Martijn v/d Horst [email protected]

April 24, 2023 Martijn v/d Horst, [email protected]/e Computer Science, System Architecture and Networking

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Martijn v/d [email protected]

Parallel Implementation of IIR Filters


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Outline• IIR Filters• Implementation Methods

– Look Ahead– Block-State– Incremental Block-State– Extra buffer

• Comparison• All-pass Filters• Conclusion and Future Work


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IIR Filters

InputOutput


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Describing Filters

• Transfer Function:

22

11

22

110

1)(

zbzbzazaazH

• Difference Equation:)2()1()2()1()()( 21210 nybnybnuanuanuany

)()()(

)()()1(

ndunxcny

nbunAxnxT

0202101

21

01

01adbaabaac

bbb

A

T

• State space form:


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ImplementationWe 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

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

nn-2 n-1 nn-P-1 n-P

P


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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-2 n-1 nn-2P n-P

PP


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Block-State

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

)()()(

)()()1(

ndunxcny

nbunAxnxT

)()()(

)()())1((

nPuDnPxCnPy

nPuBnPxAPnx

PPPP

PPP

jibAcjidji

D

Ac

Acc

C

bbAbAB

jiTijP

PT

T

T

P

PPP

11

21

0

...

...


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Block-State Architecture

PA PC

PD

PB

Input

OutputState

P

P

P

P

N N


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State update


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Block-State

PA PC

PD

PB

P

P

P

P

N N


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I

Incremental Block-State

PA IC

ID

PB

I

P

N N

I

IA

IB

I

IC

ID

I

I

IA

IB

I

IC

ID

I

I

IA

IB

JI

JIC

JID

JI

I

I

I

JI

N NN


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• 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


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ComparisonEfficiency: 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

2040

6080

100

P20

40

60

80

100

N0

0.250.5

0.751

2040

6080

100

P

2040

6080

100

P20

40

60

80

100

N0

0.250.5

0.751

2040

6080

100

P

2040

6080

100

P20

40

60

80

100

N0

0.250.5

0.751

2040

6080

100

P

2040

6080

100

P20

40

60

80

100

N0

0.250.5

0.751

2040

6080

100

P

ScatteredLook ahead

Block-state

IncrementalBlock-state

IncrementalBlock-statewith extrabuffer


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20 40 60 80 100

0.2

0.4

0.6

0.8

1

20 40 60 80 100

0.2

0.4

0.6

0.8

1

20 40 60 80 100

0.2

0.4

0.6

0.8

1

20 40 60 80 100

0.2

0.4

0.6

0.8

1

EfficienciesScatteredLook ahead

Block-state

IncrementalBlock-state

IncrementalBlock-statewith extrabuffer

N=8 N=16

N=32 N=64


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All-pass Filters

22

11

21*1

*2

1)(

zczczzcczH

• Also called phase shifters

• Theoretical optimum is P N


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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


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Future Work• Finding more efficient all-pass

implementations• Adaptive Equalizers• Other signal processing algorithms


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Questions?


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Farewell Thang

Martijn v/d Horst [email protected]

Documents

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