qrd-based multichannel adaptive lattice algorithms for the parameter identification problem

Vo1.13 No.3 J O U R N A L OF E L E C T R O N I C S July 1996

Q R D - B A S E D M U L T I C H A N N E L A D A P T I V E LATTICE A L G O R I T H M S FOR THE P A R A M E T E R

I D E N T I F I C A T I O N PROBLEM*

Ouyang Shah Fang Huijun

(Gu//in/~titute o f Electronic Techno/ogy, Guilln 541004)

A b a t r a c t A pair of multichannel recursive least squares (RLS) adaptive lattice algorithms

based on the order recurmve of lattice filters and the superior numerical properties of Givens

aigorithnm is derived in this paper. The derivation of the first edgorithm is baaed on QR decom-

pcmition of the input data matrix directly, and the Givens rotations approach is used to compute

the QR decomposition. Using first a prerotation of the input data matrix and then a repetition

of the single eh,mnel Givens lattice algorithm, the second algorithm can be obtained. Both algo-

ritlmm have superior nunu~ical properties, particularly the robustness to wordlength limitations.

The ~ vector to he estimated can he extracted directly from interned variables in the

present algorithms without a backsolve operation with an extra triangular array. The results

of computer simulation of the parameter identification of a two-channel system are presented to

confirm efficiently the derivation.

Key words Recursive least squares lattice algorithm; QR decomposition; Multichmmel sig-

nab; Adaptive paramet~ identification

I. Introduction

In recent years, a major effort in least squares (LS) adaptive filtering has been directed

toward the solving LS problems using QR decomposition (QRD) method based on the Givens

rotations. The resulting adaptive algori thm is called QRD recursive LS (RLS) algorithm.

An impor tant reason is tha t the QRD RLS algorithm has superior numerical propert ies over

conventional RLS algorithma, and it is well suitable for parallel processing implementat ion in

a systolic array| l -s] . In paxticular, the adaptive filter error can be obtained directly without

explicitly computing the coefficient vectors. If only the adaptive filter error is of interest, such

as adaptive noise cancellation, beamforminE, etc., it is special attractive. Several reseachers

have presented various QRD RLS algorithms for the single channel case, and the extension of

the single channel algorithms to the multichannel case have been also developed in Ref.[6-

9]. However, those multichannel QRD RLS algorithms can not directly obtain the filter

coefficients. To compute the coefficients, a backsolve operat ion has to be required. Hence

those algorithms require a computat ional complexity of O(N2p2), where N is the number

of coefficients in each channel and p is the number of channels. To avoid the backsolve

*Suppor ted by the Foundation of the Academy of Electronic Science, China

202 JOURNAL OF ELECTRONICS Vol.13

operation, we propose a pair of multichannel fast QRD RLS lattice algorithmR in this paper.

Both algorithms can extract directly the filter coefficients without a backsolve operation.

The computational complexity of our algorithms is O(Np 3) and O(NF~), respectively.

I I . P r o b l e m D e s c r i p t i o n

Let us consider p scalar data sequences {:Vl(n)), {z2(n)},. . . , {zp(n)) and a scalar de-

sired sequence {y(n)} (n -- 0 to k). At time k, we form the LS estimation of Y(k) by the

linear combination of zj(k) (j = 1 to p) and its delayed version, and mlnlmi~,~e the cost

function

r = s (1)

where the superscript T denotes transposition. The LS estimation error vector is given by

�9 (k) = r(k) - X(k)W(k)

where W(k) is an/Yp x 1 coefficient vector,

Y(k) = [,~k/2y(0),..., A'/2y(k - 1), y(k)] T

(2)

(3)

where 0 ~'~ ,~ _< 1 is an exponential forgetting factor, z ( k ) = [z,(k),x2(t),...,=,(k)], and the uumber of coefficients N in each channel is assumed to be the same.

