adaptive signal processing , ebook . ingle

8/11/2019 Adaptive Signal Processing , ebook . Ingle

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Adaptive Signal Processing Adaptiivne signaalittlus

Leon H. SibulKevadsemester, 2007


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Course Outline - ppekava

I Introduction Overview of applications and basic concepts

of adaptive signal processing.1. Brief overview of applicationsa. Linear prediction. b. Speech coding

c. Noise cancellationd. Echo cancellatione. Adaptive filteringf. System identificationg. Equalization and deconvolutionh. Adaptive beamforming and array processingi. Signal separation.


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2. Introduction to basic concepts of optimization and adaptive signal processing.

a. Optimization criteria.Mean square error Minimum variance.Maximum signal to noise ratioMaximum likelihood..Bit error.

b. Introduction to basic adaptive algorithms.Gradient search.The least mean-square (LMS) algorithm.

Stochastic approximation. Nonlinear algorithms.Linear algebra and orthogonal decomposition algorithms.

3. Matrix notation and basic linear algebra.


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II Theory of optimum and adaptive systems.

1. Review of discrete-time stochastic processes.2. Mean-square error3. Finite impulse response Wiener filters.4. Gradient decent algorithm.5. Stability, convergence and properties of error surfaces.6. Examples of applications.

III Basic adaptive algorithms and their properties.1. The least mean-square (LMS) algorithm.

a. Derivation of basic LMS algorithm.

b. Learning curve, time constants, misadjustment, and stability.c. Step size control.d. Variations of LMS algorithm.


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2. Recursive least-squares algorithm.3. Lattice algorithms.4. Linear algebra and orthogonal decomposition algorithms.5. Frequency domain algorithms.

IV Applications.1. Linear prediction and speech coding.2. Noise cancellation.4. Echo cancellation.5. Adaptive beamforming and array processing.

a. Linear adaptive arrays. b. Constrained adaptive arrays.

Minimum variance desired look constraint.Frost beamformer

c. Generalized sidelobe canceller.d. Robust adaptive arrays.


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Bibliography

1. Vary, P. and Martin, R., Digital Speech Transmission- Enhancement,Coding and Error Concealment, John Wiley & Sons, LTD., Chichester,England, 2006.

2. Schobben, D. W. E., Real Time Concepts in Acoustics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

3. Poularikas, A. D. and Ramadan, z. M., Adaptive Filter Primer with MATLAB, CRC, Taylor & Francis, Boca Raton, FL., USA, 2006.

4. Haykin, S., Adaptive Filter Theory, Third Ed., Prentice Hall, UpperSaddle River, NJ, USA, 1996.

5. Alexander, S.T., Adaptive Signal Processing, Theory and Applications,Springer-Verlag, New York, USA, 1986.

6. Widrow, B. and Sterns, S.D., Adaptive Signal Processing, Prentice-Hall,Englewood Cliffs, NJ, USA, 1985.


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7. Adaptive Signal Processing, Edited by L.H Sibul, IEEE Press, NewYork, USA 1987.

8. Manzingo R.A. and Miller, T.W., Introduction to Adaptive Arrays, John

Wiley-Interscience, New York, USA, 1980.9. Swanson C. D.,Signal Processing for Intelligent Sensors, Marcel

Dekker, New York, USA, 2000.10. Colub,G.H., and Van Loan, Matrix Computations, The Johns Hopkins

University Press, Baltimore, MD, USA, 1983.11. Tammeraid, Ivar, Lineaaaralgebra rakendused, TT Kirjastus, Tallinn,

Estonia, 1999. 2. Lineaaralgebra avutusmeetodid. 2.3 Singulaarlahutus.12. Van Trees, H.L.,Optimum Array Processing, Part IV of Detetection,

Estimation and Modulation Theory, Wiley-Interscience, New York,USA, 2002. Chapter 6 Optimum Waveform Estimation, Chapter 7-Adaptive Beamformers, A- Matrix Operations.


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13. Allen, B., and Ghavami, M., Adaptive Array Systems; Fundamentals and Applications, Wiley, Chichester, England, 2005.

14. Cichocki, A., and Amari, S-I, Adaptive Blind Signal and Image

Processing, Wiley,West Sussex, England, 2002.


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ppenuded ja hindamine:

1. Semestri t ja aruanne: Rakenduslesande

lahendus kasutades adaptiivset signaalittlust jaMATLABi. Teema valik oleneb pilase huvidest jaoskustest. 60% hindest.

