an adaptivealgorithmfor fast identification systems

Proceedings of the WeAO4.244th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 12-15, 2005

An Adaptive Algorithm for Fast Identification of IIR SystemsDa-Zheng Feng and Wei Xing Zheng

Abstract-This paper considers the problem of adaptiveidentification of IIR systems when the system output iscorrupted by noise. The standard recursive least squaresalgorithm is known to produce biased parameter estimates inthis case. A new type of fast recursive identification algorithmis proposed which is built upon approximate inverse poweriteration. The proposed adaptive algorithm can recursivelycompute the total least squares solution for unbiased adaptiveidentification of IIR systems. It is shown that the proposedadaptive algorithm has global convergence. The significantfeatures of the proposed adaptive algorithm include efficientcomputation of the fast gain vector, adaptation of the inverse-power iteration, and rank-one update of the augmentedcovariance matrix. The proposed adaptive algorithm issuperior to the standard recursive least squares algorithm andother recursive total least squares algorithms in such aspects asits ability for unbiased parameter estimation, its lowercomputational complexity, and its good long-term numericalstability. Computer simulation results that corroborate thetheoretical findings are presented.

I. INTRODUCTION

DURING the past decade, adaptive infinite impulseresponse (IIR) filters have found extensive applications

in many areas, such as system identification, adaptive noisecancellation, spectral estimation, channel estimation andequalization in communication systems and so on [1], [2],[3]. Adaptive IIR filters are considered as the efficientreplacements for adaptive finite-impulse-response (FIR)filters when the desired filter can be more economicallymodeled with poles and zeros only than with the all-zeroform of an FIR tapped-delay line. The possible benefits inreduced complexity and improved performance haveenhanced the applicability of adaptive IIR filters.

This paper is concerned with adaptive identification ofIIR systems when the system output is contaminated bynoise. There are generally two main classes of methods foradaptive identification of IIR systems in this scenario. Thefirst class of adaptive identification algorithms is the output-error (OE) method [4]. But the convergence as well asstability of the OE method is guaranteed only under theassumption that certain system transfer function is strictlypositive real [5]. Given the fact that it is a highly nonlinearalgorithm, the OE method has the difficulty in providing

This work was supported in part by the National Natural ScienceFoundation of China and in part by a research grant from the AustralianResearch Council.

D.-Z. Feng is with the National Laboratory for Radar Signal Processing,Xidian University, Xi'an, 710071, China.

W. X. Zheng is with the School of QMMS, University of WesternSydney, Penrith South DC NSW 1797, Australia.

unbiased parameter estimates. The second class of adaptiveestimation algorithms is the equation-error (EE) method [2].Since the system model adopted is linear, the EE method foradaptive IIR identification can operate in a stable mannerwhen the step size is properly selected. Moreover, the EEmethod has such attractive features as a unimodal errorsurface, good convergence and guaranteed stability whencompared with the OE method. However, the EE method(including particularly the standard recursive least squares(RLS) algorithm) usually produces biased parameterestimates for IIR systems subject to output noise [6]. Someefficient algorithms to remove the bias in adaptive EEidentification of IIR systems have been developed. Forexample, there are the instrumental variable algorithms [7],the hybrid algorithms [8], the unit-norm EE algorithms [9],the total least squares (TLS) algorithms [10] and the totalleast mean squares algorithm [11], to just mention a few.Among them, the Rayleight quotient (RQ) based TLSalgorithms [10] have proved to be a viable alternative forachieving unbiased estimates for adaptive identification ofIIR systems with output noise.

Regarding the computational complexity, the recursiveTLS (RTLS) algorithms normally involve O(L2) operations

per iteration, where L denotes the number of modelparameters in an IIR system. Similarly, other adaptivealgorithms (for instance, the inverse-power (IP) method

[12]) also require O(L2) operations per eigenvector update.For on-line solving the TLS problem in adaptiveidentification, the fast RTLS algorithm proposed in [10] isbased on the gradient search for the generalized RQ alongthe Kalman gain vector. This algorithm can fast track theeigenvector associated with the smallest eigenvalue of theaugmented autocorrelation matrix, since the Kalman gainvector can be fast estimated by taking advantage of the shiftstructure of the input data vector. The RTLS algorithm in[10] has computational complexity of O(L) per iteration,but is dependent on the fast computation of the Kalman gainvector. Unfortunately, it is a well-known fact that thecomputation of the Kalman gain vector may be potentiallyunstable [13].

