a prediction problem

Professor A G

Constantinides© 1

AGCDSP

A Prediction Problem Problem: Given a sample set of a stationary

processes

to predict the value of the process some time into the future as

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

]}[],...,2[],1[],[{ Mnxnxnxnx

])[],...,2[],1[],[(][ Mnxnxnxnxfmnx

Professor A G

Constantinides© 2

AGCDSP

A Prediction Problem Linear Prediction dates back to Gauss in

the 18th century. Extensively used in DSP theory and

applications (spectrum analysis, speech processing, radar, sonar, seismology, mobile telephony, financial systems etc)

The difference between the predicted and actual value at a specific point in time is caleed the prediction error.

Professor A G

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AGCDSP

A Prediction Problem The objective of prediction is: given

the data, to select a linear function that minimises the prediction error.

The Wiener approach examined earlier may be cast into a predictive form in which the desired signal to follow is the next sample of the given process

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Forward & Backward Prediction

If the prediction is written as

Then we have a one-step forward prediction

If the prediction is written as

Then we have a one-step backward prediction

])[],...,2[],1[(][ˆ Mnxnxnxfnx

])1[],...,2[],1[],[(][ˆ MnxnxnxnxfMnx

Professor A G

Constantinides© 5

AGCDSP

Forward Prediction Problem

The forward prediction error is then

Write the prediction equation as

And as in the Wiener case we minimise the second order norm of the prediction error

][ˆ][][ nxnxne f

M

kknxkwnx

1][][][ˆ

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Thus the solution accrues from

Expanding we have

Differentiating with resoect to the weight vector we obtain

}])[ˆ][{(min}])[{(min 22 nxnxEneEJ f ww

}])[ˆ{(])[ˆ][{(2}])[{(min 22 nxEnxnxEnxEJ w

}][ˆ][ˆ{2)][ˆ][{(2iii wnxnxE

wnxnxE

wJ

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AGCDSP


However

And hence

or

][][ˆ inxwnx

i

]}[][ˆ{2])[][{(2 inxnxEinxnxEwJ

i

]}[][][{2])[][{(21

inxknxkwEinxnxEwJ M

ki

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On substituting with the correspending correlation sequences we have

Set this expression to zero for minimisation to yield

M

kxx

i

kirkwirwJ

1][][2][2

Miirkirkw xx

M

kxx ,...,3,2,1][][][

1

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AGCDSP


These are the Normal Equations, or Wiener-Hopf , or Yule-Walker equations structured for the one-step forward predictor

In this specific case it is clear that we need only know the autocorrelation propertities of the given process to determine the predictor coefficients

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AGCDSP

Forward Prediction Filter Set

And rewrite earlier expression as

These equations are sometimes known as the augmented forward prediction normal equations

MmMmmw

mmaM

0,..,1][01

][

Mkkr

kmrma xxM

mxxM ,...,2,10

0]0[][][

0

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AGCDSP

Forward Prediction Filter The prediction error is then given

as

This is a FIR filter known as the prediction-error filter

M

mMf knxkane

0][][][

MMMf zMazazazA ][...]2[]1[1)( 21

1

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Backward Prediction Problem

In a similar manner for the backward prediction case we write

And

Where we assume that the backward predictor filter weights are different from the forward case

][ˆ][][ MnxMnxneb

M

kknxkwMnx

1]1[][~][ˆ

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AGCDSP


Thus on comparing the the forward and backward formulations with the Wiener least squares conditions we see that the desirable signal is now

Hence the normal equations for the backward case can be written as

][ Mnx

MkkMrkmrmw xx

M

mxx ,...,3,2,1]1[][][~

1

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This can be slightly adjusted as

On comparing this equation with the corresponding forward case it is seen that the two have the same mathematical form and

Or equivalently

MkkrmkrmMw xx

M

mxx ,...,3,2,1][][]1[~

1

MmmMwmw ,...,2,1]1[~][

MmmMwmw ,...,2,1]1[][~

Professor A G

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AGCDSP

Backward Prediction Filter Ie backward prediction filter has the same

weights as the forward case but reversed.

This result is significant from which many properties of efficient predictors ensue.

Observe that the ratio of the backward prediction error filter to the forward prediction error filter is allpass.

