{ 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

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

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

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

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

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

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

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

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

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

Signal Flow Graphs Canonical form: Let ie and 11

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

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

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

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

Signal Flow Graphs Hence And thus 16

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

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

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

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

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

Signal Flow Graphs Set Hence 22

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

Signal Flow Graphs Alternative SFG with three real multipliers + 24