1 / 24

Signal Flow Graphs

T or z -1. A. y k. or. y k. x k. x k. A. x k. x k -1. y k = A.x k. 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. +. X + Y.

rob
Download Presentation

Signal Flow Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Tor z-1 A yk or yk xk xk A xk xk-1 yk= A.xk 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 1

  2. + X + Y X + Y 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). 2

  3. b yk xk z-1 yk-1 a1 yk-2 a2 Signal Flow Graphs • For example 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. z-1 z-1 z-1 n delays a0 a1 a2 an + + + + W(z) Signal Flow Graphs • For which • Moreover 8

  9. W(z) Y(z) + + - - - - z-1 b1 z-1 b2 z-1 b3 z-1 bm m delays Signal Flow Graphs • For which 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. + + a0 a1 a2 Y(z) X(z) + + + + - W(z) an - - b1 b2 bm Signal Flow Graphs • Hence SFG (n > m) 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. U(z) V(z) 2 1 3 4 Y(z) X(z) Linear T-I Discrete System Signal Flow Graphs • Sensitivity: Consider the effect of changing a multiplier on the transfer function • Set • With constraint 15

  16. Signal Flow Graphs • Hence And thus 16

  17. X2(z) X1(z) Linear Systems S T(z) Y1(z) Y2(z) Signal Flow Graphs • Two-ports 17

  18. x1(n) y1(n) M x2(n) y2(n) Signal Flow Graphs • Example: Complex Multiplier 18

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

More Related