Feedback (“Mixing”) Matrix Stability

• A “3-channel” FDN feedback matrix can be represented as:
  $$\mathbf{M} = \left[ \begin{array}{lll} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{array}\right]$$

• The inner loop calculations of the FDN shown in Fig. 12 can then expressed as:
  $$\left[ \begin{array}{l} x_{1}(n) \\ x_{2}(n) \\ x_{3}(n) \end{... ...] + \left[ \begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right] u(n)$$
  and the loop output given by
  $$v(n) = \left[ \begin{array}{lll} c_{1} & c_{2} & c_{3} \end{a... ...in{array}{l} x_{1}(n-M_1) \\ x_{2}(n-M_2) \\ x_{3}(n-M_3) \end{array}\right]$$

• These expressions can also be written in frequency-domain vector notation as
  $$\begin{eqnarray*} \mathbf{X}(z) &=& \mathbf{G}\mathbf{M}\mathbf{D}(z)\mathbf{X}(... ...U}(z) \\ \mathbf{V}(z) &=& \mathbf{C}\mathbf{D}(z)\mathbf{X}(z) \end{eqnarray*}$$
  where
  $$\mathbf{D}(z) \ensuremath{\stackrel{\Delta}{=}}\left[ \begin{arr... ...{-M_1} & 0 & 0 \\ 0 & z^{-M_2} & 0 \\ 0 & 0 & z^{-M_3} \end{array}\right]$$

• The matrix $\mathbf{A} = \mathbf{G}\mathbf{M}$ is called the state transition matrix. $\mathbf{G}$ is typically a diagonal matrix of lowpass filters, each having gain no greater than 1.

• Stability of the FDN is assured when the norm of the state vector $\mathbf{x}[n]$ decreases over time when the input signal is zero:
  $$\Vert \mathbf{x}(n + 1) \Vert < \Vert \mathbf{x}(n) \Vert,$$
  for all $n \geq 0$, where
  $$\mathbf{x}(n + 1) = \mathbf{A} \left[ \begin{array}{l} x_{1}(n-M_1) \\ x_{2}(n-M_2) \\ x_{3}(n-M_3) \end{array}\right].$$

• Stable feedback matrices can thus be parameterized in terms of $\mathbf{A} = \mathbf{G}\mathbf{M}$, where $\mathbf{M}$ is any orthogonal matrix and $\mathbf{G}$ is a diagonal matrix having entries less than 1 in magnitude.

• A feedback matrix $\mathbf{M}_N$ is lossless if and only if its eigenvalues have modulus 1 and its $N$ eigenvectors are linearly independent.