docs: def whitener_lss(self):
          r"""
          This function takes the linear state space system
          that is an input to the Kalman class and it converts
          that system to the time-invariant whitener represenation
          given by
  
          .. math::
  
              \tilde{x}_{t+1}^* = \tilde{A} \tilde{x} + \tilde{C} v
              a = \tilde{G} \tilde{x}
  
          where
  
          .. math::
  
              \tilde{x}_t = [x+{t}, \hat{x}_{t}, v_{t}]
  
          and
  
          .. math::
  
              \tilde{A} =
              \begin{bmatrix}
              A  & 0    & 0  \\
              KG & A-KG & KH \\
              0  & 0    & 0 \\
              \end{bmatrix}
  
          .. math::
  
              \tilde{C} =
              \begin{bmatrix}
              C & 0 \\
              0 & 0 \\
              0 & I \\
              \end{bmatrix}
  
          .. math::
  
              \tilde{G} =
              \begin{bmatrix}
              G & -G & H \\
              \end{bmatrix}
  
          with :math:`A, C, G, H` coming from the linear state space system
          that defines the Kalman instance
  
          Returns
          -------
          whitened_lss : LinearStateSpace
              This is the linear state space system that represents
              the whitened system
          """
          K = self.K_infinity
  
          # Get the matrix sizes
          n, k, m, l = self.ss.n, self.ss.k, self.ss.m, self.ss.l
          A, C, G, H = self.ss.A, self.ss.C, self.ss.G, self.ss.H
  
          Atil = np.vstack([np.hstack([A, np.zeros((n, n)), np.zeros((n, l))]),
                            np.hstack([dot(K, G), A-dot(K, G), dot(K, H)]),
                            np.zeros((l, 2*n + l))])
  
          Ctil = np.vstack([np.hstack([C, np.zeros((n, l))]),
                            np.zeros((n, m+l)),
                            np.hstack([np.zeros((l, m)), np.eye(l)])])
  
          Gtil = np.hstack([G, -G, H])
  
          whitened_lss = LinearStateSpace(Atil, Ctil, Gtil)
          self.whitened_lss = whitened_lss
  
          return whitened_lss