Example State Space Filter Transfer Function

In this example, we consider a second-order filter ( ) with two inputs ( ) and two outputs ( ):

so that

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

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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