This yields the lattice predictor structures. More on this later

MMMMb zzMazMaMazA ...]2[]1[][)( 21

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Levinson-Durbin Solution of the Normal Equations The Durbin algorithm solves the following

Where the right hand side is a column of as in the normal equations.

Assume we have a solution for

Where

mmm rwR

R

mkkkk 1rwRT

kk rrrr ],...,,,[ 321r

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Levinson-Durbin For the next iteration the normal equations

can be written as

Where

Set

110

kk

k

rrw

JrrJR

kTk

*kk

11

k

kk r

rr

k

kk

zw 1

kJIs the k-order counteridentity

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Levinson-Durbin Multiply out to yield

Note that

Hence

Ie the first k elements of are adjusted versions of the previous solution

** rJRwrJrRz kkkkkkkkkkk11 )(

11 kkkk RJJR*wJwz kkkkk

1kw

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Levinson-Durbin The last element follows from the

second equation of

Ie

10 k

k

k

kk

rrrw

JrrJR

kTk

*kk

)(1 10

kkkkk rr

zJrT

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Levinson-Durbin

The parameters are known as the reflection coefficients.

These are crucial from the signal processing point of view.

k

Professor A G

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AGCDSP

Levinson-Durbin

The Levinson algorithm solves the problem

In the same way as for Durbin we keep track of the solutions to the problems

byR m

kkk byR

Professor A G

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AGCDSP

Levinson-Durbin

Thus assuming , to be known at the k step, we solve at the next step the problem

10 k

k

k

kk

brbv

JrrJR

kTk

*kk

kw ky

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Constantinides© 23

AGCDSP

Levinson-Durbin

Where

Thus

k

kk

vy 1

** yJyrJbRv kkkkkkkkkk )(1

*0

1

kTk

kkTkk

k rb

yryJr

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AGCDSP

Lattice Predictors Return to the lattice case. We write

or)()()(zAzAzT

f

bM

MMM

MMMM

M zMazazazzMazMaMazT

][...]2[]1[1...]2[]1[][)( 21

1

21

Professor A G

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AGCDSP

Lattice Predictors The above transfer function is allpass of order

M. It can be thought of as the reflection coeffient

of a cascade of lossless transmission lines, or acoustic tubes.

In this sense it can furnish a simple algorithm for the estimation of the reflection coefficients.

We strat with the observation that the transfer function can be written in terms of another allpass filter embedded in a first order allpass structure

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Lattice Predictors This takes the form

Where is to be chosen to make of degree (M-1) .

From the above we have

)(1)()(

11

1

11

1

zTzzTzzT

M

MM

1 )(1 zTM

))(1()()(1

11

1 zTzzTzT

M

MM

Professor A G

Constantinides© 27

AGCDSP

Lattice Predictors And hence

Where

Thus for a reduction in the order the constant term in the numerator, which is also equal to the highest term in the denominator, must be zero.

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

12

11

11

111

MMMM

MMM

M zMazazazzzMaMazT

][1][][][

1

11 Ma

rMararaM

MMM

Professor A G

Constantinides© 28

AGCDSP

Lattice Predictors This requirement yields The realisation structure is

][1 MaM

)(zTM

)(1 zTM 1z

1

Professor A G

Constantinides© 29

AGCDSP

Lattice Predictors There are many rearrangemnets that can be

made of this structure, through the use of Signal Flow Graphs.

One such rearrangement would be to reverse the direction of signal flow for the lower path. This would yield the standard Lattice Structure as found in several textbooks (viz. Inverse Lattice)

The lattice structure and the above development are intimately related to the Levinson-Durbin Algorithm

Professor A G

Constantinides© 30

AGCDSP

Lattice Predictors The form of lattice presented is not the

usual approach to the Levinson algorithm in that we have developed the inverse filter.

Since the denominator of the allpass is also the denominator of the AR process the procedure can be seen as an AR coefficient to lattice structure mapping.

For lattice to AR coefficient mapping we follow the opposite route, ie we contruct the allpass and read off its denominator.

Professor A G

Constantinides© 31

AGCDSP

PSD Estimation It is evident that if the PSD of the

prediction error is white then the prediction transfer function multiplied by the input PSD yields a constant.

Therefore the input PSD is determined. Moreover the inverse prediction filter

gives us a means to generate the process as the output from the filter when the input is white noise.

a prediction problem

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