#### Efficiency of Diagonalized State-Space Models

Note that a general th-order state-space model Eq.(1.8) requires around multiply-adds to update for each time step (assuming the number of inputs and outputs is small compared with the number of state variables, in which case the computation dominates). After diagonalization by a similarity transform, the time update is only order , just like any other efficient digital filter realization. Thus, a diagonalized state-space model (modal representation) is a strong contender for applications in which it is desirable to have independent control of resonant modes.

Another advantage of the modal expansion is that frequency-dependent characteristics of hearing can be brought to bear. Low-frequency resonances can easily be modeled more carefully and in more detail than very high-frequency resonances which tend to be heard only ``statistically'' by the ear. For example, rows of high-frequency modes can be collapsed into more efficient digital waveguide loops (§8.5) by retuning them to the nearest harmonic mode series.

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Typical State-Space Diagonalization Procedure