In this paper, the problem of adaptive identification of IIRsystems subject to output noise is investigated from a newpoint of view. A fast recursive identification algorithm isderived by approximating the well-known inverse poweriteration along the data vector. Further, a fast scheme isdeveloped to compute the gain vector that is used inadaptation. With guaranteed global convergence, theproposed approximate inverse power (AIP) algorithm is ableto conduct recursive computation of the TLS solution so as

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to achieve unbiased identification of IIR systems. Theproposed algorithm is equipped with attractive features, suchas efficient computation of the fast gain vector, adaptation ofthe inverse-power iteration, and rank-one update of theaugmented covariance matrix. The proposed AIP algorithmis significantly advantageous over the standard RLSalgorithm because the RLS algorithm produces biasedestimates with computational complexity of O(L2) periteration while the AIP algorithm gives unbiased estimateswith computational complexity of O(L) per iteration.Moreover, the proposed algorithm outperforms the fastRTLS algorithm in [10]. First, the computational complexityof the AIP algorithm is much less than that of the fast RTLSalgorithm. Second, the long-term numerical stability of theproposed algorithm is much better than that of the fast RTLSalgorithm, thanks to its efficient computation scheme for thefast gain vector and its no use of the inverse of thecovariance matrix. The performance of the proposed AIPalgorithm is evaluated via simulation and is compared withthe standard RLS algorithm, the fast RTLS algorithm andthe IP method.

II. PRELIMINARIES

d(t) =[d(t -1), ,d(t-N +1)]T (2.6)Moreover, we have

d(t) =dN- (t- 1)+nN1(t 1) (2.7a)

nN-1 (t -)=[n(t - 1),n(t - 2), ,. n(t - N + 1)]T (2.7b)

dN- (t 1)= [d(t- 1),d(t - 2),...,d(t -N + 1)]T (2.7c)At time t the augmented data vector is defined as

-r(t) [d(t),rT(t)]T [d (t),xT(t)] (2.8a)d(t) = dN(t) +nN(t) = [d(t),dT(t)]T (2.8b)

The covariance matrix of the data vector is given by

R = E{r(t)r(t)}R T R R ± 2 LIN 0

where

The covarizdescribed by

where

I T dd dx

R = E{r(t)r (t)} RT

a dx mr oaince matrix of the augmented

R = E{r(t)r (t)}T=R

(2.9)

(2.10)

data vector is

I (2.11)

A. System DescriptionConsider an unknown IIR system and assume that only

the output is corrupted by the additive white Gaussian noise.Our task is to use an EE adaptive IIR filter to estimate theIIR system from the observations of the input and output.The parameter vector of the unknown IIR system is given by

h [al, a2, *, aN1, bo0Ib,I-, , bM_ ]ITE RLXl (2.1)where N and M are the denominator order and thenumerator order of the IIR system transfer function,respectively, and L = N +M -1. h may be time varying.Assume that the system orders N and M are known.

The noise-free system output is given byd(t) = rT(t)h (2.2)

where the noise-free data vector ri(t) is given by

r(t) = [dT(t) T (t)]T with d(t) = [d(t - 1 ,d(t -N +1)]T

and x(t) = [x(t), .., x(t M + 1)]T.The observation output can be represented as

At) Td(t) = d(t) + n(t) r (t)h + n(t) (2.3)where the measurement noise n(t) is a zero-mean Gaussian

white noise with variance u 2, independent of the inputsignal x(t). The output for a sufficient-order EE adaptiveIIR filter is given by

N-1 M-1y(t) Yiamd(t - m) + bmx(t - m) = rT(t)w (2.4)

mi=l m=0

where w =[aT, bT]T is the adjustable parameter vector,

r(t) = [dT (t), XT (t)]T is the noisy data vector, and

a = [al, aN-1 ]I b = [bo, ,bMl ]T

g=E{r(t)d(t)} dd = Rh

c E{d(t)d(t)} = hTRh +±70

(2.12)

(2.13)So the covariance matrix of the augmented data vector canbe written as

_~ ChTRh±ob hTR] R± (2.14)

where R* ! I 'R and R0= oK ]. It is easy to

verify that RL hi 0. This means that if R has full rank,

then R* is rank-one deficient.

B. Equation Error MethodThe equation error (EE) [6], [9] is defined as

e(t) y(t) -d(t) wTr(t) -d(t) w r(t) (2.15)where w = [1,wT]T = [-1 aT bT]T E R(L+i)xl Under thenatural 'monic' constraint, the variance of the EE is given by

E{e2(t)} w Rw wTR*w+ [aTa (2.16)where a = [a o, aT]T . The first term on the right-hand side of(2.16) is associated with the noise-free situation, while thesecond term is seen to add an interference that will producethe estimation bias. This shows that the least-squares-typecost function (2.16) based on the 'monic' constraint does notyield the unbiased estimate for IIR systems subject to outputnoise. If aTa is constrained to 1, then (2.16) becomes

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E{e2(t)} w R*W + (2.17)Since the presence of the measurement noise adds only aconstant term to the EE variance, the solution obtained byminimizing the cost function (2.17) does not vary with thenoise variance. However, the final solution has to beobtained by the following scaling operation

W =-[W]2,L+ La0 (2.18)

