Pipelining and Parallel Processing in

IIR Digital Filters•

Juo Yu Lee

Department of Electrical Engineering, Stanford University

Outline

• Introduction• Pipelining in 1st-Order IIR Digital Filters• Pipelining in Higher-Order IIR Digital Filters• Parallel Processing for IIR Filters• Combined Pipelining and Parallel

Processing for IIR Filters

Introductionconsider a 1st-order LTI IIR Filter

)()()1( nunayny +=+

iteration period: Tm + Ta

Tm: multiplication time

Ta: addition time

Interleaved Time Series• a naïve try:

inefficient interleaving and slow sample rate

Time(n)

0 1 2 3 4 5 6 7 8 9 10

Statex(n)

y(1)(0) y(2)(0) y(3)(0) y(4)(0) y(5)(0) y(1)(1) y(2)(1) y(3)(1) y(4)(1) y(5)(1) y(1)(2)

Look-Ahead Computation

• more iterated recursion

iteration bound: 2 (Tm + Ta)/2

the same as the previous version

Look-Ahead Computation

• another equivalent recursion

iteration bound: (Tm + Ta)/2

Look-Ahead Computation

• general case: applying M-1 steps of look-ahead

iteration bound: (Tm + Ta)/Mlinear increase in complexity

Interleaved Time Series• pipelined interleaving M = 5

Time(n) 0 1 2 3 4 5 6 7 8 9

Statex(n) y(-4) y(-3) y(-2) y(-1) y(0) y(1) y(2) y(3) y(4) y(5)

Pipelining in 1st-Order IIR Digital Filters

• revisit the 1st-order IIR filter

)()()1( nunayny +=+

• Example: 3-stage pipelining

adding poles and zeros at

Look-Ahead Pipelining with Power-of-2 Decomposition

consider a 1st-order LTI IIR Filter

applying the decomposition technique

sets of transformation

logarithmic increase in hardware complexity

pipelinedIIR

( M = 8 )

originalIIR

pipelinedIIR

withDecomposition

( M = 8 )

Finite Precision Problems• pole position sensitivity to filter coefficients

more sensitive for small value of a

• inexact pole/zero cancellation

Look-Ahead Pipelining with General Decomposition

• the 1st-order IIR filter again

• 12-stage pipelined: 2x3x2 decomposition

pipelinedIIR

( M = 12 )

pipelinedIIR

withDecomposition

( M = 12 )

Look-Ahead Pipelining with General Decomposition

• 2x2x3 decomposition

• 3x2x2 decomposition

Pipelining in Higher-order IIR Digital Filters

• Clustered Look-Ahead Pipeliningconsider a 2nd-order IIR filter:

with poles at1/2 and 3/4

using 2-stage pipelining

Pipelining in Higher-order IIR Digital Filters

• Clustered Look-Ahead Pipelining

if using higher value of M (ex. M = 3): multiplying

linear increase hardware complexity

Instability Problems

pipelinedIIR

( M = 3 )

pipelinedIIR

( M = 2 )original

IIR

numerical method to find M for stability

Pipelining in Higher-order IIR Digital Filters

• Scattered Look-Ahead Pipeliningrevisit the 2nd-order IIR filter:

guaranteed stability if the original filter is stable

using decomposition to obtain area efficiency

Parallel Processing in IIR Filters

• consider a 1st – order IIR filter

1

1

1 −

−

−=

azzH(z) )()()1( nunayny +=+

)34()24()14()4()4()44( 234

+++++++=+

kukaukuakuakyaky

L = 4

A Straightforward Structure

L = 4

Hardware complexity : Multiply-add operation2L

Incremental Block Processing

L = 4

Hardware complexity : 12 −L multiply-add operation

Round-off Noise Robustness• pole movement

4az=az = V.S.

211a−

∝Round-off noise

for one pole IIR filter

Parallel Processing in IIR Filters

• consider a 2nd – order IIR filter

( )21

21

83

451

1−−

−

+−

+=

zz

zH(z)

)2()1(2)()(

)()2(83)1(

45)(

−+−+=

+−−−=

nunununf

nfnynyny

Round-off Noise Robustness• pole movement

33

43,

21

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=z

43,

21

=z V.S.

Combined Pipelining and ParallelProcessing For IIR Filters

• revisit the 1st – order IIR filterwith L = 4 and M = 3

)()()1( nunayny +=+

)3()123( 12 kyaky =+

)123()93()63()3()123()113()103()93()83()73()63()53()43()33()23()13(

113

2612

2

345

678

91011

++++++=

++++++

++++++

++++++

++++++

kfkfakfakyakukaukuakuakuakuakuakuakuakuakuakua

⇒== 1&4 NMQ

4 poles : 3333 ,,, jajaaa −−

⇒= 3LQ

pole distance :3a

The multiplication complexity :

Combined Pipelining and ParallelProcessing For IIR Filters

( )21

21

83

451

1−−

−

+−

+=

zz

zH(z)• revisit the 2nd – order IIR filter

loop update L = 3 and M = 2

( ) ( )( ) FkYAFkYAAFkYAkY +=+••=++•=+ )3()3()33()63( 212

( )TkykykY )13()3()3( +=where

66

43,

21

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛:2AThe eigenvalues of

3333

43,

43,

21,

21

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛⇒ Take the square root :

Case Analysis• Example: 4-th order Chebyshev low-pass filter

with M =4

)1)(1()1()( 2121

41

−−−−

−

+++++

=EzDzCzBz

zAzH

simple model:

fVCtotalPVthVk

eCchVT

••=

−••

=

2

2)(arg

result (if using V=5 and Vth=1): V’ = 2.38power ratio = 58.91%

Case Analysis• Example: 2nd order IIR filter with L =3

( )21

21

83

451

1−−

−

+−

+=

zz

zH(z)

result (if using V=5 and Vth=1): V’ = 2.3365power ratio = 29.116%