6-a prediction problem.ppt

8/17/2019 6-A Prediction Problem.ppt

1/31

Professor A G 1

AGC

DSP

AGC

DSP A Prediction Problem

Problem: Given a sample set of a stationaryprocesses

to predict the value of the process some timeinto the future as

The function may be linear or non-linear. Weconcentrate only on linear prediction functions

]}[],...,2[],1[],[{ M n xn xn xn x −−−

])[],...,2[],1[],[(][ M n xn xn xn x f mn x −−−=+


2/31

Professor A G 2

AGC

DSP

AGC


inear Prediction dates bac! to Gauss inthe "#th century.

$%tensively used in DSP theory andapplications &spectrum analysis' speechprocessin(' radar' sonar' seismolo(y'mobile telephony' )nancial systems etc*

The di+erence bet,een the predictedand actual value at a speci)c point intime is caleed the prediction error.


3/31

Professor A G 3

AGC

DSP

AGC


The obective of prediction is: (iventhe data' to select a linear function

that minimises the prediction error. The Wiener approach e%amined

earlier may be cast into a

predictive form in ,hich the desiredsi(nal to follo, is the ne%t sampleof the (iven process


4/31

Professor A G 4

AGC

DSP

AGC

DSP

or,ard / 0ac!,ard

Prediction 1f the prediction is ,ritten as

Then ,e have a one-step for,ardprediction

1f the prediction is ,ritten as

Then ,e have a one-step bac!,ardprediction

])[],...,2[],1[(][ˆ M n xn xn x f n x −−−=

])1[],...,2[],1[],[(][ˆ −−−−=− M n xn xn xn x f M n x


5/31

Professor A G 5

AGC

DSP

AGC

DSP

or,ard Prediction

Problem The for,ard prediction error is then

Write the prediction e2uation as

And as in the Wiener case ,eminimise the second order norm ofthe prediction error

][ˆ][][ n xn xne f −=

∑ −==

M

k

k n xk wn x1

][][][ˆ


6/31

Professor A G 6

AGC

DSP

AGC

DSP

or,ard Prediction

Problem Thus the solution accrues from

$%pandin( ,e have

Di+erentiatin( ,ith resoect to the,ei(ht vector ,e obtain

}])[ˆ][{(in}])[{(in22 n xn x E ne E J f −==

ww

}])[ˆ{(])[ˆ][{(2}])[{(in22 n x E n xn x E n x E J +−=

w

}][ˆ

][ˆ{2)][ˆ

][{(2

iii w

n xn x E

w

n xn x E

w

J

∂

∂+

∂

∂−=

∂

∂


7/31 Professor A G !

AGC

DSP

AGC

DSP

or,ard Prediction

Problem 3o,ever

And hence

or

][][ˆ

in xw

n x

i

−=∂

∂

]}[][ˆ{2])[][{(2 in xn x E in xn x E

w

J

i

−+−−=

∂

∂

]}[][][{2])[][{(2

1

in xk n xk w E in xn x E

w

J M

k i

−∑ −+−−=

∂

∂

=


8/31 Professor A G "

AGC

DSP

AGC

DSP

or,ard Prediction

Problem 4n substitutin( ,ith the

correspendin( correlation

se2uences ,e have

Set this e%pression to 5ero forminimisation to yield

∑ −+−=∂

∂

=

M

k xx

i

k ir k wir w

J

1

][][2][2

M iir k ir k w xx M

k xx ,...,3,2,1][][][

1

==∑ −=


9/31 Professor A G #

AGC

DSP

AGC

DSP

or,ard Prediction

Problem These are the 6ormal $2uations' or

Wiener-3opf ' or 7ule-Wal!er e2uations

structured for the one-step for,ardpredictor

1n this speci)c case it is clear that ,eneed only !no, the autocorrelationpropertities of the (iven process todetermine the predictor coe8cients


10/31 Professor A G 1$

AGC

DSP

AGC

DSP or,ard Prediction ilter

Set

And re,rite earlier e%pression as

These e2uations are sometimes !no,n asthe au(mented for,ard prediction normale2uations

