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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:

$\displaystyle \left[\begin{array}{c} y_1 \\ [2pt] y_2 \end{array}\right]
\eqsp \frac{1}{A}\left[\begin{array}{cc} B_1 & B_2 \\ [2pt] B_3 & B_4 \end{array}\right]\left[\begin{array}{c} u_1 \\ [2pt] u_2 \end{array}\right]
\eqsp \left[\begin{array}{cc} B_1 & B_2 \\ [2pt] B_3 & B_4 \end{array}\right]\left\{\frac{1}{A}\left[\begin{array}{c} u_1 \\ [2pt] u_2 \end{array}\right]\right\}

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 [452, p. 360]. This will be discussed a bit further in §8.5. In a diagonalized state-space model, the $ A$ matrix is diagonal.2.11 The $ B$ matrix provides routing and scaling for all the input signals driving the modes. The $ C$ 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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University