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Efficiency of Diagonalized State-Space Models

Note that a general $ N$th-order state-space model Eq.$ \,$(1.8) requires around $ N^2$ 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 $ A\underline{x}(n)$ computation dominates). After diagonalization by a similarity transform, the time update is only order $ N$, 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