## Transposition of a State Space Filter

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

*transpose*of this equation gives

*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

*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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Transfer Function of a State Space Filter