Husheng Li, UTK-EECS, Fall 2012

 Study how to implement the LTI discrete-time systems.  We first present the block diagram and signal flow graph.  Then, we derive a number of basic equivalent structures for implementing a causal LTI.

 The basic blocks include adders, scalars and delay registers.

 A block diagram can be rearranged or modified without changing the overall system function.  In the left system, the two cascading sub- systems can be reversed while the system function is still the same.

 The graph representation consists of source, sink, intermediate nodes, additions, scalings and delays.

 A variety of theoretically equivalent systems can be obtained by simply pairing the poles and zeros in different ways.  When the computation precision is finite, the performance could be quite different for different realizations.

 We can express a rational system as a partial faction expansion and thus obtain the parallel form of an IIR.

 Transposition of a flow graph is accomplished by reversing the directions of all branches in the network while keeping the branch transmittances as they were and reversing the roles of the input and output.  For SISO systems, the transposition does not change the system function.

Direct form Transposition

M is even M is odd

 The basic building block is called a two-port flow graph. The system is achieved by cascading the blocks.

 The coefficients of the filters can be determined by the coefficients-to-k-parameters algorithm.

 A lattice structure for the all-pole system can be developed from FIR lattice by realizing that it is the inverse filter of an FIR system.