1. Lecture 9: Structure for Discrete-Time System XILIANG LUO 2014/11 1

2. Block Diagram Adder, Multiplier, Memory, Coefficient 2

3. Example 3

4. General Case 4 Direct Form 1

5. Rearrangement 5

6. 6 Zeros 1 st Poles 1 st

7. Canonic Form 7 Minimum number of delay elements: max{M, N} Direct Form 2

8. Signal Flow Graph 8 A directed graph with each node being a variable or a node value. The value at each node in a graph is the sum of the outputs of all the branches entering the node. Source node: no entering branches Sink node: no outputs

9. Signal Flow Graph 9

10. Structures for IIR: Direct Form 10

11. Structures for IIR: Direct Form 11

12. Structures for IIR: Cascade Form 12 Real coefs: Combine pairs of real factors/ complex conjugate pairs

13. Structures for IIR Cascade Form 13 2 nd –order subsystem

14. Structures for IIR Parallel Form 14 Group real poles in pairs: Partial fraction expansion:

15. Structures for IIR Parallel Form 15

16. Feedback Loops 16 If a network has no loops, then the system function has only zeros and the impulse response has finite duration! Loops are necessary to generate infinitely long impulse responses! Loop: closed path starting at a node and returning to same node by traversing branches in the direction allowed, which is defined by the arrowheads input unit impulse, the output is:

17. Transposed Form 17 Transposition: 1.reverse direction of all branches 2.keep branch gains same 3.reverse input/output For SISO, transposition gives the same system function!

18. Transposed Form 18 Transposed direct form II: poles first zeros first

19. Structures for FIR Direct Form 19 Tapped delay line

20. Structures for FIR Cascade Form 20

21. Structures for FIR with Linear Phase 21 Impulse response satisfies the following symmetry condition: or So, the number of coefficient multipliers can be essentially halved! Type-1:

22. Lattice Filters 22 2-port flow graph

23. Lattice Filters: FIR 23

24. Lattice Filters: FIR 24 Input to i-th nodes: Recursive computation of transfer functions!

25. Lattice Filters: FIR 25 To obtain a direct recursive relationship for the coefficients, or the impulse response, we use the following definition:

26. Lattice Filters: FIR 26 From k-parameters to FIR impulse response:

27. Lattice Filters: FIR 27 From FIR impulse response to k-parameters:

28. Lattice Filters: FIR 28 From FIR impulse response to k-parameters:

29. Lattice Filters: FIR 29 Direct Form Lattice Form

30. Lattice Filters: IIR 30 Invert the computations in the following figure:

31. Lattice Filters: IIR 31 Derive from FIR: IIR:

32. Lattice Filters: IIR 32 Derive from

33. Lattice Filters: IIR 33