Transposition of a State Space Filter

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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.