1. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 1 Martijn v/d Horst M.G.v.d.Horst@tue.nl Block Implementations of Recursive Filters

2. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 2 Outline Implementation Methods – Look Ahead – Block-State – Incremental Output Computation Comparison All-pass Filters Conclusion

3. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 3 Describing Filters Transfer Function: Difference Equation: State space form:

4. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 4 Implementation We want: Sample rates exceeding processing rates This means parallel inputs and outputs, also called block implementations Implementations which scale well

5. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 5 Clustered Look Ahead Increase the size of the recursive loop The order of the filter increases Might become unstable nn-2n-1n n-P-1 n-P P

6. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 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 nn-2n-1nn-2Pn-P PP

7. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 7 Block-State The state space form can be rewritten into a state space form using input and output vectors:

8. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 8 Block-State Architecture Input Output State

9. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 9 State update

10. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 10 Block-State

11. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 11 Incremental Block-State

12. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 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. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 13 Efficiencies Scattered Look ahead with Incremental Output Computation Block-state Incremental Block-state Clustered Look ahead with Incremental Output Computation

14. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 14 Efficiencies Scattered Look ahead with Incremental Output Computation Block-state Incremental Block-state N=8 N=16 N=32N=64 Clustered Look ahead with Incremental Output Computation

15. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 15 All-pass Filters Also called phase shifters Theoretical optimum is P N

16. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 16 Efficiencies w.r.t. IIR Scattered Look ahead with Incremental Output Computation Block-state Incremental Block-state Clustered Look ahead with Incremental Output Computation

17. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 17 Efficiencies w.r.t. All-pass Scattered Look ahead with Incremental Output Computation Block-state Incremental Block-state Clustered Look ahead with Incremental Output Computation

18. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 18 Efficiencies adapted Block-state for All-pass Incremental Block-state for All-pass Clustered Look ahead for All-pass

19. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 19 Efficiencies Clustered Look ahead with Incremental Output Computation Incremental Block-state Incremental Block-state for All-pass N=8 N=16 N=32N=64 Clustered Look ahead for All-pass

20. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 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. 25 July, 2014 Martijn v/d Horst, M.G.v.d.Horst@tue.nl TU/e Computer Science, System Architecture and Networking 21 Questions?