2. Kodutd ja harjutused. 20% hindest Kodutd jaharjutused peavad olema sooritatud, etlppeksamile pseda.

3. Suuline lppeksam. Ksitab peamiselt semestri td ja phiteooriat. pilane vib kasutada kuni 20leheklge enda tehtud mrkmeid. 20% hindest.


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Semestrit ja aruande nuded.1. Sissejuhastus- lesande definitsioon, selle rakendus, selle

thtsus ja lhike aruande levaade.2. Teooria ja algoritmi tuletus.3. Kasutatud algoritm ja kuidas see lahendab kesoleva

rakenduslesande.

4. MATLABIi programm.5. Graafikud ja nende selgitus.6. Tulemuste anals, selgitused ja jreldused.

7. Kokkuvte.8. Kirjandus.

Mrkused: Word, Powerpoint, PDF, umbes 10 kuni 15 leheklge, eesti vi inglisekeeles.


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Basic Concepts of Adaptive Signal Processing.

Applications.

Optimization Criterion or Performance Measures.

Adaptive or Learning Algorithms .

Improved System Performance.

Noise reduction, echo cancellation,..

Performance Measures.

Signal andnoiseenvironment


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Linear Prediction Filter of Order n.

T T T T

1a

1

n

i=

2a 3a na

d(k ) x(k )

( ) x k

( ) { }

{ } ( ) ( ){ }

1

22

( ) Prediction error ( ) ( ) ( ) .

Adaptive algorithms minimize mean square prediction error:

( )

n

ii

x k a x k i d k x k x k

E d k E x k x k

=

= =

=

V Vary and Martin, 2006, Ch. 6, Haykin,1996, Ch. 6.


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Optimum Linear Prediction.

{ } ( )( )

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

{ } ( ){ }

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

22

2

2 22

2

1

Minimize mean square error: ( ) ( ) .

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

( )2 0 Minimum.

=

n

ii

E d k E x k x k

E d k d k E d k E d k x k n

a a

E d k E x k

a

E d k x k E x k a x k i x k

E

=

=

= = = =

=

=

( ) ( ){ } ( ) ( )1

1

{ }

( ) ( ) 0

n

ii

n

xx xxi

x k x k a E x k i x k

R R i

=

=

= =


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Linear Prediction and Speech Coding.

0

Noisegenerator

Variable

Filter

Impulsegenerator

( ), ( )h k H z

Filter parameters

Discrete time speech production model.

( )u k ( )v k ( ) x k

g

S

0 : pitch period

: voiced/unvoiced : gain( ) : impulse response( ) : excitation signal

( ) :speech signal

N

S g

h k

v k

x k i

1

Autoregressive (AR) model for speech.1( )1 ( )

C(z)= - cm

i

i

H z C z

z

=

=


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Example of application of Linear Predictor to SpeechCoding.

Transmitter Channel Receiver

-+

++++

( )k a ( )k a

Linear Prediction Filter Coefficients.

Adaptive Analysis Filter Speech Synthesis

x(k)

( ) x k d(k) y(k)

Vary and Martin, 2006, Ch. 8.


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Model-Based Speech Coding.

Speech production LP encoder Channel LP Decoder Model

1 ( )1 ( )

V z C z

( ) z 1 ( ) z

( ) D z 11 ( ) z

( )Y z

( ) z

1 ( )( ) ( ) if ( ) ( ) then ( ) ( ) (excitation)1 ( )

1 1Synthesis filter is: ( ) , ( ) ( ) ( ) ( ) ( ).1 ( ) 1 ( )

Bit rate of encoded speech: 2 bit/sample,

sampling frequenc s

s

A z D z V z A z C z D z V z

C z

H z Y z H z D z V z X z A z C z

Bw

f

f

= = =

= = =

=

y=8 kHz, transmision rate in bits/sec. BVary and Martin,2006, Ch. 8.


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Adaptive Noise Canceller.

Primary inputSignal

Source s(k)

NoiseSourcen(k)

+

AdaptiveFilter

LMS algorithm

Noise Reference

Noise estimate( )n k

Output s(k)+n(k)

n(k)

Auxiliary noise sensor obtains signal free noise sample.

Widrow and Sterns, 1985.


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Echo Cancellation for Hands-free Telephone Systems .

Local speaker M

LS

A/D

D/A

( ) distant speaker echo x t

s(t)

n(t)

Adaptive algorithm .

