) tensor([0.1423, 0.4154, 0.6549, 0.8413, 0.9595, 1.0000, 0.9595, 0.8413, 0.6549, 0.4154]) """.format( **window_common_args, ), ) def cosine( M: int, *, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False ) -> Tensor: if dtype is None: dtype = torch.get_default_dtype() _window_function_checks('cosine', M, dtype, layout) if M == 0: return torch.empty((0,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) start = 0.5 constant = torch.pi / (M + 1 if not sym and M > 1 else M) k = torch.linspace(start=start * constant, end=(start + (M - 1)) * constant, steps=M, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) return torch.sin(k)
[docs]@_add_docstr( r""" Computes a window with a gaussian waveform. The gaussian window is defined as follows: .. math:: w_n = \exp{\left(-\left(\frac{n}{2\sigma}\right)^2\right)} """, r""" {normalization} Args: {M} Keyword args: std (float, optional): the standard deviation of the gaussian. It controls how narrow or wide the window is. Default: 1.0. {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric gaussian window with a standard deviation of 1.0. >>> torch.signal.windows.gaussian(10) tensor([4.0065e-05, 2.1875e-03, 4.3937e-02, 3.2465e-01, 8.8250e-01, 8.8250e-01, 3.2465e-01, 4.3937e-02, 2.1875e-03, 4.0065e-05]) >>> # Generates a periodic gaussian window and standard deviation equal to 0.9. >>> torch.signal.windows.gaussian(10, sym=False,std=0.9) tensor([1.9858e-07, 5.1365e-05, 3.8659e-03, 8.4658e-02, 5.3941e-01, 1.0000e+00, 5.3941e-01, 8.4658e-02, 3.8659e-03, 5.1365e-05]) """.format( **window_common_args, ), ) def gaussian( M: int, *, std: float = 1.0, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False ) -> Tensor: if dtype is None: dtype = torch.get_default_dtype() _window_function_checks('gaussian', M, dtype, layout) if std <= 0: raise ValueError(f'Standard deviation must be positive, got: {std} instead.') if M == 0: return torch.empty((0,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) start = -(M if not sym and M > 1 else M - 1) / 2.0 constant = 1 / (std * sqrt(2)) k = torch.linspace(start=start * constant, end=(start + (M - 1)) * constant, steps=M, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) return torch.exp(-k ** 2)
[docs]@_add_docstr( r""" Computes the Kaiser window. The Kaiser window is defined as follows: .. math:: w_n = I_0 \left( \beta \sqrt{1 - \left( {\frac{n - N/2}{N/2}} \right) ^2 } \right) / I_0( \beta ) where ``I_0`` is the zeroth order modified Bessel function of the first kind (see :func:`torch.special.i0`), and ``N = M - 1 if sym else M``. """, r""" {normalization} Args: {M} Keyword args: beta (float, optional): shape parameter for the window. Must be non-negative. Default: 12.0 {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric gaussian window with a standard deviation of 1.0. >>> torch.signal.windows.kaiser(5) tensor([4.0065e-05, 2.1875e-03, 4.3937e-02, 3.2465e-01, 8.8250e-01, 8.8250e-01, 3.2465e-01, 4.3937e-02, 2.1875e-03, 4.0065e-05]) >>> # Generates a periodic gaussian window and standard deviation equal to 0.9. >>> torch.signal.windows.kaiser(5, sym=False,std=0.9) tensor([1.9858e-07, 5.1365e-05, 3.8659e-03, 8.4658e-02, 5.3941e-01, 1.0000e+00, 5.3941e-01, 8.4658e-02, 3.8659e-03, 5.1365e-05]) """.format( **window_common_args, ), ) def kaiser( M: int, *, beta: float = 12.0, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False ) -> Tensor: if dtype is None: dtype = torch.get_default_dtype() _window_function_checks('kaiser', M, dtype, layout) if beta < 0: raise ValueError(f'beta must be non-negative, got: {beta} instead.') if M == 0: return torch.empty((0,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) if M == 1: return torch.ones((1,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) # Avoid NaNs by casting `beta` to the appropriate dtype. beta = torch.tensor(beta, dtype=dtype, device=device) start = -beta constant = 2.0 * beta / (M if not sym else M - 1) end = torch.minimum(beta, start + (M - 1) * constant) k = torch.linspace(start=start, end=end, steps=M, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) return torch.i0(torch.sqrt(beta * beta - torch.pow(k, 2))) / torch.i0(beta)
