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Transposition of a State Space Filter

Above, we found the transfer function of the general state-space model to be

$\displaystyle H(z) = D + C \left(zI - A\right)^{-1}B.

By the rules for transposing a matrix, the transpose of this equation gives

$\displaystyle H^T(z) = D^T + B^T \left(zI - A^T\right)^{-1}C^T.

The system $ (A^T,C^T,B^T,D^T)$ may be called the transpose of the system $ (A,B,C,D)$ . The transpose is obtained by interchanging $ B$ and $ C$ in addition to transposing all matrices.

When there is only one input and output signal (the SISO case), $ H(z)$ is a scalar, as is $ D$ . In this case we have

$\displaystyle H(z) = D + B^T \left(zI - A^T\right)^{-1}C^T.

That is, the transfer function of the transposed system is the same as the untransposed system in the scalar case. It can be shown that transposing the state-space representation is equivalent to transposing the signal flow graph of the filter [75]. The equivalence of a flow graph to its transpose is established by Mason's gain theorem [49,50]. See §9.1.3 for more on this topic.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University