M m

M mmw

m

ma M

>

=−

=

=

$

,..,1][

$1

][

M k

k r

k mr ma

xx M

m xx M ,...,2,1$

$]$[

][][$ =

==∑ −

=


11/31

Professor A G 11

AGC

DSP

AGC

DSP or,ard Prediction ilter

The prediction error is then (ivenas

This is a 19 )lter !no,n as theprediction-error )lter

∑ −= =

M

m M f k n xk ane $ ][][][

M M M f z M a z a z a z A

−−− ++++= ][...]2[]1[1)( 211


12/31

Professor A G 12

AGC

DSP

AGC

DSP

0ac!,ard Prediction

Problem 1n a similar manner for the bac!,ard

prediction case ,e ,rite

And

Where ,e assume that the bac!,ardpredictor )lter ,ei(hts are di+erentfrom the for,ard case

][ˆ][][ M n x M n xneb −−−=

∑ +−=−=

M

k

k n xk w M n x1

]1[][%][ˆ


13/31

Professor A G 13

AGC

DSP

AGC

DSP

0ac!,ard Prediction

Problem Thus on comparin( the the for,ard and

bac!,ard formulations ,ith the Wienerleast s2uares conditions ,e see that thedesirable si(nal is no,

3ence the normal e2uations for thebac!,ard case can be ,ritten as

][ M n x −

M k k M r k mr mw xx M

m xx ,...,3,2,1]1[][][

%1

=+−=∑ −=


14/31

Professor A G 14

AGC

DSP

AGC

DSP

0ac!,ard Prediction

Problem This can be sli(htly adusted as

4n comparin( this e2uation ,ith thecorrespondin( for,ard case it is seen thatthe t,o have the same mathematical form

and

4r e2uivalently

M k k r mk r m M w xx M

m xx ,...,3,2,1][][]1[

%

1

==∑ −+−=

M mm M wmw ,...,2,1]1[%][ =+−=

M mm M wmw ,...,2,1]1[][% ==+−=


15/31

Professor A G 15

AGC

DSP

AGC

DSP 0ac!,ard Prediction ilter

1e bac!,ard prediction )lter has the same,ei(hts as the for,ard case but reversed.

This result is si(ni)cant from ,hich manyproperties of e8cient predictors ensue.

4bserve that the ratio of the bac!,ardprediction error )lter to the for,ardprediction error )lter is allpass.

This yields the lattice predictor structures. ore on this later

M

M M M b z z M a z M a M a z A −−−

++−+−+= ...]2[]1[][)( 21


16/31

Professor A G 16

AGC

DSP

AGC

DSP evinson-Durbin

Solution of the 6ormal $2uations

The Durbin al(orithm solves the follo,in(

Where the ri(ht hand side is a column ofas in the normal e2uations.

Assume ,e have a solution for

Where

mmm rwR =R

mk k k k ≤≤= 1rwR T

k k r r r r ],...,,,[ 321=r


17/31

Professor A G 1!

AGC

DSP

AGC

DSP evinson-Durbin

or the ne%t iteration the normal e2uationscan be ,ritten as

Where

Set

11

$++ =

k k

k

r rwJr

rJR

k

T

k

*

k k

=

++ 11

k

k

k r

r

r

=+

k

k

k α

zw 1

k J1s the !-order

counteridentity


18/31

Professor A G 1"

AGC

DSP

AGC

DSP evinson-Durbin

ultiply out to yield

6ote that

3ence

1e the )rst ! elements of areadusted versions of the previous solution

**rJR wrJrR z k k k k k k k k k k k

11 )( −− −=−= α α

11 −− = k k k k R JJR

*wJwz k k k k k α −=

1+k w


19/31

Professor A G 1#

AGC

DSP

AGC

DSP evinson-Durbin

The last element follo,s from thesecond e2uation of

1e

=

+1$ k

k

k

k k

r r

rw

Jr

rJR

k

T

k

*

k k

α

)(1

1

$

k k k k k r r

zJrT−= +α


20/31

Professor A G 2$

AGC

DSP

AGC

DSP evinson-Durbin

The parameters are !no,n asthe re;ection coe8cients.