In a QR algorithm, an orthogonal (k + 1) x (k + 1) matrix Q(k) = [Ql(k)Q2(k)] T is introduced to triangularize the weighted data matrix X(k). Due to Q(k) being the

orthogonal matrix, the minimization of Eq.(1) is equivalent to minimizing the norm of

rotated version of Eq.(2). Using Q(k) to premultiply Eq.(2), we have

Q(.).(.> = L.,,(k)] - [ ~ ) ] w(.) (5) where R(k) is an Np x Np upper triangular matrix. One can find that

X(~)R-~(k) = Ox (k) (e)

It is noticed that the LS solution satisfies the normal equations and the matrix Q(k) is

orthogonal. Thus the estimation of Y(k) can be obtained as follows:

Np--1 ~'~/C) = (~l(/C)Q1T(k)u ---- ~l(k)Yq(k) -~ ~ qj(k)yqj(k) (7)

j=o

where (qj(k)} is the elements of the Ql(k) and {yqj(k)} is the elements of Yq(k), called the

backward prediction error vectors and the tap coet~icients of a normalized lattice filter [w],

respectively.

[" ,~k/2=(O) 0 . - . 0 ]

x(k) = [ xc~-~)/2z(1) ~ /2=(~ "'" 0

l : : : (4)

/L •ik) = ( k - l ) . . . = ( k - N + 1 )

No.3 QRD-BASED MULTICHANNEL ADAPTIVE LATTICE ALGORITHMS 203

To obtain the tap coefficients {yqj(k)} (j = 0 to Np- I), the following sections will

propose two different approaches. The first approach is based on the QR decomposition of

the multichannel data matrix directly, and the corresponding parameters are updated by the

application of forward and backward prediction process to the variables of the extended di-

mension. The second approach is using first a prerotation of the multichannel data matrix [11]

and then using directly the conclusion for the single QRD RLS lattice algorithm [I~

III. T h e D e r i v a t i o n of Algorithms

For simplicity of illustration, we only restrict attention to the p = 2 in following deriva-

tion. The extension to that case of more than two channels will be straightforward.

1. A lgor i thm I

The derivation of a fast RLS algorithm makes good use of the Toeplitz nature of input

data matrix to reduce the computational complexity. Because the input data matrix in the

mtdtichannel case has a block Toeplitz nature, the QR decomposition of the multichannel

data matrix can be applied directly.

(1) Backward l inear p red ic t ion

For the N-th order backward linear prediction problem, the estimation of x(k - N) =

[zl(k - N) , z2(k - N)I is performed from a linear combination of { z ( k ) , z ( k - 1),. . . ,z (k -

N + 1)}. Appending a~(k - N) to the fight of the data matrix X2N(k) (where the subscript

2N denotes the number of columnR of matrix X2N(k)), we obtain

X2N+2(k) = [=2N(k) X(k -- N)] (8)

Now suppose that an orthogonal matrix Q(k) triangularizes the data matrix X2N(k), denoted as R2/v(k). Eq.(8) can be triangularized by premultiplying the cascade of two orthog-

onal matrices q2,b(k), ql,b(k). We thus find

Qa,b(k)Ql,b(k)Q(k)XaN+2(k) = [ PczN~a(k) ] _

where R2N+2(k) Can be denoted as follows: �9 1/2 1/2 Ellb,0(k ) E12b,0(k ) r=

I/2 0 E22b,0(k ) r~

1/2 0 0 Ellb,1 (k) 0 0 O /Z2N+~(k) =

"S2M(k) Xl,b(k) X2,b(k) ] 1/2 1/2

0 E11b,N(k) E12b,lq(k)] 1/2 ] 0 0 E~2 (k)

o o

0 0 0

0 0 0

(9)

~ z " " "

7 " z . - .

1/2 El2b, 1 ( k ) . . .

1/2 E~b,l(k ) . . .

1/2 0 ..- Ellb,A,(k ) 0 "." 0

~ 1 7 6 1 7 6

~

� 9 1 7 6

~ 1 7 6 1 7 6

I/2 E, ab,N(k) l

1/2 E22b,N ( k ) J

0o) In Eq.(10) rffi denotes a quantity of no specific intel~st. According to the fact that the

rotated vectors have the same norm as their unrotated counterparts because the matrix


Q(k) is orthogonal, the zeroth-order backward prediction error energy can be found:

k

(11a) i = O

k

i---o

So, the order update procedure from P~N(k) to /i~N+2(k) has been completed from

the above equations. In effect, the order update equations for the backward prediction error

energy have been obtained from Eq.(9). It will play a key role in updating time and order

for the normalized backward prediction errors and in updating rotation angles below.

(2) Forward l inear p red ic t ion

For N-th order forward prediction problem, the estimation of a:(k + 1) -- [xl(k +

1), z2(k + 1)] is performed from a linear combination of the data {x(k), z(k - 1), . .- , x(k -

N + 1)}. Appending z(k + 1) to the left of the data matrix X2N(k), the following equation

can be obtained:

X2lv+2(k + 1)= [~v(k + 1) 0-..0 ] (12) X2N(k)

[ o o ] By a sequence of orthogonal rotations of X2N+2(k + 1) ' one finds that

[ 4 Elo~,N(k) ~ EI,~,N(k) o-..0 i/2 i/2

(~(k) X~N§ + 1) = X l t + 1) X~1(k + 1) ~,N(k) (13) 0

L elf ,N(k "[- 1) e2I,N(k + 1) O

Now, by again a sequence of orthogonal rotations, the elements, elLN(k + 1),e2LN(k +

1) ,Xl l (k + 1) and XaL~v(k + 1), can be rotated into the first 2 rows and the original

positions of the elements become zero, namely

O1)]Q,~l(k+l)Q~(k+l).Eq.(13): [R'N+~k+l) 1 (14)

where Q~(k + 1) is two orthogonal 2 x 2 Givens matrices, Q . l (k + 1) and Q~,,(k + 1) is 2N 2 x 2 Givens matrices, respectively, the submatrix may be denoted as:

cos a i j sin a i j

Q~,,~ (k + l) -- l j -1 , ( i = l , 2 ; j = l , - . . , 2 N ) (15)

L - s m a~,j cos a~j

Now Q~(k + 1) and Qa,.,(k + 1)(i = 1,2;j = 1 , . . . , 2N) caa be obtained from Eq.(14).

As has pointed out in Ref.[10], the - s i n a i j is the reflection coefficient of a normalized

prediction lattice filter.

(3) U p d a t i n g ro ta t ion nmt r ix Q(k + 1)


In a QR algorithm, the rotation matrix Q(k + 1) can be computed recursively [11. Several

versions of QRD RLS algorithm have been developed by using different methods to imple-

ment the recursive computation[3-5,1~ To obtain (~(k + 1), let us consider the following

rotations

+ ~) = ,~(k + ~)x~,,(~, + 1) : ,~(~ + 1) [ Q!~)-- R2N(/r t U

=r

• ] X2N(k + 1)

. . ~l,2b,O~,. / r l " �9 "

Zl(k-b 1) z2(k + 1) �9 - �9

(16)

A series of 2N 2 x 2 Givens rotation angles in (~(k + 1) can be computed according to

Eq.(16), i. e., the update of Q(k + 1) has been finished.

In similar to Ref.[10], for the multichannel normalized prediction lattice the recursive

equation for a normaliT~fd posterior error can be obtained as follows:

0

sina2#(k + 1)

cos a~,,(k + 1)

o + i)i F + I)I 1 o ] /~s,(k + 1) / 0 ~l,,(k + 1) L ~,,_1(k) J

(17)

I ~ 1 1 , - ~ ( ~ + 1) ] { 1 o ~2.f,/,--1(~ "Jr" 1)[ = o cosa2,,(k -I- 1) ~b.i+l(k + 1) J 0 -.in a2,~(k + 1)

X [ Cosal,~(k-F1)

- sin al,i(k % 1)

Note that Ei/,,(k + I) and E21,i(k + I) in Eq.(17) have been obtained from the forward

prediction procedure. Thus ~bj+l(k + I) can be computed according to the normal~,ed

backw~d prediction error g~,~-1(k) (i = 2N,..., 1) at time k. And we have

~b,0(k+l) =~11,0(k+1), ~b,l(k-i-1) -~21,o(k-}-I) (18)

Now the likelihood variable can be updated in the form:

2N--1 2 N - - I

~f2N-l(k "~- 1) -----[1 - ~ Eb,i(k + 1)] I12 ---- H cosO,(k + 1) (19) i = 0 i = 0

A complete dual-channel QRD RLS lattice algorithm is sllmmaa'ized in Tab.1. The

resulting lattice structfire is depicted in Fig.1.

Tab.1 A l g o r i t h m I

(1) I n i t i a l i ~ t i o n :

~/2N-I(-1) ----- 1; 0$(--1) := Zlyj(--1) = z2yj(--1) = yqj(--1); ~bj(-1) = 0,j ---- 0,.--,2N-1; i/2 i/2 i/2 EnyaN(-1) = EI2~.2N(--D = E~j,2~(--I) = 0 (exact ~t);

I/2 1/2 I/2 or EIll,2N(--1 ) = E ~ / , ~ N ( - - 1 ) = amal l p<mitive con ten t , E121,2N(--1) = 0, ( sof t s t a r t ) .

k > 0, I n p u t ~ n a k : zl(k),z2(k);

D~d .~: v(~); ~11,oC~) = x~Ck), ~21,o(k) = ~2(k), ~,,o(k) = v(k),

(2) P red i c t i on sec t ion:


(a) j = 0 , 1 , - . - , 2 N - 1

exL~+~(k ) e~/j+~(tk)J l - s i n O J ( k - 1) coseC(k- 1)

x [ x~/==~yJ(~- ~) ~:/a==y,~(~_ ~)]

(b) j = 2~v ~/~ Zl=

-sin~o~ e _ ~ L - -S in~ 0 c ~ . ~

,~ Et~Y0~ ( 1) x AZI~E ~I~ ~=I'~N(~ -- ~)/

L ez~,~N(~=) e~,~N(/=) J

(c) y = 2~v,..., 2,1 . ~/~ ~/~

S'~J-d~)0~ E~,~_,(~),E~=~'~-'(~)]0 lj = [i

(d) j = 2N,- . . ,2 ,1

0

co.,-.=A~ ) - ,in a ~ ( k : )

x [ C==o ~(~)

L -=~,===~ (~)

[ I [i o

l.~b,~- ~ (ll -- ILl

~b,o(ll) = ~#,o(ll); ib,~ (ll) = ~#,o(ll)

")'~N_ 1 (k) = - L - 2_,~_-o ebjL~)] (,,) .j = 2,v - ~ , . . . , ~ , o

-,i~0~(k) o=o~(k)J L~b,~(~).l

o l coo.-,=.~ (k) J

~ =: l [ ' , : , 0 ~a~j(k)J Lzls,y-~(~)

112 smj(k) ") ~22k',j~ J I

=29,~_~(k) J

o

c~a,j(k)J L-mkua1#(k) 0 ~aly(k)J

(3) Joint p ~ c e ~ section:

(f) j = 0,1,.-.,2N- I

1


I ~ Cffl~ "% / tanas. JM

2 I T 2 1 / \ ~ - , o

f - ' ' t / seCa~o

El '(#)

X~ i}

J

1 !

E.I/z(k)

Y . .

~bCall [ i

tano~ ~ Urea11 [ - tlngto M a l l ~ 3

/ J,

i

Fig.l A ducal-channel n ~ joint process Isttice structure

2. A l g o r i t h m II

The intrinsic feature of the QRD RLS algorithm is that the estimation error can be

obtained directly from the Givens rotations without explicitly computing estimated tap

coefficients. Thus ~ ( k ) may be considered as a desired response and can be estimated

based on xl(k) and its delayed version similar to the joint process estimation of a single

channel QRD RLS algorithm. Such an operation may be called as prerotation [11]. Using

such a simple prerotation of the multichannel input data the QRD RLS algorithm derived

for a single channel case can be used to deal with the multicbannel problem.

Consider the prerotation equation

~I/~x2(k) ] (20) X2(k + 1)] = Q,(k + 1) [ z2(k + 1) J [e2(k ~c 1) J

where Q~(k+ 1) is N 2 • 2 Givens rotations matrices ~n corresponding with the m-st channel. With respect to the second channel the forw',cd prediction problem can be expre~ed as:

[X~l(k + 1) ] A1/2X21(k) e21(k + I) J = (~'(k) [e2(k 1)1 + 1)9'l,N-l(k +

For the joint process estimation problem, one can obtain

eq(k + 1) = ~2(k + 1)Ql(k + 1) L[ A1/2yq(k)]y(k + 1) J

(21)

(22)


where the orthogonal m a t r i c e s Q l ( k -t- 1) And Q 2 ( k -j- 1) can be updated from the same

procedure developed for s single channel[Z~ respectively.

Now, the Klter output error is given by

e(~-.f-1) =~,N-Z(~-I-1)~,N-Z(~+I)eq(k-I-I) (23)

By using the results given in Ref.[10] directly, a complete dual-channel QRD RLS lattice

algorithm is s , m m e ~ d in Tab.2. The prerotation method is discussed in detail in Ref.[ll].

Tab.2 Algor i thm II

(1) Initialization: (as same as Algorithm I);

> 0 Input signals: zz(k), z2(k); Desired signal: y(k); ezi,0(k ) = zz(k), e2~,,0(k) = z~(k)

(2) Prediction section:

i-----1

(a) j ---- 0 , 1 , . - . , N - 1

e~l,~+l(t)J L-sine,,~(t- z) cms,,~(k- 1) [ e,],~(t ) (b) j = N

[ E1/,O(k)] = [ c~ - 1) me,.N(k -- 1) ] [.xl/2E:,/,~,(' -- 1)] L-,,i~e,,.,v(k- z) ~e~, jv(k- 1) L ~t,~(,~)

(c) j = N , . . . , 2 , 1

1 L--~,,,~,i-1(k) o~,,,,j-1(J:).l L-,s,~-lq:)i

(d) j = N,-.. ,~, z

: [ [ 1 E,~,~(~) J L-sin~,,~(~) ,~o,o,~,~q:)j LZ~j_1(k)j

z~,,o(~) = ~,,~(~) "Y~,U- x (k) ---- [1 - )-~;_.~x Eibj (k)] 1/2

(e) j---- N - 1 , . . . , 1 , 0

(f) j---- 0 , 1 , . . . , N - I

e2j+~(k)J L - m e ~ j ( k ) co, e~,i(~)J L e2,j(k) J (g) i ---- 2, e2t.o(k) --- e2,N(k)~f/,N-z(k), repeat (s),,~(f) stelm.

(3) Joint process section:

.,(k)J t v(~)

I V . Sim~dAtions a n d D i s c u s s i o n

To show the algorithmic properties of the identification of parmneters, we consider a

system with two input variables[ s]

y(k) -- a0z l (k ) + alZ~(k - 1) + b0~2(k) + bt~2(k - 1) + ~(k)


where the input sequence8 {Zl(~)) and (z2(k)} are a second-order autoregressive process

produced by the drive of two uncorrelated Gaussian white noise, respectively. And v(k)

is an additive Gaussian sequence. The signal-to-noise ratio is 0dB for each of two input

sequences. ,~ = 0.99, N = 2. a0 = al = 1, b0 = 0.5, bl = 1.5.

For the Algorithm I, the LS estimated tap coei~cients are given as:

aj = y~,2~(k)/E~/2(k), ~,j = Z,~,2~+,(k)/E]/~(k)

For the Algorithm II, the LS estlrmtted tap coefficients are given as:

The average resuits of 20 individual runs are plotted in Fig.2. The simulation results

show that both multichannel QRD RLS algorithms proposed in this paper have performed

the LS estimation of the tap coefficients well.

1 ~ . . ~ " ~ . . . . . ~ . . . . , c , g 11, . " . . . . . - . .

Nsmkw d ~mmlms Nsmkm d IMmUims

Fig.2 The psrsmeter identification results for two channel signs~

The computational complexity is one of interesting problems for an adaptive algorithm.

The computational load proposed algorithmR is given in Txb.3 (for p-channel). In general,

the existing multichannel LS algorithm~ have a computational complexity of O(Np2). If the

LS estimated tap coefficients have to be computed in a QRD RLS algorithm~ however, an

extra computational complexity of O(N2p 2) has to be required to extract the tap coefficients

with a backsolve operation. We see that the computational complexity in those algorithn~

is very high in that case where the N is larger, as often is the case. Furthermore, the

backsolve process is not suitable for real-time recursive. The present Algorithm I requires a

computational complexity of O(Np3). The algorithm hM a higher computational complexity

in where the p is larger. For a smaller p, such as the application of parallel image processing,

adaptive equalization (p : 2), etc., however, it is still a computationally efficient algorithm

because the algorithm can in real-time obtain the tap coetiicients directly without a backsolve

operation. The present Algorithm II has a lower computational complexity of O(Np2). However, the algorithm is not suitable for the parallel processing due to the algorithm being

210 JOURNAL OF ELECTRONICS VoL13

in effect a cascade computation.

Tab.3 Computat ional complex i ty o f the p r ~ e n t a lgori thms Number of op~stionb Algorithm I Algorithm II

• (~ps + 10f + ~ ) ~ + 4 f +~p 2p2~+ lspN + zp § ~ N + p N +p 2pN +p ~/ p= N + pN -l- p 2pN

V . C o n c l u s i o n

In this paper, two QRD RLS lattice &]gorithms based on two dilferent approaches have

been proposed for the multichannel signals. The tap coefficients to be estimated in each

algorithm can be extracted directly from internal variables without a backsolve operation.

Hence the algorithn~ may be well suitable for the applications of the parameter identi-

fication. The superior numerical properties and a highly modular structure are expected

due to the derivation of the edgorithmn based solely on Givens rotations. Not only may be

implemented the algorithmg with VLSI technique, but with the commercinl digital signal

processor. It is expecte<i that this paper may expand the range of applications of the QRD

RLS algorithms.

References

[1] S. Haykin, Adaptive Filter Theory, Englewood Cli~, N J: Prantice Hall, 1985.

[2] K. EL. Liu, et s l . , /EEE Trims. on CA$, C&S-38(1991)6, 62~636.

[3] J. M. Ciom, [EEE Time. on ASSP, ASSP-345(1990)4, 631-653.

[4] F. Lin~ IEEE Tr~ms. on SP, SP-39(1991)7, 1541-1r~51.

[5] I. K. Prouddler, e t el., IEE Proc.-F, 138(1991)4, 341-3.53.

[6] P. S. Lewis, /EEE ~'4m~. on ASSP, ASSP-38(1990)3, 421-432.

[7] K. Zhao, et al., Mtdtk2mnnel Givens lattice adaptive algorithm, in Proc. IEEE ICASSP~I, Ontario,

Canad~ 1991, 184~-1852.

[8] M. G. B~l~*vr , et ~I., S/~na/Pmems/a& 3~(1991)2, 115-126.

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qrd-based multichannel adaptive lattice algorithms for the parameter identification problem

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