The transfer function can be defined as the transform of the impulse response:
Note that if there are inputs and outputs, is a transfer-function matrix (or ``matrix transfer function'').
Example State Space Filter Transfer Function
In this example, we consider a second-order filter () with two inputs () and two outputs ():
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 , 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.
Transposition of a State Space Filter