Pipelined and Parallel Recursive and Adaptive Filtersviplab.cs.nctu.edu.tw/course/VLSI_DSP2010_Fall/VLSIDSP... · 2011. 1. 4. · 2 type of digital filters for time invariant system:

VLSI Digital Signal Processing Systems

Pipelined and Parallel Recursive and Adaptive Filters

Lan-Da Van (范倫達), Ph. D.

Department of Computer Science

National Chiao Tung University

Taiwan, R.O.C.

Fall, 2010

[email protected]

http://www.cs.nctu.tw/~ldvan/


Lan-Da Van VLSI-DSP-10-2

Outline

Introduction

Pipeline Interleaving in Digital Filter

Parallel Processing for IIR Filter

Combined Pipelining and Parallel Processing for IIR

Filters

Low-Power IIR Filter Design Using Pipelining and

Parallel Processing

Pipelined Adaptive Digital Filters

Conclusions



Introduction

2 type of digital filters for time invariant system:

FIR

IIR

IIR is preferred since it can be implement in a much

lower order.

Pipelining technique:

look-ahead computation and incremental block processing

techniques

relaxed look-ahead transformations for pipelining of LMS

and lattice adaptive filters



Outline

Introduction




Filters


Parallel Processing


Conclusions



Pipeline interleaving in Digital Filters

Inefficient single/multi-channel interleaving

Efficient single-channel interleaving

Efficient multi-channel interleaving



Inefficient Single/Multi-channel Interleaving

Consider

y(n+1)=ay(n)+bu(n)

Iteration period= am TT



Inefficient Single/Multi-channel Interleaving(cont.)

M-stage pipeline version:insert (M-1) additional

latched

Clock period decrease M times(ex:M=5)

For a single time series this array will be useful for

only 20% of the time

If 5 independent time series are available ==> fully

utilized

Consequence:

A sample rate M times slower than the clock rate

inefficient utilization of processing elements



Inefficient Single/Multi-channel Interleaving(cont.)



Efficient Single-Channel Interleaving

Using look-ahead transform.

Consider:

y(n+2)=a[ay(n)+bu(n)]+bu(n+1)

Iteration bound=2(Tm+Ta)/2



Efficient Single-channel Interleaving(cont.)

Another recursion equivalent equation:

y(n+2)=a2y(n)+abu(n)+bu(n+1)

Iteration period bound: (Tm+Ta)/2




(M-1)steps of look-ahead

Iteration bound: (Tm+Ta)/M

1

0

)1()()(M

i

iM iMnbuanyaMny




In a causal system, to perform as before:

y(1)=ay(0)+b(0)

IF M=5

y(1)=a5y(-4)+bu(0)

(u(-4),u(-3),…,u(-1)=0 for causality)

y(-4)=a-4y(0)

y(-i)=a-iy(0), i=1,2,….,(M-1)

The iteration bound can be achieved by retiming or cutset transform



Efficient Multi-channel Interleaving



Outline

Introduction




Filters


Parallel Processing


Conclusions



Example 10.5.1 (1/3)

Consider the following transfer function

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

4

)3()2()1()()()4(

)1()()(

4)ure(Larchitecht parallel-4consider , )()()1(

1)(

1234

1234

1

0

1

1

kukuakuakuakyaky

knngsubstituti

nunuanuanuanyany

iMnbuanyaMnyfrom

nunayny

az

zzH

M

i

iM



Example 10.5.1 (2/3)

pole: z=a ==> z=a4

since |a|



u(4k+2)

u(4k+1)

u(4k)

Next:u(4k+7),u(4k+8),u(4k+9),u(4k+10)



Parallel Processing for IIR Filters

Need L2 multiply-add operations: hardware cost is

high

use incremental block processing to simplify

hardware:

use y(4k) to compute y(4k+1)

use y(4k+1) to compute y(4k+2)

use y(4k+2) to compute y(4k+3)



Example 10.5.2 (1/2)

incremental block processing

since y(n+1)=ay(n)+u(n)

y(4k+1)=ay(4k)+u(4k)

y(4k+2)=ay(4k+1)+u(4k+1)

y(4k+3)=ay(4k+2)+u(4k+2)



Example 10.5.3 (1/3)

System Transfer Function:

L=3



Example 10.5.3 (2/3)



y(3k+4)

y(3k+3)

Example 10.5.3 (3/3)



