#### State Space to Modal Synthesis

The partial fraction expansion works well to create a modal-synthesis
system from a transfer function. However, this approach can yield
inefficient realizations when the system has multiple inputs and
outputs, because in that case, each element of the transfer-function
*matrix* must be separately expanded by the PFE. (The poles are
the same for each element, unless they are canceled by zeros, so it is
really only the residue calculations that must be carried out for each
element.)

If the second-order filter sections are realized in direct-form-II or
transposed-direct-form-I (or more generally in any form for which the
poles effectively precede the zeros), then the poles can be
*shared* among all the outputs for each input, since the poles
section of the filter from that input to each output sees the same
input signal as all others, resulting in the same filter
state. Similarly, the recursive portion can be shared across all
inputs for each output when the filter sections have poles implemented
after the zeros in series; one can imagine ``pushing'' the identical
two-pole filters through the summer used to form the output signal.
In summary, when the number of inputs exceeds the number of outputs,
the poles are more efficiently implemented before the zeros and shared
across all outputs for each input, and vice versa. This paragraph
can be summarized symbolically by the following matrix equation:

What may not be obvious when working with transfer functions alone is
that it is possible to share the poles across all of the inputs
*and* outputs! The answer? Just *diagonalize* a state-space
model by means of a *similarity transformation* [449, p.
360]. This will be discussed a bit further in
§8.5. In a diagonalized state-space model, the
matrix is diagonal.^{2.11} The matrix provides
routing and scaling for all the input signals driving the modes. The
matrix forms the appropriate linear combination of modes for each
output signal. If the original state-space model is a physical model,
then the transformed system gives a parallel filter bank that is
excited from the inputs and observed at the outputs in a physically
correct way.

**Next Section:**

Force-Driven-Mass Diagonalization Example

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Zero-Input Response of State Space Models