Signal from distant speaker.

( ) ( ) ( ) ( ) y k s k n k x k = + +

( ) x k

( ) x k

-( ) s k

[ ] ( ) ( ) ( ) ( ) ( ) s k s k n k x k x k = + +

( ) y k +

Vary and Martin, 2006, Ch. 13


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System Identification and Modeling.

Excitation signal

Plant or unknownsystem

Adaptive processor

x(k) d(k)

e(k)

+- y(k)

x(k) must be the persistent excitation.


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System Identification.

Clark, G., JASA 2007


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Blind Equalization

Channel h(n)

Unobserved datasequence

Blind equalizer x(n)

v(n)noise

u(n)

Minimize intersymbol interference in unknown multipath channels.

x^(n)

+


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Bussgang Algorithm for Blind Equalization

Transversalfilter

{w^(n)}

Transversalfilter

{w^(n)}

Zero-memorynonlinear estimator

g(.)

Zero-memorynonlinear estimator

g(.)

LMSAlgorithm

LMSAlgorithm

Receivedsignalu(n) y(n)

x^(n)

+-

e(n)Error

(Haykin,1996)


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Finite Impulse Response (FIR) Wiener Filters.

( ) s k

( )n k

( ) x k ( ), ( )h k H z

( ) s k

( ) ( )d k s k =( )k

( ){ } ( ){ }0

22

( ) ( ) ( )

( ) ( )

N

l

s k h l x k l

E k E s k s k

=

=

=


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Finite Impulse Response (FIR) Wiener Filters.

( ){ }( ) ( )

( )( ) ( ) ( )

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

( ) ( ) ( )

[ ]

2

1

0

( )

by WSS assumption 0 0

in matrix notation where ( )

N

l

N

xx sxl

il xx

E k s k s k E s k E s k h i h i h i

E h l x k l x k i E s k x k i

h l R i l R i i N

R R i l

=

=

=

=

= =

= =

=

sx xx

1xx sx

R R h

h R R


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MATLAB Example of System Identification.

varx=100; x=sqrt(varx)*randn(1,20); >> v=randn(1,20); >> r=xcorr(x,1,'biased')

r =

10.5420 80.7933 10.5420

>> rx=[80.7933 10.5420]; >> Rx=toeplitz(rx)

Rx =

80.7933 10.5420 10.5420 80.7933

>> y=filter([1 0.38],1,x);

>> dn=y+v; >> pdx=xcorr(x,dn,'biased'); >> p=pdx(1,19:20)

p =

39.0133 83.4911

>> w=inv(Rx)*p'

w =

0.3541 0.9872


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Frequency Domain Wiener Filter.

( ) ( ) ( )

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

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

( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( )

( )

22

2

( )

XX XS SX SS

SX SX XX

XX XX

SX SS

XX

h k x k H X

E E S S

H E X X H H E X S

H E X S E S S

H P H H P H P P

P P H P H

P P P

P P

=

=

+

= +

=

+


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The Mean Square Error (MSE) Performance Criterion.

{ } { }{ }

{ }

{ }

{ }

2 2

2 2( )

1

( )

2

Error signal: ( ) ( ) ( )

Squared error: ( ) ( ) 2 ( ) ( ) ( ) ( )MSE: ( ) ( ) 2

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

d t

d t i

N

t d t t

t d t d t t t t E t E d t

E x t d t

E x t d t E t t

E x t d t

E

=

= += +

= =

T

T T T

T Tx xx

Tx xx

w

w x

w x w x x ww R w R w

R R x x

{ } ( ) ( )( )

( ) 2 2 0 Wiener-Hopf equation :

"Wiener solution":d t opt d t

opt d t

t

= + = =

=x xx xx x

1xx x

R R w R w R

w R R


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Minimum Variance (MV) Optimization.

Array Steering Adaptive Weights

1

2

1w

2w

N w

1 x 1 z

2 x 2 z

N x N z { }

( ) ( )( ) exp ( )i i i

y t t z t j x t

==

T

w z

( )t x ( )t z w Adaptive array steered to the signal direction.


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Minimum Variance (MV) Optimization

[ ] [ ] ( )

( )

Array input: ( ) ( ) ( ) ( ) ( )Signal direction vector:

1, exp , exp 2 , ,exp 1

2 sin , sensor distance between linear array elements,

wavelength.

