Transposition of a State SpaceFilter

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

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

The system may be called the transpose of the system . The transpose is obtained by interchanging and in addition to transposing all matrices.

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

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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Poles of a State Space Filter
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Transfer Function of a State Space Filter