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Martijn v/d Horst M.G.v.d.Horst@tue.nl

Block Implementations of Recursive Filters. Martijn v/d Horst M.G.v.d.Horst@tue.nl. Outline. Implementation Methods Look Ahead Block-State Incremental Output Computation Comparison All-pass Filters Conclusion. Describing Filters. Transfer Function:. Difference Equation:.

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Martijn v/d Horst M.G.v.d.Horst@tue.nl

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  1. Block Implementations of Recursive Filters Martijn v/d Horst M.G.v.d.Horst@tue.nl

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

  3. Describing Filters • Transfer Function: • Difference Equation: • State space form:

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

  5. Clustered Look Ahead • Increase the size of the recursive loop • The order of the filter increases • Might become unstable n-2 n-1 n n n-P-1 n-P P

  6. Scattered Look Ahead • 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 n-2 n-1 n n-2P n-P n P P

  7. Block-State The state space form can be rewritten into a state space form using input and output vectors:

  8. Block-State Architecture Input State Output

  9. State update

  10. Block-State

  11. Incremental Block-State

  12. 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.

  13. Efficiencies Scattered Look ahead with Incremental Output Computation Clustered Look ahead with Incremental Output Computation Incremental Block-state Block-state

  14. Efficiencies Scattered Look ahead with Incremental Output Computation N=16 N=8 Clustered Look ahead with Incremental Output Computation N=32 N=64 Block-state Incremental Block-state

  15. All-pass Filters • Also called phase shifters • Theoretical optimum is P N

  16. Efficiencies w.r.t. IIR Scattered Look ahead with Incremental Output Computation Clustered Look ahead with Incremental Output Computation Incremental Block-state Block-state

  17. 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

  18. Efficiencies adapted Clustered Look ahead for All-pass Incremental Block-state for All-pass Block-state for All-pass

  19. Efficiencies Clustered Look ahead with Incremental Output Computation N=16 N=8 Clustered Look ahead for All-pass N=32 N=64 Incremental Block-state Incremental Block-state for All-pass

  20. 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

  21. Questions?

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