[docs]@_add_docstr( r""" Computes the Hamming window. The Hamming window is defined as follows: .. math:: w_n = \alpha - \beta\ \cos \left( \frac{2 \pi n}{M - 1} \right) """, r""" {normalization} Arguments: {M} Keyword args: {sym} alpha (float, optional): The coefficient :math:`\alpha` in the equation above. beta (float, optional): The coefficient :math:`\beta` in the equation above. {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric Hamming window. >>> torch.signal.windows.hamming(10) tensor([0.0800, 0.1876, 0.4601, 0.7700, 0.9723, 0.9723, 0.7700, 0.4601, 0.1876, 0.0800]) >>> # Generates a periodic Hamming window. >>> torch.signal.windows.hamming(10, sym=False) tensor([0.0800, 0.1679, 0.3979, 0.6821, 0.9121, 1.0000, 0.9121, 0.6821, 0.3979, 0.1679]) """.format( **window_common_args ), ) def hamming(M: int, *, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False) -> Tensor: return general_hamming(M, sym=sym, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad)
[docs]@_add_docstr( r""" Computes the Hann window. The Hann window is defined as follows: .. math:: w_n = \frac{1}{2}\ \left[1 - \cos \left( \frac{2 \pi n}{M - 1} \right)\right] = \sin^2 \left( \frac{\pi n}{M - 1} \right) """, r""" {normalization} Arguments: {M} Keyword args: {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric Hann window. >>> torch.signal.windows.hann(10) tensor([0.0000, 0.1170, 0.4132, 0.7500, 0.9698, 0.9698, 0.7500, 0.4132, 0.1170, 0.0000]) >>> # Generates a periodic Hann window. >>> torch.signal.windows.hann(10, sym=False) tensor([0.0000, 0.0955, 0.3455, 0.6545, 0.9045, 1.0000, 0.9045, 0.6545, 0.3455, 0.0955]) """.format( **window_common_args ), ) def hann(M: int, *, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False) -> Tensor: return general_hamming(M, alpha=0.5, sym=sym, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad)
[docs]@_add_docstr( r""" Computes the Blackman window. The Blackman window is defined as follows: .. math:: w_n = 0.42 - 0.5 \cos \left( \frac{2 \pi n}{M - 1} \right) + 0.08 \cos \left( \frac{4 \pi n}{M - 1} \right) """, r""" {normalization} Arguments: {M} Keyword args: {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric Blackman window. >>> torch.signal.windows.blackman(5) tensor([-1.4901e-08, 3.4000e-01, 1.0000e+00, 3.4000e-01, -1.4901e-08]) >>> # Generates a periodic Blackman window. >>> torch.signal.windows.blackman(5, sym=False) tensor([-1.4901e-08, 2.0077e-01, 8.4923e-01, 8.4923e-01, 2.0077e-01]) """.format( **window_common_args ), ) def blackman(M: int, *, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False) -> Tensor: if dtype is None: dtype = torch.get_default_dtype() _window_function_checks('blackman', M, dtype, layout) return general_cosine(M, a=[0.42, 0.5, 0.08], sym=sym, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad)
[docs]@_add_docstr( r""" Computes the Bartlett window. The Bartlett window is defined as follows: .. math:: w_n = 1 - \left| \frac{2n}{M - 1} - 1 \right| = \begin{cases} \frac{2n}{M - 1} & \text{if } 0 \leq n \leq \frac{M - 1}{2} \\ 2 - \frac{2n}{M - 1} & \text{if } \frac{M - 1}{2} < n < M \\ \end{cases} """, r""" {normalization} Arguments: {M} Keyword args: {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric Bartlett window. >>> torch.signal.windows.bartlett(10) tensor([0.0000, 0.2222, 0.4444, 0.6667, 0.8889, 0.8889, 0.6667, 0.4444, 0.2222, 0.0000]) >>> # Generates a periodic Bartlett window. >>> torch.signal.windows.bartlett(10, sym=False) tensor([0.0000, 0.2000, 0.4000, 0.6000, 0.8000, 1.0000, 0.8000, 0.6000, 0.4000, 0.2000]) """.format( **window_common_args ), ) def bartlett(M: int, *, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False) -> Tensor: if dtype is None: dtype = torch.get_default_dtype() _window_function_checks('bartlett', M, dtype, layout) if M == 0: return torch.empty((0,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) if M == 1: return torch.ones((1,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) start = -1 constant = 2 / (M if not sym else M - 1) k = torch.linspace(start=start, end=start + (M - 1) * constant, steps=M, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) return 1 - torch.abs(k)