Beam steering matri

t t t s t t

j j j N

d d

= + = +

=

=

T

x s n d n

d

[ ]

1

k 2

N-1

1 0 0 00

x: = exp .0 00 0

jk

=


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Sidelobe Cancellation (SLC) System.

Main channelMain beam

Auxiliary sensors1w

N w

+

Adaptive weightAdjustment.*

- -

*Minimize cross-correlation between main and auxiliary channels.


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Generalized Sidelobe Canceller.

Sensor Data

B

daptiveaw

Fixed Beamformer cw

B Blocking matrix.


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Example of Blocking Matrix for Sidelobe Canceller.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

2

4

6

8

10

12

14

16

Main beam

Auxiliary beam from

blocking matrix.

For blocking of signal from desired look direction blocking matrix must beorthogonal to "desired look" steering vector .

10 1 1 1 1 1 Example:

10

1

= = = =

H1

H H

HN

B

d

b d

B d 0 d B

b

1 1 1 1


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Blind Source Separation

IndependentSources

Array ofSensors

Uncorrelated Normalized Data

SeparatedSources

MixingMatrix

A

Linear Preprocessor

T

SourceSeparation

W . .

.

.

.

.

.

.

.

.

.

.

Nonlinear Adaptive

Algorithm

PCA ICA

( )t1s( )t2s

( )ts( )ts ( )tx( )t1x( )t2x

( )t1v( )t2v

1u

2u

u

( )tr s

WMr

( )tMx ( )tr vr

u


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Beamforming & Source Separation

SVD or ULVD

1r

H

r U

W

Subspace Filter Source Separation

Subspace

Estimation

Eigenstructure

&ParameterEstimation

TA Estimates

1s2s

3s

Estimated SourceSignals

sX

M21 ,..,


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Optimization Criteria and Basic Algorithm Minimize or maximize a scalar performance measure ( ) Basic Adaptive Algorithm:( 1) ( ) ( ) ( )( ) search direction

( ) step sizeExamples:Steepest decent - ( ) ( )LMS - estimated gradient

J

k k k k

k

k

k J k

+ = +

=

w

w w d

d

d

Stochasic Approximation

Newton's and Quasi-Newton


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Common Adaptive Algorithms.

( )( )

( ) ( )

Steepest Decent:

( 1) ( ) -

Least-Mean Squares (LMS) algorithm:( 1) ( ) 2

Estimation and Direct Matrix Inversion (DMI).Recursive Least- Squares (RLS).Affine Projection.

k k J k

k k k k

+ = +

+ = +

Ww w w

w w x


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Error Performance Surface.

020

4060

80100

0

50

100440

450

460

470

480

490


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Derivation of the LMS Algorithm .

2

0 0

2

( ) ( ) ( ) ( )

LMS algorithm assumes that performance measure is ( ) ( ).

( ) ( )

( ) 2 ( ) 2 ( ) ( ).( )( )

LMS weight adjustment alg L L

k d k k k

k

k k w w

J k k k

k k ww

=

=

= = =

T

2

w

x w

w

w x

[ ][ ]

0

orithm is:( 1) ( ) ( )

( ) 2 ( ) ( ) step size.( ) ( ), , ( ) filter weights at time .

( ) ( ), ( 1), , ( ) input data.

T

L

T

k k J

k k k k w k w k k

k x k x k x k L

+ =

= +=

=

ww w w

w xw

x


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The LMS Algorithm for M-th Order Adaptive

Filter.

Inputs: M=filter length =step-size factor ( ) input data to the adaptive filter (0) int alize the

n

==

x

w weight vector=0

Outputs: ( ) adaptive filter output= ( ) ( ) ( ) ( ) ( ) ( ) error Algorithm: ( 1) ( ) 2 ( ) ( )

n n n d n

n d n y n

n n n n

= == =+ = +

Tw x

w w x


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LMS Function.function[w,y,e,J]=lms(x,dn,mu,M) N=length(x);y=zeros(1,N);w=zeros(1,M);for n=M:N

x1=x(n:-1:n-M+1);%for each n% the vector x1 of length M with produced from x

%with elements in reverse order.y(n)=w*x1';e(n)=dn(n)-y(n);w=w+2*mu*e(n)*x1;w1(n-M+1,:)=w(1,:);

end;J=e.^2;


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Convergence of the Mean Weight Vector of LMS

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )

( )

( )( ) [ ] [ ]

{ }

2

1

1 2 0

2 2 1 2 0

1 2 0

1 2 0 00

1 20

0 0 0 1 2

1lim 0 if 1 1 2 1 0

lim ( /

k

k

k

m

k

L

MAX MAX k MAX

OPT k

k

k tr tr

E k

=

= =

+ =

=

= < < < < =

=

xx

xx xx

xx

xx

h I h

h I h I h

h I h

h R

w w


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Convergence Rate of the LMS Algorithm.