Parallel Processing for IIR Filters

For N-th order IIR filter, its L-level incremental parallel

processing architecture can be obtained by

computing the first N output samples

y(Lk),y(Lk+1),…,y(LK+N-1) independently using loop

update equations obtained by clustered look-ahead

technique and then computing the remaining L-N

samples y(Lk+N)…,y(LK+L-1) incrementally using the

previous N output samples



Outline

Introduction



Combined Pipelining and Parallel Processing for

IIR Filters


Parallel Processing


Conclusions



Example 10.6.1 (1/3)

Consider: M=4,L=3

11

1)(

azzH



Example 10.6.1 (2/3)



f1(3k+9) f2(3k+6)



Outline

Introduction




Filters


Parallel Processing


Conclusions



Example 10.7.1 (1/2)

Consider: V0=5V,Vt=1V

)5175.01337.11)(1777.04085.01()(

)7194.05524.01)(4216.01188.11(

)8482.04996.11)(6493.05548.11(

)1(001836.0)(

:iondecomposit 2-of-power with techniqueahead-look scatteredby

)8482.04996.11)(6493.05548.11(

)1(001836.0)(

8484

4242

2221

41

2121

41

zzzzzD

zzZz

zzZz

zzN

zzzz

zzH



5891.05

38.2

5

13

13 where,38.238.2

5 where, 55

38.2,476.0,

)15(

5

4

1where,tagesupply vollower use can

ecapacitancsmaller path creticalshorter :pipeline level-4

)15(

5

)(

2

22)(

22)(

0

2

arg

0

arg

0

0

seq

pip

pipsMpip

pip

pip

totalpar

seqsMseq

seq

totalseq

pip

ech

pip

ech

t

total

P

PRatio

mfCmT

CP

mfCmT

CP

VVTT

k

CT

V

k

C

VVk

VCT

Example 10.7.1 (2/2)



Consider L=3

%116.293

4

43

)(12

3

3365.2,4673.03 since

)(

)(

delay npropogatio

6.0,5

)2()1(2)()2(8

3)1(

4

5)(

2

222

2

2

2

seq

par

soMs

oMpar

soMseq

oseqpar

to

oMpar

to

oMseq

to

P

PRatio

fVCf

VCP

fVCP

VVTT

VVk

VCT

VVk

VCT

VVVV

nunununynyny

Example 10.7.2 (1/2)



Example 10.7.2 (2/2)



Outline

Introduction




Filters


Parallel Processing


Conclusions



Introduction to an Adaptive Algorithm

Adaptive filter: filter with adaptive coefficient

general filter block

coefficient update block

Widely used in communication, DSP, and control system

Deterministic gradient / least square algorithm

Steepest descent algorithm

RLS algorithm

Stochastic gradient algorithm

LMS algorithm, DLMS algorithm

Block LMS algorithm

Gradient Lattice algorithm

Difficult to pipeline due to the presence of long feedback loops

Relaxed look-ahead transformation technique is used to pipeline

the adaptive filter with little or no increase in hardware at the

expense of marginal degradation in the adaptive behavior



Adaptive Applications

Channel equalizer

System identification

Image enhancement

Echo canceller

Noise cancellation

Predictor

Line enhancement

Beamformer

MIMO-ODFM: V-BLAST



Notation

)Error: e(n

Factor: μAdaptation

tor: W(n)Weight Vec

tput: d(n)Desired Ou

al: X(n)Input Sign

.5

.4

.3

.2

.1

: .10

.9

.8

.7

.6

MatrixDiagonal

:λEigenvalue

ix: Ration MatrAutocorrel

: NTap Number

ent: MMisadjustm adj



Steepest Descent Algorithm

)()()( nXnWny T

TNnxnxnxnXwhere )]1( ... )1( )([)(

TN nwnwnwnW )]( ... )( )([)( 110

Note:

SSS (Strict-sense stationary): A stochastic process x(t) is called

SSS if its statistical properties are invariant to a shift of the origin.

WSS (Wide-sense stationary):

)()}()({

)}({

*

RtxtxE

txE



Steepest Descent Algorithm (1/8)

)()()(

)()()(

)()()(

nWnXnd

nXnWnd

nyndne

T

T

)()()(

)()()(2)()( 22

nWnXnXW

nWnXndndne

TT

T

The error at the n-th time is




)()()(2)}({

)()}()({)(

)()}()({2)}({

)}({ )(

2

2

2

nRWnWnWPndE

nWnXnXEnW

nWnXndEndE

neEnJ

TT

TT

T

)1()(

.

