adaptive overview talk

8/6/2019 Adaptive Overview Talk

http://slidepdf.com/reader/full/adaptive-overview-talk 1/22

An Overview of Optimal and Adaptive

Digital Signal Processing

Dr. Michael T. JohnsonMarquette University

Department of Electrical and Computer Engineering



Marquette University

Optimal filtering

Basic purpose of filtering: get rid of undesired noise, keep the desired signal.³Optimal´ filter: A filter whose parametersminimize a selected error criterion (MMSE)

OptimalFilter 7

+y

In put

d

e

u -

Error

Refere nce




Adaptive filtering

When signal or noise components arenon-stationary, the optimal filter isconstantly changing.³Adaptive´ filter: A filter which has time-varying parameters (like an FIR filter withweights that slowly change over time

Ad aptiveFilter 7

+yIn putd

e

u -

Error

Refere nce




Classes of adaptive filter problems

System Identification

Inverse Modeling

u=filter input, d=desired signal, e=error, y=output

Unkn ownSystem

Ad aptiveFilter

7-

+

y

In put d

e

u

Dela y

Ad aptive

Filter 7 -

+

y

In put d

e

uUnkn own

System




Prediction

Noise suppression

Ad aptiveFilter 7

+yIn put

d

e

uDela y-

Pred

iction error Pred icted signal

Ad aptiveFilter 7+yIn put

d

e

u -

Output

Refere nce




Room acoustic identification




Echo cancellation




Noise cancellation




Channel equalization




Active Noise Control




Narrowband supression




Notation

u(n)=[u[n] u[n-1] « u[n-M+1] ] T , the input window vector r(m) = R XX[m]=E{u(n) u(n-m)}, the m th autocorrelation coefficientr =[r *(1) r *(2) « r *(M)]T , the autocorrelation vector of the inputR = E{u(n) u(n) H} = the autocorrelation matrix of the input

(0) (1) ( 1)

( 1) (0)

( 1) (0)

r r r M

r r R

r M r

« »¬ ¼¬ ¼! ¬ ¼¬ ¼ ½

L

M O

OptimalFilter 7

+yIn put

d

e

u -

Error

Refere nce




Optimality: Wiener filters

_ a*

1 12 * *

0 0

arg mi n [ ] [ ] [ ] [ ] [ ]M M

o i i

i i

J E e n E d n w u n i d n w u n i

! !

¨ ¸® ¾¨ ¸¨ ¸± ±© ¹! ! ! ¯ ¿© ¹© ¹© ¹ª º ª º± ±° Àª º§ §

ww

10 !w R p

where w 0 is the vector of M filter weightsR is the correlation matrixp is the cross-correlation vector between u and d

OptimalFilter 7

+yIn put

d

e

u -

Error

Refere nce




Basic Adaptation Theory

Can analyze error of optimal filter

And use concept of gradient descent

To determine a theoretical update formula

_ a*

1 12 * *

0 0

2

[ ] [ ] [ ] [ ] [ ] M M

i i

i i

d

J e n d n u n i d n u n i

W

! !

® ¾¨ ¸¨ ¸! ! ¯ ¿© ¹© ¹ª º ª º À

!

§ §H H Hw p p w w Rw

_ a

12

*

( 1) ( ) ( ( ))

( ) [ ] [ ]

( ) ( )

n n J n

n E n e n

n n

Q

Q

Q

!

!

!

w w w

w u

w p Rw

1( 1) ( ) ( )

2n n n Q ! w w g




L east Mean Square ( L MS)

A pp roxim ate p =E {u (n )d* (n )} and R=

E {u (n )u* (n )} us ing o ne wi nd ow of the s ignal:

To g iv e the LMS update r ule:

*

Ö

( ) ( ) ( )Ö ( ) ( ) ( )

H n n n

n n d n

!!

R u u

p u

*

*

Ö Ö( 1) ( ) ( ) ( ) ( ) ( )Ö ( ) ( ) ( )

H n n n d n n n

n n e n

Q

Q

!

! w w u u w

w u




Update process

Step-by-step:1. Filter output2. Error signal3. Weight-update

Ö( ) ( ) ( ) H y n n n! w u

( ) ( ) ( )e n d n y n! *Ö Ö( 1) ( ) ( ) ( )n n n e n Q!w w u

z - 1I

I

u (n) u H(n)

7 w [n]w [n+ 1]

7Q -

d *

(n)

u(n)




Analysis of adaptive filtersL

MS converges to optimal Wiener solution plusrandom component that acts as Brownian motionCan control learning rate and window size M

Stability

Maximum learning rate a function of dataConvergence

Convergence speed is directly proportional to

MisadjustmentRandom misadjustment is directly proportional to

and to M

Must balance convergence and misadjustment




Adaptive equalizer example




As function of data characteristics




As function of learning rate




L MS variants

Many variants to L MSNormalized L MS (N L MS)Regularized L MS ( -N L MS)L eaky L MS

Sign-error L MSFX-L MS (compensation for active noise control)Affine Projection Algorithm (APA)Block L MS in frequency domain

DCT- L MSSubband- L MSRecursive L east Squares (R L S)




Example results

adaptive overview talk

Documents