5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 1 Martijn v/d Horst Parallel Implementation of IIR Filters

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 2 Outline IIR Filters Implementation Methods – Look Ahead – Block-State – Incremental Block-State – Extra buffer Comparison All-pass Filters Conclusion and Future Work

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 3 IIR Filters Input Output

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 4 Describing Filters Transfer Function: Difference Equation: State space form:

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 5 Implementation We want: Sample rates exceeding processing rates This means parallel inputs and outputs, also called block implementations Implementations which scale well

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 6 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

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 7 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

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 8 Block-State The state space form can be rewritten into a state space form using input and output vectors:

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 9 Block-State Architecture Input Output State

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 10 State update

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 11 Block-State

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 12 Incremental Block-State

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 13 Transform a Mealey into a Moore Machine Filter order increases by one P - 1 multipliers saved for Block-State P - P div I multipliers saved for Incremental Block-State Extra buffer State Output State Output

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 14 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.

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 15 Efficiencies Scattered Look ahead Block-state Incremental Block-state Incremental Block-state with extra buffer

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 16 Efficiencies Scattered Look ahead Block-state Incremental Block-state Incremental Block-state with extra buffer N=8N=16 N=32N=64

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 17 All-pass Filters Also called phase shifters Theoretical optimum is P N

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 18 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

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 19 Future Work Finding more efficient all-pass implementations Adaptive Equalizers Other signal processing algorithms

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 20 Questions?

5 August, 2014 Martijn v/d Horst, TU/e Computer Science, System Architecture and Networking 21 Farewell Thang