[docs]@_add_docstr( r""" Computes the general cosine window. The general cosine window is defined as follows: .. math:: w_n = \sum^{M-1}_{i=0} (-1)^i a_i \cos{ \left( \frac{2 \pi i n}{M - 1}\right)} """, r""" {normalization} Arguments: {M} Keyword args: a (Iterable): the coefficients associated to each of the cosine functions. {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric general cosine window with 3 coefficients. >>> torch.signal.windows.general_cosine(10, a=[0.46, 0.23, 0.31], sym=True) tensor([0.5400, 0.3376, 0.1288, 0.4200, 0.9136, 0.9136, 0.4200, 0.1288, 0.3376, 0.5400]) >>> # Generates a periodic general cosine window wit 2 coefficients. >>> torch.signal.windows.general_cosine(10, a=[0.5, 1 - 0.5], sym=False) tensor([0.0000, 0.0955, 0.3455, 0.6545, 0.9045, 1.0000, 0.9045, 0.6545, 0.3455, 0.0955]) """.format( **window_common_args ), ) def general_cosine(M, *, a: Iterable, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False) -> Tensor: if dtype is None: dtype = torch.get_default_dtype() _window_function_checks('general_cosine', M, dtype, layout) if M == 0: return torch.empty((0,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) if M == 1: return torch.ones((1,), dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) if not isinstance(a, Iterable): raise TypeError("Coefficients must be a list/tuple") if not a: raise ValueError("Coefficients cannot be empty") constant = 2 * torch.pi / (M if not sym else M - 1) k = torch.linspace(start=0, end=(M - 1) * constant, steps=M, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad) a_i = torch.tensor([(-1) ** i * w for i, w in enumerate(a)], device=device, dtype=dtype, requires_grad=requires_grad) i = torch.arange(a_i.shape[0], dtype=a_i.dtype, device=a_i.device, requires_grad=a_i.requires_grad) return (a_i.unsqueeze(-1) * torch.cos(i.unsqueeze(-1) * k)).sum(0)
[docs]@_add_docstr( r""" Computes the general Hamming window. The general Hamming window is defined as follows: .. math:: w_n = \alpha - (1 - \alpha) \cos{ \left( \frac{2 \pi n}{M-1} \right)} """, r""" {normalization} Arguments: {M} Keyword args: alpha (float, optional): the window coefficient. Default: 0.54. {sym} {dtype} {layout} {device} {requires_grad} Examples:: >>> # Generates a symmetric Hamming window with the general Hamming window. >>> torch.signal.windows.general_hamming(10, sym=True) tensor([0.0800, 0.1876, 0.4601, 0.7700, 0.9723, 0.9723, 0.7700, 0.4601, 0.1876, 0.0800]) >>> # Generates a periodic Hann window with the general Hamming window. >>> torch.signal.windows.general_hamming(10, alpha=0.5, sym=False) tensor([0.0000, 0.0955, 0.3455, 0.6545, 0.9045, 1.0000, 0.9045, 0.6545, 0.3455, 0.0955]) """.format( **window_common_args ), ) def general_hamming(M, *, alpha: float = 0.54, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False) -> Tensor: return general_cosine(M, a=[alpha, 1. - alpha], sym=sym, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad)
[docs]@_add_docstr( r""" Computes the minimum 4-term Blackman-Harris window according to Nuttall. .. math:: w_n = 1 - 0.36358 \cos{(z_n)} + 0.48917 \cos{(2z_n)} - 0.13659 \cos{(3z_n)} + 0.01064 \cos{(4z_n)} where :math:`z_n = \frac{2 \pi n}{M}`. """, """ {normalization} Arguments: {M} Keyword args: {sym} {dtype} {layout} {device} {requires_grad} References:: - A. Nuttall, "Some windows with very good sidelobe behavior," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 1, pp. 84-91, Feb 1981. https://doi.org/10.1109/TASSP.1981.1163506 - Heinzel G. et al., "Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows", February 15, 2002 https://holometer.fnal.gov/GH_FFT.pdf Examples:: >>> # Generates a symmetric Nutall window. >>> torch.signal.windows.general_hamming(5, sym=True) tensor([3.6280e-04, 2.2698e-01, 1.0000e+00, 2.2698e-01, 3.6280e-04]) >>> # Generates a periodic Nuttall window. >>> torch.signal.windows.general_hamming(5, sym=False) tensor([3.6280e-04, 1.1052e-01, 7.9826e-01, 7.9826e-01, 1.1052e-01]) """.format( **window_common_args ), ) def nuttall( M: int, *, sym: bool = True, dtype: Optional[torch.dtype] = None, layout: torch.layout = torch.strided, device: Optional[torch.device] = None, requires_grad: bool = False ) -> Tensor: return general_cosine(M, a=[0.3635819, 0.4891775, 0.1365995, 0.0106411], sym=sym, dtype=dtype, layout=layout, device=device, requires_grad=requires_grad)

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