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Introduction to Digital Filters
State Space Filters
Transfer Function of a State Space Filter
Example State Space Filter Transfer Function
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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/filters/img2075.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/filters/img2076.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/filters/img2077.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
norm, as can be readily checked.
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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Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.
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