( )

( )

2

maxmax

max min

2

LMS weight convergence is geometric with geometric ratiofor coordinate:

1 1 11 2 exp 12!

For large

Conditio

:1

n number) o

11

f

2 12

1 (

p

p p p p p

p

p p p p p

r

p th

r

r

= +

=

xxR


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Misadjustment Due to Gradient Noise

Estimated gradient: ( ) 2 ( ) ( ) ( ) ( )( ) true gradient( ) zero-mean gradient estimation noise

At minimum mean-square error (mse) point ( ) =0 and ( ) 2 ( ) ( ) ( )

Gradient noise

k k k k k

k

k

k

k k k k

= = +

= =

x n

n

x n

{ } { } { } { }2 2min

covariance:( ) ( ) 4 ( ) ( ) ( ) 4 ( ) ( ) ( )

4 ( )

( ) and ( ) ar

k k E k k k E k E k k

k

k k

= =

=

H H H

xx

n n x x x x

R

x e uncorrelated


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( ) ( )( ) ( )

( ) ( ){ }

LMS a lgorithm with noisy gradient:( 1) ( ) ( ) ( ) ( ) ( )

1 ( ) 2 ( ) ( )

Transforming by :( 1) 2 ( ) ( ) ( )

At close to optimum ( ) 0 (learning transients hav

k k k k k k

k k k k

k k k k k

E k

+ = + = + +

+ = + +

+ = + =

=

xx

T

T

w w w n

v v R v n

U

h I h n n U n

h

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

{ } { }

2

e died out)

Using the fact that ( ) ( ) =0 covariance of ( ) is:( 1) ( 1) 2 ( ) ( ) 2 ( ) ( )

Close to optimum value ( ) are wide-sense stationary

( 1) ( 1) ( ) ( )

E k k k

k k E k k E k k

k

E k k E k k

+ + = +

+ + =

T

T T T

T T

h n h

h h I h h I n n

h

h h h h

{ } ( ) { } ( ){ } [ ]1 mi

2

n

2min( ) ( ) 2

( ) )

( ( )

(

) 4 E k k E

E k

k

k

k

= +

= = T

T T

hh

h h I

h I

h h

R h


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{ } ( ){ }21

min1

1min

Excess MSE= ( ) ( )

Average excess MSE= ( ) ( ) ( )

=1

average excess MSEM=misadjustment =1

If 1 for all (usual cas

N

p p p

N p

p p

N p

p p

p p

k k

E k k E h k

=

=

=

=

=

T

T

h h

h h

[ ]

pmse

1

1

1 1 14

e), th

long filter large misadjustment.

fast convergence large m

en

M=

isadjustment.

1,

1 1

2F

M4

or

N

p p

p

p pmse p

pmse

N

p

tr

=

=

= =

=

=

xxR


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Sensitivity of Square Linear Systems.

Results from numerical analysis.

( )

( )

( )( )

1Let be a nonsingular matrix and:

The solution to approximates the solution of with error estimate:

1

where denotes norms and is the

=

<

<


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Leaky LMS Algorithm.Wiener optimum weight calculation requiresinverting possibly an ill-conditiond matrix:

this causes numerical errors and slow convergence.if the mode (1 2 ) does not converge.

12

opt d

p p

p p

=

1xx xw R R

slow convergence for small .

Leaky LMS algorithm:

( 1) (1 2 ) ( ) 2 ( ) ( ) 0.use ( ) ( ) ( ) ( ) we have:

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

p

k k k k k d k k k

k k k k d k k

+ = + >=

+ = + +

T

T

w w xx w

w I x x I w x


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Block LMS Algorithm[ ]input signal ( ) ( ), ( 1), , ( 1

block 1

0 B 2B 3B B time samples

n x n x n x n M

k

= +x

[ ]0 1 11

1

, , # ,

, 0,1, , 1 0,1,

( ) ( ), ( ), ,( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) (

M

M

n

k block time n sample time B samples in a block M FI length

sample time n kB i i M k

filter weights k w k w k w

output y n y kB i k kB i w k x kB i

error n y n d n

=

= + = =

== + = + = +

=

T

T

w

w x

) ( ) ( ) ( )kB i d kB i y kB i + = + +


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Block LMS Algorithm

1

0

1

0

:( 1) ( ) ( ) ( )

