Modal Representation

When the state transition matrix $A$ is diagonal, we have the so-called modal representation. In the single-input, single-output (SISO) case, the general diagonal system looks like

$$\left[\!\!\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \\ [2pt] \vdots \\ [2pt] x_{N-1}(n+1)\\ [2pt] x_N(n+1)\end{array}\!\!\right] =\!\!\!\! \left[\! \begin{array}{ccccc} \lambda _1 & 0 & 0 & \cdots & 0 \\ [2pt] 0 & \lambda _2 & 0 & \cdots & 0 \nonumber \\ [2pt] \vdots & \vdots & \ddots & \vdots & \vdots \nonumber \\ [2pt] 0 & 0 & 0 & \lambda _{N-1} & 0 \nonumber \\ [2pt] 0 & 0 & 0 & 0 & \lambda _N \nonumber \\ [2pt] \end{array}\!\right] \left[\!\!\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \\ [2pt] \vdots \\ [2pt] x_{N-1}(n)\\ [2pt] x_N(n)\end{array}\!\!\right] + \left[\!\!\begin{array}{c} b_1 \\ [2pt] b_2 \\ [2pt] \vdots \\ [2pt] b_{N-1}\\ [2pt] b_N\end{array}\!\!\right] u(n)\nonumber$$

$$y(n) = C {\underline{x}}(n) + du(n)$$

$$= [c_1, c_2, \dots, c_N]{\underline{x}}(n) + d u(n). \protect$$ (G.21)

Since the state transition matrix is diagonal, the modes are decoupled, and we can write each mode's time-update independently:

$$\begin{eqnarray*} x_1(n+1) &=& \lambda _1 x_1(n) + b_1 u(n)\\ x_2(n+1) &=& \lambda _2 x_2(n) + b_2 u(n)\\ &\vdots& \\ x_N(n+1) &=& \lambda _N x_N(n) + b_N u(n)\\ [5pt] y(n) & = & c_1 x_1(n) + c_2 x_2(n) + \dots + c_N x_N(n) + d u(n) \end{eqnarray*}$$

Thus, the diagonalized state-space system consists of $N$ parallel one-pole systems. See §9.2.2 and §6.8.7 regarding the conversion of direct-form filter transfer functions to parallel (complex) one-pole form.