Transposition (Flow graph reversal)

Transposition leaves the overall system function in between input and output unchanged and leads to a set of transposed system structures that provide some useful alternatives.

  Transposition of the flow graph is accomplished by reversing directions of all the branches while keeping the branch transmittances as they were and reversing roles of the input and output such that source nodes become sink nodes and vice versa. Certainly, if a signal flow graph configuration is transposed, the number of delay branches and coefficients remain the same. Therefore a canonic structure remains canonic in the course of transposition.

  The result of applying transposition theorem to signal flow graph of the direct

form II structure (and reorienting diagram such that the input and output still appear on the left and right sides, respectively) is shown as follows. This will be referred to as "direct form II transposed".
 

By the comparison of direct form II with its transposed version shows that

• Direct form II structure implements poles 1st and then the zeros
• Transposed direct form II structure implements zeros 1st and then the poles

These differences can become significant in the presence of quantization in finite precision digital implementations or in presence of noise in the discrete-time analog implementations.

The transposition theorem is applied to parallel or cascade structures, the individual second-order systems are replaced by the transposed structures. 

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