1. { Storage, Scaling, Summation }
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 xk
xk-1 xk A yk or
yk = A.xk 1

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 + 2

3. Signal Flow Graphs
For example xk b yk a1 a2
z-1
yk-1
yk-2 3

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) 4

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. 5

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 6

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

8. Signal Flow Graphs For which Moreover z-1 a0 a1 a2 an W(z) 8 n delays+ 8

9. Signal Flow Graphs
For which
W(z) +
Y(z) - b1
z-1 b2 b3 bm
m delays 9

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 10

11. Signal Flow Graphs
Canonical form: Let ie
and 11

12. Signal Flow Graphs Hence SFG (n > m) + 12 a0 a1 a2 Y(z) X(z) W(z)-
W(z) a0 a1 a2 an b1 b2 bm 12

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 13

14. Signal Flow Graphs Parallel form: Let
with Hi(z) as in cascade but a0i = 0
With Transposition many more structures can be derived. Each will have different performance when implemented with finite precision 14

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

16. Signal Flow Graphs
Hence
And
thus 16

17. Signal Flow Graphs Two-ports S X2(z) X1(z) Linear Systems T(z) Y1(z)17

18. Signal Flow Graphs Example: Complex Multiplier M x1(n) y1(n) x2(n)18

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

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 20

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

22. Signal Flow Graphs
Set
Hence 22

23. Signal Flow Graphs With and , the oscillation frequency Set then and
We obtain
Hence x1(n) and x2(n) correspond to two sinusoidal oscillations at 90 w.r.t. each other 23

24. Signal Flow Graphs
Alternative SFG with three real multipliers + 24