#### State Space Approach to Modal Expansions

The preceding discussion of modal synthesis was based primarily on
fitting a sum of biquads to measured frequency-response peaks. A more
general way of arriving at a modal representation is to first form a
*state space model* of the system [449], and then convert
to the modal representation by *diagonalizing* the state-space
model. This approach has the advantage of preserving system behavior
between the given inputs and outputs. Specifically, the similarity
transform used to diagonalize the system provides new input and output
gain vectors which properly excite and observe the system modes
precisely as in the original system. This procedure is especially
more convenient than the transfer-function based approach above when
there are multiple inputs and outputs. For some mathematical details,
see [449]^{9.7}For a related worked example, see §C.17.6.

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