Signal Flow Graphs

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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

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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).

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Signal Flow Graphs

• For example

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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)

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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.

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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

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Signal Flow Graphs

• Direct form 1 Consider the transfer function
• So that
• Set

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Signal Flow Graphs

• For which
• Moreover

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Signal Flow Graphs

• For which

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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

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Signal Flow Graphs

• Canonical form Let
• ie
• and

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Signal Flow Graphs

• Hence SFG (n gt m)

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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

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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

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Signal Flow Graphs

• Sensitivity Consider the effect of changing a
  multiplier on the transfer function
• Set
• With constraint

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Signal Flow Graphs

• Hence
• And
• thus

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Signal Flow Graphs

• Two-ports

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Signal Flow Graphs

• Example Complex Multiplier

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Signal Flow Graphs

• So that
• Its SFD can be drawn as

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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

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Signal Flow Graphs

• Example Oscillator
• Consider and
  externally impose the constraint
• So that
• For oscillation

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Signal Flow Graphs

• Set
• Hence

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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

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Signal Flow Graphs

• Alternative SFG with three real multipliers

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Signal Flow Graphs

• Example Oscillator