The remaining two direct forms are obtained by formally transposing directforms I and II [60, p. 155]. Filter transposition may also be called flow graph reversal, and transposing a SingleInput, SingleOutput (SISO) filter does not alter its transfer function. This fact can be derived as a consequence of Mason's gain formula for signal flow graphs [49,50] or Tellegen's theorem (which implies that an LTI signal flow graph is interreciprocal with its transpose) [60, pp. 176177]. Transposition of filters in statespace form is discussed in §G.5.
The transpose of a SISO digital filter is quite straightforward to find: Reverse the direction of all signal paths, and make obviously necessary accommodations. ``Obviously necessary accommodations'' include changing signal branchpoints to summers, and summers to branchpoints. Also, after this operation, the input signal, normally drawn on the left of the signal flow graph, will be on the right, and the output on the left. To renormalize the layout, the whole diagram is usually leftright flipped.
Figure 9.3 shows the TransposedDirectFormI (TDFI) structure for the general secondorder IIR digital filter, and Fig.9.4 shows the TransposedDirectFormII (TDFII) structure. To facilitate comparison of the transposed with the original, the input and output signals remain ``switched'', so that signals generally flow righttoleft instead of the usual lefttoright. (Exercise: Derive forms TDFI/II by transposing the DFI/II structures shown in Figures 9.1 and 9.2.)


The difference equation implementing TDFII can be written by inspection from Fig.9.4:
Using this update order, no temporary storage elements are required, i.e., is updated after it is used, using , and then is updated.
We can refer to and the state of the filter at time . It is typical in statespace formulations to label delayelement outputs as state variables, and write update equations for the state at time as we have done here. See Appendix G for more about statespace formulations.