.

)1()(

)()(

)}()({

Nnxnd

nxnd

nxnd

EnXndEP




)0(...)2()1(

....

.....

......

)2(...)0()1(

)1(...)1()0(

)}()({

**

*

rLrLr

Lrrr

Lrrr

nXnXER T

)}()({ )( * nxnxEr




RWPJ 22

0J

PRWopt1

Take “gradient”

Let

Wiener-Hopf equation (1949) or Wiener Filter can

be obtained!!




opt

TWPndEJ )}({ 2min

)})(())({( min optT

opt WnWRWnWEJJ

optWnWnC )( )(

)}()({ min nRCnCEJJT




SignalInput for

Definitive Positive and Symmetric :Assume

RQ

diag

QQQQR

L

T

ofmatrix orthogonal are and

]... [ where 110

1

)()( where

}{

)}()({)(

1

min

1

min

nCQnV

VVEJ

nCQQnCEJnJ

T

T




)()(

))((2

1)()1(

RWPnW

nJnWnW

Each change in the weight vector proportional to

the negative of the gradient vector




opt

opt

WnWnC

PRW

nJnWnW

)()(

)(2

1)()1(

1

)0()-(IV(n)

0)()I()1(V

)()()1(

V

nVn

nVnVnV

n

kk allfor 1)1(0 max

10



LMS Algorithm

))(ˆ(2

1)()1( nJnWnW

)()(2)(ˆ nXnenJ

μe(n)X(n)W(n))W(n 1

w0’, w1, w0

An efficient implementation in software of steepest

descent using measured or estimated gradients

The gradient of the square of a single error sample



)()()( nyndne

Summary of LMS Adaptive Algorithm (1960)

)()()( nnny T xw

)()( )()1( nnenn xww



Block Diagram of an Adaptive FIR Filter Driven by the LMS Algorithm

1z)(nx 1z 1z

)(0 nw )(2 nw)(1 nw )(1 nwN

)1( nx )2( nx )1( Nnx

)(nd)(ny

)(ne



Adaptive Digital Filter Structure



Relaxed Look-Ahead (1/6)

Consider:

)1()1()]1([

)()1()(

)()()()1(

1

1

1

0

1

0

MnuiMnujMna

nyiMnaMny

nunynany

M

i

i

j

M

i




Product Relaxation

1

0

1

0

1

0

)1()()))1(1(1()(

)1()1(])1([

zero toclose is)( if

))1(1(1)1(1

))1(1()1()(

unity toclose is a(n) if

M

i

i

j

MMM

i

iMnunyMnaMMny

iMnuiMnujMna

inu

MnaMMnM

MnMnaina




Another approximation:




Sum relaxation

if u(n) varies slowly over M cycles then

if u(n) is close to zero the Mu(n) can be approximated by u(n)

1

0)()1(

M

inMuiMnu



Delay relaxation

M-level look-ahead pipelined version

assume that the product a(n)u(n) is more or less constant over

M’ samples




Pipelined LMS Adaptive Filter (1/3)

1'

0

112

221

1

0

112

1

0

2

2

2

2

2

)()()()(

:M'M relaxation sum usesamples,Mover

much changenot does e(n)U(n) estimategradient that theassume

)()()()(

:relaxationdelay

)()()()(

ahead-look stage-

)()()1()(

e(n) minimize to

)()1()()(ˆ)()(

M

i

M

i

M

i

T

iMnUiMneMnWnW

iMnUiMneMnWnW

inUineMnWnW

M

nUnenWnW

nUnWndndndne




)()()()(

)(by)1( replace and smallly sufficient is assume

)(])1()1(

)1([)(

)()1()()(ˆ)()(

2

22

1'

0

11

2

2

nUMnWndne

MnWMnW

nUiMnUiMne

MnWnd

nUnWndndndne

T

M

i

T

T



Conclusions

Introduce the pipeline filter design

Introduce the parallel filter design

Explore Relax-Look ahead techniques

Demonstrate the concept and design of adaptive

filters

Documents

Pipelined and Parallel Recursive and Adaptive Filtersviplab.cs.nctu.edu.tw/course/VLSI_DSP2010_Fall/VLSIDSP... · 2011. 1. 4. · 2 type of digital filters for time invariant system: