Array Lattice Filters

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T h e nterpretations we provided for the coefficients{ K M i),KLi), KLi)} in ~ e c .2.1,

in terms

o

solutions to first-order least-squares problems, can be used to m otivate yet an-

other lattice implementation in array form. We discussed array methods and their advan-

tages in some detail in Chapter 33. We show here that such array methods can also be

developed for orde r-recursive problem s.

Thus, recall that in Sec. 42.1 we in troduced the angle-norm alized estimation errors

and the corresp onding angle-norm alized error vectors

{ b L f i ,i, , a , bh,A

We then argued that th e reflection coefficients {

KM (i),

KLi),

K

(i)} can be interpreted

as the solu tions to three simple (regularized) projection proble ms, namely

K E . M ( ~ ) rojects T L ~nto

bh,i

K (i) projects jh,a nto

/c&(i)projects bLf,i onto fh,i

That is, each of these reflection coefficients solves the problem of projecting one angle-

norma lized error vector onto ano ther. More specifically, they solve the following regular-

ized least-squares problems:

where

i j = q X M f 2 and

f j = q A 2

The above interpretations were used in Sec. 42.1 to show that the reflection coefficients

{ K M ~ ) ,

a ( i ) ,& ( i ) }an be time-updated by resorting to the

RLS

algorithm in each

case.

680

Adaptive Filters,by Ali H. Sayed

Copyright

@ 2008 John Wiley Sons,

Inc.


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Now in Cha pter

33,

we arg ued that least-squares solutions can also be updated in array

form,

e.g., by using the

QR

algorithm of Sec.

35.2.

The

QR

method can th erefore be used

here to d evelop array m ethods for up dating the reflection coefficients themselves.

43.1

ORDER-UPDATE

OF

OUTPUT ESTIMATION ERRORS

We start with the reflection coefficient K M ~ ) .Comparing the cost function for

I C M ~ )

n

(43.1) with the one that appears in the statemen tof the QR method in Alg. 35.2 we see that

we can make the following identifications:

and

Q i

+

d * A,b

= f j-lXi+l + (i) =

CL(i)

M , i

M , z

If we now write down the QR equation s of Alg. 35.2 for these new variab les, we arrive at

the follow ing statement. Define the normaliz ed reflection coefficient

Then sta rt with Cz2- 1 = v nd q M - 1)= 0, and repeat for i

2 0.

At each

iteration, find a 2 x 2 unitary matrix OM,^ that generates the zero entry in the post-array

shown below, along with a lea ding positive entry in the first row and a positive entry s. The

entries in the post-array wou ld then correspon d to:

where, as was the case with Alg. 35.2, the scalar quantities {s z} can be determined from

the identities:

The first identity follows by eq uating the inner products of the second and third lines of the

arrays, while the se cond identity follow s from equating the norm s of the last lines of the

arrays. It is easy to see that the first expression leads to

sz*

=

(i) h Z ) K M i )

689

SECTION

43.1

ORDER-UPDATE

OF

OUTPUT

ESTIMATION

ERRORS


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112 1/ 2

whereas the second expression leads to

s

= yM+ l

i ) / y M

i ) and, correspondingly, =

i;C;+l(i). he array algorithm then becomes

690

CHAPTERRRAY

43

ATTICE

FILTERS

(43.4)

1

If we further multiply the last rows on both sides

of

(43.4)

by

r z 2 i )

e arriv e at the array

equation:

This step tells us how to orde r-update the angle-no rmalized variable r h ( i ) . f desired, the

reflection coefficient KM 2) can be determined from the equality

(43.6)

We now derive array methods for order-updating the angle-n ormalize d variables

{

,

(i)

bhf i ) }

by applying similar arguments to the other cost functions in (43.1).

43.2

ORDER-UPDATE

OF

BACKWARD ESTIMATION ERRORS

Consider the reflection coefficient ~ ( i ) .omparing its cost function from

(43.1)

with

the one that appears in the statement of the QR method in Alg.

