Transfer Function of a State Space Filter
The transfer function can be defined as the 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
$](http://www.dsprelated.com/josimages_new/filters/img2070.png)
Note that if there are




Example State Space Filter Transfer Function
In this example, we consider a second-order filter () with two
inputs (
) and two outputs (
):
![\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*}](http://www.dsprelated.com/josimages_new/filters/img2076.png)
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*}](http://www.dsprelated.com/josimages_new/filters/img2077.png)
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*}](http://www.dsprelated.com/josimages_new/filters/img2078.png)
Note that when , the state transition matrix
is simply a 2D
rotation matrix, rotating through the angle
for which
and
. For
, we have a type of
normalized second-order resonator [51],
and
controls the ``damping'' of the resonator, while
controls the resonance frequency
. The resonator
is ``normalized'' in the sense that the filter's state has a constant
norm (``preserves energy'') when
and the input is zero:
since a rotation does not change the

In this two-input, two-output digital filter, the input
drives state
while input
drives state
.
Similarly, output
is
, while
is
.
The two-by-two transfer-function matrix
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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Complete Response