Recalling that , the output signal from any diagonal state-space model is a linear combination of the modal signals. The two immediate outputs and in Fig.G.3 are given in terms of the modal signals and as

The output signal from the first state variable is

The *initial condition*
corresponds to modal initial
state

For this initialization, the output from the first state variable is simply

A similar derivation can be carried out to show that the output is proportional to ,

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University