1( 1) ( ) ( )22( ) ( ) ( )

B

i

B AVE

B

AVE Bi

In block LMS error signal is averaged over blocks B

k k kB i kB i

or k k k

k kB i kB i B B

=

=

+ = + + +

+ =

= + + =

w w x

w w

x


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Fast FFT Based LMS Algorithm. Fast linear convolution

Fast linear correlation

[ ]

tap weights of FIR filter are padded with zeros. point FFT 2

( )Frequency domain weight vector ( ) 1 null vector.

( ) ( ), , ( 1), ( ), , ( 1)

M

N N M

k k FFT M

k FFT kM M kM kM kM M

=

=

= +

wW 0

0X x x x x

{ }{ }[ ]

( 1)

( ) ( ), ( 1), , ( 1)

( )

.

k th block k th block

k y kM y kM y kM M

last M elements of IFFT k

element by element multiplication of matrices

= + +

=

Ty

X W


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Fast FFT Based LMS Algorithm. II

[ ]

[ ][ ]

(0) 0 00 (1)

Define: ( ) ( ) 00 0 ( 1)

( ) ( ), ( 1), , ( 1)

( ) ( )( ) ( ), ( 1), , ( 1)

( ) ( )( )

k

k

k

X

X

k diag k X M

k y kM y kM y kM M

last M elements of IFFT k k k kM kM kM M

k FFT k first M elements of IFFT

k

= =

= + +

== + +

= =

T

T

U X

y

U We

0E

e

( ) ( )

( )( 1) ( )

k k

k k k

+ = +

HU E

W W0


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Recursive Least-Squares Algorithm.1

1 2 1

1 1

1 1 1

11 1 1

samples, 1 filter coefficients.

n n n M

n n n M

n N n N n M N

n n n n M

x x x

x x x N M

x x x

x x x x

+ +

+ + +

+ ++ + + + + +

= + =

= = = +

n

n 1

n 1 nH H H H Hn 1 n 1 n 1 n n n n n n

n n

X

z

z zX X X z X z z X X

X X

( ) ( )1 11

1 1

11 1

11

Using lemma:

+ +

+ +

+ +

+ +

++

+ = + = +

=

=

11 1 1 1 1 1

1 1H H H1 1 n n n n n nH H

n 1 n 1 n n 1H Hn n n n

1Hn n n n

1Hn n n

n

A BCD A A B C DA B DA

X X z z X XX X X X

z X X z

I K z X X

X X zK

1

1 1

+

+ +

+

Hn

1H Hn n n n

z

z X X z


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66

Approximations to RLS Projection algorithm:

0 2 0 1

Stochastic approximation:

0 2

LMS algorithm:2

++

+ +

++

+ +

+ +

= < < <


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67

Constraints to Maintain Look Direction

Frequency Response.[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

1

1 2

2

Equivalent look-direction Filt

K

J

JK K K

w w w

w ww

x

x

i

i

[ ] [ ] [ ] [ ]1 2

er:

J d

f f f

s

i

signal output

( ) array output

y k

Frost, 1972


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Constrained Optimization

{ }

( ) ( )( )

( )

1

2

1

Minimize

Subject to constraint1=2

0

E y E

J

J

= = =

= =

= +

= +

=

= =

=

=

=TT T T T T T

1 T

T T Txx

T

T 1opt xx opt

w

1 T 1op

x

T T Txx

w x

x

T 1

xx

x

x

x

t x

x

w xx w w R w

C w

w

C w C w w C

w R C C R

w R C C w C R C

C

C R

w R w C w

w R w C

w

C

C C

Frost,1972


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>> %Levinson Durbin Recursion


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>> %Levinson-Durbin Recursion.

>> r=[1,0.5,0.25,0.0625]; %auto-correlation sequence>> a=levinson(r,3)

a =

1.0000 -0.5000 -0.0417 0.0833

>> h=filter(1,a,[1 zeros(1,25)]);>> stem(h)>> [bb,aa]=prony(h,3,3)

bb =

1.0000 0.0000 -0.0000 0.0000

aa =

1.0000 -0.5000 -0.0417 0.0833

>>


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adaptive signal processing , ebook . ingle

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