A1258150723abPch

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Block Implementations of Recursive Filters

Martijn v/d Horst M.G.v.d.Horst_at_tue.nl
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Outline

• Implementation Methods
• Look Ahead
• Block-State
• Incremental Output Computation
• Comparison
• All-pass Filters
• Conclusion

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Describing Filters

• Transfer Function

• Difference Equation

• State space form

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Implementation

• We want
• Sample rates exceeding processing rates
• This means parallel inputs and outputs, also
  called block implementations
• Implementations which scale well

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Clustered Look Ahead
n
n-2
n-1
n
n-P-1
n-P
P

• Increase the size of the recursive loop
• The order of the filter increases
• Might become unstable

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Scattered Look Ahead
n
n-2
n-1
n
n-2P
n-P
P
P

• Increase the size of the recursive loop
• The order of the filter increases
• Remains stable
• Can be implemented with P parallel filters
• Non-recursive part can be decomposed

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Block-State

• The state space form can be rewritten into a
  state space form using input and output vectors

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Block-State Architecture
Input
State
Output
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State update
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Block-State
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Incremental Block-State
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Comparison

• Efficiency The number of multipliers used by an
  implementation compared to the theoretical
  optimum number.
• A single input, single output implementation of
  an IIR filter of order N requires 2 N 1
  multipliers.
• Therefore the theoretical optimum for an
  implementation handling P simultaneous inputs and
  outputs is P (2 N 1) multipliers.

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Efficiencies
Scattered Look ahead with Incremental Output
Computation
Clustered Look ahead with Incremental Output
Computation
Incremental Block-state
Block-state
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Efficiencies
Scattered Look ahead with Incremental Output
Computation
N8
N16
Clustered Look ahead with Incremental Output
Computation
N32
N64
Block-state
Incremental Block-state
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All-pass Filters

• Also called phase shifters
• Theoretical optimum is P N

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Efficiencies w.r.t. IIR
Scattered Look ahead with Incremental Output
Computation
Clustered Look ahead with Incremental Output
Computation
Incremental Block-state
Block-state
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Efficiencies w.r.t. All-pass
Scattered Look ahead with Incremental Output
Computation
Clustered Look ahead with Incremental Output
Computation
Incremental Block-state
Block-state
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Efficiencies adapted
Clustered Look ahead for All-pass
Incremental Block-state for All-pass
Block-state for All-pass
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Efficiencies
Clustered Look ahead with Incremental Output
Computation
N8
N16
Clustered Look ahead for All-pass
N32
N64
Incremental Block-state
Incremental Block-state for All-pass
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Conclusion

• Efficient block implementations for IIR filters
  exist
• These implementations can be used for all-pass
  filters
• Theoretically there is room for improvement in
  implementing all-pass filters
• We can adapt some of the implementations for
  All-pass filters

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Questions?