class_docstr: This module implements the filter function for audio signals. It provides with an infinite impulse response (IIR)
  or finite impulse response (FIR) filter. This implementation is a wrapper around the `scipy.signal.lfilter`
  function in the `scipy` package.
docstr: Provide an estimate of the gradients of the defence for the backward pass.
  
  :param x: Input data for which the gradient is estimated. First dimension is the batch size.
  :param grad: Gradient value so far.
  :return: The gradient (estimate) of the defence.
docstr_tok: ["Provide","an","estimate","of","the","gradients","of","the","defence","for","the","backward","pass",".",":","param","x",":","Input","data","for","which","the","gradient","is","estimated",".","First","dimension","is","the","batch","size",".",":","param","grad",":","Gradient","value","so","far",".",":","return",":","The","gradient","(","estimate",")","of","the","defence","."]
code: def estimate_gradient(
      self, x: np.ndarray, grad: np.ndarray
  ) -> np.ndarray:
      """
      Provide an estimate of the gradients of the defence for the backward pass.
  
      :param x: Input data for which the gradient is estimated. First dimension is the batch size.
      :param grad: Gradient value so far.
      :return: The gradient (estimate) of the defence.
      """
      if (grad.shape != x.shape) and (  # pragma: no cover
          (len(grad.shape) != len(x.shape) + 1)
          or (grad.shape[2:] != x.shape[1:])
          or (grad.shape[0] != x.shape[0])
      ):
          raise ValueError(
              f"The shape of `grad` {grad.shape} does not match the shape of input `x` {x.shape}"
          )
  
      if (
          self.denominator_coef[0] != 1.0
          or np.sum(self.denominator_coef) != 1.0
      ):
          logger.warning(
              "Accurate gradient estimation is currently only implemented for finite impulse response filtering, "
              "the coefficients indicate infinite response filtering, therefore Backward Pass Differentiable "
              "Approximation (BPDA) is applied instead."
          )
          return grad
  
      # We will compute gradient for one sample at a time
      x_grad_list = []
  
      for i, x_i in enumerate(x):
          # First compute the gradient matrix
          grad_matrix = self._compute_gradient_matrix(size=x_i.size)
  
          # Then combine with the input gradient
          if grad.shape == x.shape:
              grad_x_i = np.zeros(x_i.size)
              for j in range(x_i.size):
                  grad_x_i[j] = np.sum(grad[i] * grad_matrix[:, j])
  
          else:
              grad_x_i = np.zeros_like(grad[i])
              for j in range(x_i.size):
                  grad_x_i[:, j] = np.sum(
                      grad[i, :, :] * grad_matrix[:, j], axis=1
                  )
  
          # And store the result
          x_grad_list.append(grad_x_i)
  
      if x.dtype == object:
          x_grad = np.empty(x.shape[0], dtype=object)
          x_grad[:] = list(x_grad_list)
  
      else:
          x_grad = np.array(x_grad_list)
  
      return x_grad
  
file_path: adversarial-robustness-toolbox/art/preprocessing/audio/l_filter/numpy.py
imports: from __future__ import absolute_import, division, print_function, unicode_literals
  import logging
  from typing import Optional, Tuple, TYPE_CHECKING
  from scipy.signal import lfilter
  import numpy
  from tqdm.auto import tqdm
  from art.preprocessing.preprocessing import Preprocessor
label_desc: Use image node_id 3 for calling the LFilter obj's underlying member method code with example usage: obj.estimate_gradient(x, grad) and returns: x_grad, grad