1. Professor A G Constantinides 1 Signal Flow Graphs Linear Time Invariant Discrete Time Systems can be made up from the elements { Storage, Scaling, Summation } Storage: (Delay, Register) Scaling: (Weight, Product, Multiplier T or z -1 xkxk x k-1 xkxk A ykyk or xkxk A ykyk y k = A.x k

2. Professor A G Constantinides 2 Signal Flow Graphs Summation: (Adder, Accumulator) A linear system equation of the type considered so far, can be represented in terms of an interconnection of these elements Conversely the system equation may be obtained from the interconnected components (structure). X Y X + Y + +

3. Professor A G Constantinides 3 Signal Flow Graphs For example xkxk b ykyk a1a1 a2a2 z -1 y k-1 y k-2

4. Professor A G Constantinides 4 Signal Flow Graphs A SFG structure indicates the way through which the operations are to be carried out in an implementation. In a LTID system, a structure can be: i) computable : (All loops contain delays) ii) non-computable : (Some loops contain no delays)

5. Professor A G Constantinides 5 Signal Flow Graphs Transposition of SFG is the process of reversing the direction of flow on all transmission paths while keeping their transfer functions the same. This entails: –Multipliers replaced by multipliers of same value –Adders replaced by branching points –Branching points replaced by adders For a single-input / output SFG the transpose SFG has the same transfer function overall, as the original.

6. Professor A G Constantinides 6 Structures STRUCTURES: (The computational schemes for deriving the input / output relationships.) For a given transfer function there are many realisation structures. Each structure has different properties w.r.t. i) Coefficient sensitivity ii) Finite register computations

7. Professor A G Constantinides 7 Signal Flow Graphs Direct form 1 : Consider the transfer function So that Set

8. Professor A G Constantinides 8 Signal Flow Graphs For which Moreover z -1 a0a0 a1a1 a2a2 anan n delays W(z)W(z) ++ + +

9. Professor A G Constantinides 9 Signal Flow Graphs For which W(z)W(z) + + Y(z)Y(z) - - - - b1b1 z -1 b2b2 b3b3 bmbm m delays

10. Professor A G Constantinides 10 Signal Flow Graphs This figure and the previous one can be combined by cascading to produce overall structure. Simple structure but NOT used extensively in practice because its performance degrades rapidly due to finite register computation effects

11. Professor A G Constantinides 11 Signal Flow Graphs Canonical form: Let ie and

12. Professor A G Constantinides 12 Signal Flow Graphs Hence SFG (n > m) ++ X(z)X(z) Y(z)Y(z) + - - - + + + W(z)W(z) a0a0 a1a1 a2a2 anan b1b1 b2b2 bmbm

13. Professor A G Constantinides 13 Signal Flow Graphs Direct form 2 : Reduction in effects due to finite register can be achieved by factoring H(z) and cascading structures corresponding to factors In general with or

14. Professor A G Constantinides 14 Signal Flow Graphs Parallel form: Let with H i (z) as in cascade but a 0i = 0 With Transposition many more structures can be derived. Each will have different performance when implemented with finite precision

15. Professor A G Constantinides 15 Signal Flow Graphs Sensitivity: Consider the effect of changing a multiplier on the transfer function Set With constraint X(z)X(z) 1 4 3 2 V(z)V(z) U(z)U(z) Y(z)Y(z)  Linear T-I Discrete System

16. Professor A G Constantinides 16 Signal Flow Graphs Hence And thus

17. Professor A G Constantinides 17 Signal Flow Graphs Two-ports X1(z)X1(z) Y1(z)Y1(z) X2(z)X2(z) Y2(z)Y2(z) T(z)T(z) Linear Systems S

18. Professor A G Constantinides 18 Signal Flow Graphs Example: Complex Multiplier x1(n)x1(n) x2(n)x2(n) M y1(n)y1(n) y2(n)y2(n)

19. Professor A G Constantinides 19 Signal Flow Graphs So that Its SFD can be drawn as x1(n)x1(n) x2(n)x2(n) y1(n)y1(n) y2(n)y2(n) + + + + - +

20. Professor A G Constantinides 20 Signal Flow Graphs Special case We have a rotation of t o by an angle We can set so that and This is the basis for designing i)Oscillators ii)Discrete Fourier Transforms (see later) iii)CORDIC operators in SONAR

21. Professor A G Constantinides 21 Signal Flow Graphs Example: Oscillator Consider and externally impose the constraint So that For oscillation

22. Professor A G Constantinides 22 Signal Flow Graphs Set Hence

23. Professor A G Constantinides 23 Signal Flow Graphs With and, the oscillation frequency Set then and We obtain Hence x 1 (n) and x 2 (n) correspond to two sinusoidal oscillations at 90  w.r.t. each other

24. Professor A G Constantinides 24 Signal Flow Graphs Alternative SFG with three real multipliers + + + + + +

25. Professor A G Constantinides 25 Signal Flow Graphs Example: Oscillator