35.2

we see that we can

make the following identifications

If we now write down the Q R equation s of Alg. 35.2 for these new variables, we arrive at

the followin g statemen t. Define the norm alized reflection coefficient

(43.7)

Then start with CC2 -l)

=

v nd

(-1) =

0, and repeat for i . At each

iteration, find

a

2

x 2

unitary matrix O ,i that generate s the zero entry in the post-array

below, along with a positive leading entry in the first row and

a

positive

s.

The entries in

the post-array would then correspon d to:


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92

CHAPTER

43

If we now write down the QR equation s of Alg. 35.2 for these new variab les, we arrive at

the follow ing statement. Define the normalized reflection coefficient

ARRAY

LATTICE

FILTERS

(43.12)

T h e n ~ t a r t w i t h & ~ ( - l ) = d-and&(-l) = 0,andrepeatfor i

2

0. Ateach

iteration find a 2 x

2

unitary matrix QL hat generates the zero entry in the post-array

below, along with a lead ing positive entry in the first row an d a positive

s.

The entries in

the post-array would then co rrespond to:

(43.13)

where , as was the case with Alg. 35.2, the scalar quantities {s, z} can be determined from

the identities:

The first identity follows by e quating the inner prod ucts of the secon d and third lines of the

arrays, while the seco nd identity follows from equating the norms of the last lines of the

arrays. It is easy to see that the first expression leads to

sz = (i) 6/M(i)rcR(i)

whereas the second expression gives

s

=

7 M + l i ) / 7 M

/2

1 2 (i)

and, consequently,

z

=

fLcl i). n this way, the array algorithm (43.13) becomes

If we further multiply the last rows on both sides of (43.14) by 7 z 2 i ) we arrive at the

array equation:


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693

This step tells us how to order-up date the angle-no rmalized variable (i). If desired, the

ECTION 43.4

SlGNfFlCANCE

OF DATA

STRUCTURE

reflection coefficient ,(i) can be determined from the equality

(43.16)

43.4

SIGNIFICANCE

OF

DATA STRUCTURE

As we already know, when the successive regressors have shift struc ture it holds that

b M ( i ) =

b M ( i

,

6/M(i)

=

b/M(i ,

? M ( i ) = Y M ( i

,

(i) = ( Z - 1

and we are led to the array-b ased lattice algorithm, also known as the QR D-based lattice

filter ee Fig. 43.1; the qualification QRD -based is used to indica te that the array

recursions correspond to Q R decompo sitions of the corresponding pre-arrays (recall the

third remark following the statement of Alg. 35.1).

For comparison purposes, Table 43.1 lists the estimated computational cost per itera-

tion for the various lattice filters derived in

this

chapter assuming real data. The costs are

in terms of the number of multiplications, additions, divisions, and square-roots that are

needed for each iteration. It is seen that lattice filters generally require O 2 0 M )operations

per iteration.

. . .

. . .

FlOURE 43 1

The QRD-based lattice filter.


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694

HAPTER

43

ARRAY

L AT IC E

FILTERS

Algorithm 43 1 Array lattice filter) Consider again the same setti ng of

Alg. 42.2. For each

i 2

0,

the

M - th o rder

a posteriori

estimation error,

r ~ ( i )

d ( i )

u ~ , i t u ~ , i ,

hat results from the solution of the regularized

least-squares problem

min

W M

can be computed as follows:

1.

Initialization. From m = 0 t o m = M

1

set:

Ciyy-1) =

JV

2y-q

=d

q m ( - l ) = o,

q i - 1 )

=

o,

q ( - l ) =

o,

b m ( - l )

=

o

2. For i 2

0,

repeat:

Set ~ ; / ~ ( i )

1, bb(i) =

(i)

=

u i ) ,and

rb(i) =

d i )

For

m

= 0 t o m =

M 1,

apply 2 x 2 unitary rotations

f

Om,i ,

and

Oh, i ,

with positive

2,2)

entries, in order to annihilate

the 1,2) entries of the post-arrays below:

and set rm+l(i)= ~ k + ~ ( i ) ~ ~ + ~ ( i ) ./2

bk+l(i)

TABLE 43.1

Estimated computational cost per iteration for various lattice filters.

Array Lattice Filters

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