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Transfer Function of a State Space Filter

The transfer function can be defined as the $ z$ transform of the impulse response:

$\displaystyle H(z) \isdef \sum_{n=0}^{\infty} h(n) z^{-n} = D + \sum_{n=1}^{\in... ...z^{-n} = D + z^{-1}C \left[\sum_{n=0}^{\infty} \left(z^{-1}A\right)^n\right] B $

Using the closed-form sum of a matrix geometric series,G.4we obtain

$\displaystyle \fbox{$\displaystyle H(z) = D + C \left(zI - A\right)^{-1}B.$} \protect$ (G.5)

Note that if there are $ p$ inputs and $ q$ outputs, $ H(z)$ is a $ p\times q$ transfer-function matrix (or ``matrix transfer function'').

Example State Space Filter Transfer Function

In this example, we consider a second-order filter ($ N = 2$) with two inputs ($ p=2$) and two outputs ($ q=2$):

\begin{eqnarray*} A &=& g\left[\begin{array}{rr} c & -s \\ [2pt] s & c \end{arra... ... \left[\begin{array}{cc} 0 & 0 \\ [2pt] 0 & 0 \end{array}\right] \end{eqnarray*}

so that

\begin{eqnarray*} \left[\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \end{array}\... ...left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \end{array}\right]. \end{eqnarray*}

From Eq.$ \,$(G.5), the transfer function of this MIMO digital filter is then

\begin{eqnarray*} H(z) &=& C(zI-A)^{-1}B = (zI-A)^{-1} = \left[\begin{array}{cc}... ...z^{-2}}{\displaystyle 1-2gcz^{-1}+g^2z^{-2}} \end{array}\right]. \end{eqnarray*}

Note that when $ g=1$, the state transition matrix $ A$ is simply a 2D rotation matrix, rotating through the angle $ \theta$ for which $ c=\cos(\theta)$ and $ s=\sin(\theta)$. For $ g<1$, we have a type of normalized second-order resonator [51], and $ g$ controls the ``damping'' of the resonator, while $ \theta = 2\pi f_r/f_s$ controls the resonance frequency $ f_r$. The resonator is ``normalized'' in the sense that the filter's state has a constant $ L2$ norm (``preserves energy'') when $ g=1$ and the input is zero:

$\displaystyle \left\Vert\,{\underline{x}}(n+1)\,\right\Vert \isdef \sqrt{x_1^2(... ...x}}(n)\,\right\Vert \equiv \left\Vert\,{\underline{x}}(n)\,\right\Vert \protect$ (G.6)

since a rotation does not change the $ L2$ norm, as can be readily checked.

In this two-input, two-output digital filter, the input $ u_1(n)$ drives state $ x_1(n)$ while input $ u_2(n)$ drives state $ x_2(n)$. Similarly, output $ y_1(n)$ is $ x_1(n)$, while $ y_2(n)$ is $ x_2(n)$. The two-by-two transfer-function matrix $ H(z)$ contains entries for each combination of input and output. Note that all component transfer functions have the same poles. This is a general property of physical linear systems driven and observed at arbitrary points: the resonant modes (poles) are always the same, but the zeros vary as the input or output location are changed. If a pole is not visible using a particular input/output pair, we say that the pole has been ``canceled'' by a zero associated with that input/output pair. In control-theory terms, the pole is ``uncontrollable'' from that input, or ``unobservable'' from that output, or both.

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Transposition of a State Space Filter
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