These are crucial from the si(nalprocessin( point of vie,.

k α


21/31

Professor A G 21

AGC

DSP

AGC

DSP evinson-Durbin

The evinson al(orithm solves theproblem

1n the same ,ay as for Durbin ,e!eep trac! of the solutions to theproblems

byR =m

k k k byR =


22/31

Professor A G 22

AGC

DSP

AGC

DSP evinson-Durbin

Thus assumin( ' to be!no,n at the k step' ,e solve atthe ne%t step the problem

=

+1$ k

k

k

k k

br bv

JrrJR

k

T

k

*

k k

µ

k w k y


23/31

Professor A G 23

AGC

DSP

AGC

DSP evinson-Durbin

Where

Thus

=

+

k

k

k

µ

vy 1

**yJyrJbR v k k k k k k k k k k µ µ −=−=

− )(1

&$

1

k T k

k k T k k

k r

b

yr

yJr

−

−= + µ


24/31

Professor A G 24

AGC

DSP

AGC

DSP attice Predictors

9eturn to the lattice case.

We ,rite

or

)(

)()(

z A

z A z T

f

b M =

M

M M

M

M M M M

z M a z a z a

z z M a z M a M a z T

−−−

−−−

++++

++−+−+=

][...]2[]1[1

...]2[]1[][)(

21

1

21


25/31

Professor A G 25

AGC

DSP

AGC


The above transfer function is allpass of orderM.

1t can be thou(ht of as the re;ection coe8entof a cascade of lossless transmission lines' oracoustic tubes.

1n this sense it can furnish a simple al(orithmfor the estimation of the re;ection coe8cients.

We strat ,ith the observation that the transferfunction can be ,ritten in terms of anotherallpass )lter embedded in a )rst order allpassstructure


26/31

Professor A G 26

AGC

DSP

AGC


This ta!es the form

Where is to be chosen to ma!eof de(ree (M-1) .

rom the above ,e have

)(1

)()(

11

1

11

1

z T z

z T z z T

M

M M

−−

−−

+

+

= γ

γ

1γ )(1 z T M −

))(1(

)()(

11

11

z T z

z T z T

M

M M

γ

γ

−

−= −−


27/31

Professor A G 2!

AGC

DSP

AGC


And hence

Where

Thus for a reduction in the order theconstant term in the numerator' ,hich isalso e2ual to the hi(hest term in thedenominator' must be 5ero.

)][...]2[]1[1(

...]1[][(

)( 121111

111

M M M M

M M M

M z M a za za z

z z M a M a

zT −−−−−−−

−−

−−

++++

++−+=

][1

][][][

1

11

M a

r M ar ar a

M

M M M

γ

γ

−

−−=

−


28/31

Professor A G 2"

AGC

DSP

AGC


This re2uirement yields

The realisation structure is

][1 M a M =γ

)( z T M

)(1 z T M −1− z

1γ


29/31

Professor A G 2#

AGC

DSP

AGC


There are many rearran(emnets that can bemade of this structure' throu(h the use ofSi(nal lo, Graphs.

4ne such rearran(ement ,ould be toreverse the direction of si(nal ;o, for thelo,er path. This ,ould yield the standardattice Structure as found in several

te%tboo!s &vi5. 1nverse attice* The lattice structure and the above

development are intimately related to theevinson-Durbin Al(orithm


30/31

Professor A G 3$

AGC

DSP

AGC


The form of lattice presented is not theusual approach to the evinson al(orithm inthat ,e have developed the inverse )lter.

Since the denominator of the allpass is alsothe denominator of the A9 process theprocedure can be seen as an A9 coe8cientto lattice structure mappin(.

or lattice to A9 coe8cient mappin( ,efollo, the opposite route' ie ,e contruct theallpass and read o+ its denominator.


31/31

Professor A G 31

AGC

DSP

AGC

DSP PSD $stimation

1t is evident that if the PSD of theprediction error is ,hite then the

prediction transfer function multipliedby the input PSD yields a constant.

Therefore the input PSD is determined.

oreover the inverse prediction )lter(ives us a means to (enerate theprocess as the output from the )lter,hen the input is ,hite noise.

6-a prediction problem.ppt

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