Signal Flow Graphs Linear Time Invariant Discrete Time

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

Signal Flow Graphs • Summation: (Adder, Accumulator) • + X X+Y + Y • 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). 2 Professor A G Constantinides

Signal Flow Graphs • For example xk b yk z-1 a 2 3 yk-1 yk-2 Professor A G Constantinides

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 Professor A G Constantinides

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 Professor A G Constantinides

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 Professor A G Constantinides

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

Signal Flow Graphs • For which a 0 z-1 a 1 z-1 a 2 n delays an + + • Moreover 8 W(z) Professor A G Constantinides

Signal Flow Graphs • For which W(z) Y(z) + + - z-1 b 2 z-1 b 3 z-1 9 bm m delays Professor A G Constantinides

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 Professor A G Constantinides

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

Signal Flow Graphs • Hence SFG (n > m) a 0 a 1 X(z) + - - a 2 W(z) + + an + + Y(z) b 1 b 2 bm 12 Professor A G Constantinides

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 Professor A G Constantinides

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

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

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

Signal Flow Graphs • Two-ports X 1(z) Y 1(z) 17 Linear Systems X 2(z) T(z) S Y 2(z) Professor A G Constantinides

Signal Flow Graphs • Example: Complex Multiplier x 1(n) y 1(n) M x 2(n) 18 y 2(n) Professor A G Constantinides

Signal Flow Graphs • So that • Its SFD can be drawn as x 1(n) x 2(n) 19 + + - + + + y 1(n) y 2(n) Professor A G Constantinides

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

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

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

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

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