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

        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.
        
art/preprocessing/audio/l_filter/numpy.py