Note that for a vector u [ull,U2, ... , UL ]T E RLx1, we define

[U]m n = [um, um.+l .. un ]TER(n-m+l)xl for 1<m<n<L. In

order to find the solution for adaptive IIR filtering, the least-squares-type cost function in [6], [9] is to minimize

E{e2(t)} wTRw subject to a 1 (2.19)On the other hand, to efficiently seek the TLS solution foradaptive estimation of IIR systems, the following Rayleighquotient (RQ) is established in [10]:

J{w(t)}= wT (2.20)w Dw

where D =diag{lINxN OMxM}. Clearly, (2.20) is like (2.19).Now consider minimizing the RQ J(w). Forcing thegradient of J(w) with respect to w to be equal to zeroleads to the special generalized eigenvalue decompositionassociated with the matrix pair (R, D) , that is,

Rw -Da 0, aTa = 1 (2.21)This shows that the unbiased parameter estimate of IIRsystems can be also achieved by finding the generalizedeigenvector associated with the smallest generalizedeigenvalue of the matrix pair (R, D) .

C. Inverse Power IterationThe inverse-power (IP) iteration for finding the

eigenvector associated with the smallest generalizedeigenvalue of the matrix pair (R,D) is as follows [14].Randomly produce the initial value w(0), for t 1, 2,-.solve a set oflinear equations

Rw(t) = Dw(t -1) (2.22)andperform thefollowing normalization

w(t) = w(t)/ lw(t)ll (2.23)It can be seen that the core of the IP iteration is (2.22).

Moreover, it is shown in [14] that the IP iteration globallyexponentially converges to the eigenvector associated withthe smallest generalized eigenvalue of the matrix pair(R,D)). Unfortunately, the IP iteration is an algorithm with

computational complexity O(L3). Note that if the inverse

update formula of R is used like the RLS algorithms, the IPiteration will become an algorithm with computationalcomplexity O(L2). However, the potential instability of theinverse update formula of the covariance matrix may causethe potential instability of such IP iteration.

Since we expect that the non-normalized eigenvectorassociated with the smallest generalized eigenvalue of the

matrix pair (R, D) has the first entry to be fixed as -1, wemay implement an IP iteration in conjunction with monicnormalization, which is given as follows.

Randomly produce the initial value w(O) and let[w(0)] l = -1, for t = 1,2,

solve a set oflinear equationsRwv(t) = Dw(t -1) (2.24)

andperform thefollowing monic normalizationwv(t) = w(t)/[w^(O)]lI (2.25)

Using the eigenvalue decomposition (EVD), it can beshown that w(t) given by the simple IP iteration (2.24) and(2.25) globally exponentially converges to the trueaugmented parameter vector [- 1, hrT T as t -> oo.

For implementation convenience, we may combine (2.24)and (2.25) together to get the following compact IP iteration.Randomly produce the initial value w(O), for t = 1, 2,solve a set oflinear equations

-1 wT T - T T (2.26)Note that it is straightforward to establish the equivalence

of (2.26) to (2.24) and (2.25).

III. THE APPROXIMATE INVERSE POWER ALGORITHM

In this section, we develop an efficient algorithm forfinding the TLS solution of the adaptive IIR filteringproblem. This algorithm is an approximate inverse-power(AIP) iteration. Choosing update direction to be the datavector will also give rise to the computationally efficientalgorithm with computational complexity O(L).

The basic idea is to update the parameter vector in (2.26)by the following rank-one updating formula

w(t) = w(t - 1) + ,(t)r(t) (3.1 a)or equivalently

[_I, WT (t)]T = [_I, WT (t-_1)]T + ,8(t)[O, r T(t)]T (3. lb)Substituting (3.1) into (2.26) yields the approximate formula

R(t)[-l,wT(t-1) 8,6(t)r (t)] y(t)D[ 1,wT (t 1)]T(3.2)

where ,8(t) and y(t) can be efficiently determined in O(L)multiplications by the following equations

[O,r (t)]{R(t)[-1,(w(t-1) + ,(t)r(t))T ] (3

-y(t)D[-1,wT(t-1)]T} 0

[-1, wT (t-1)] {R(t)[-, (w(t-1) + p(t)r(t))T ]_-/(t)D[_1, wT (t-_1)]T I = o

(3.3b)

Notice that R(t) in adaptive identification is computed by arecursive formula

R(t)= () P)=R(t _1)+ (t)-T(t)Lg(t) R(t)jrwhere

c(t) = uc(t -1) + d(t)d(t)

(3.4)

(3.5a)

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g(t) pug(t - 1) + d(t)r(t) (3.5b)R(t) puR(t - 1) + r(t)rT (t) (3.5c)

The ,u in (3.4) and (3.5) is the forgetting factor.Substituting (3.4) and (3.5) into (3.3) yields

[r (t)g(t), rT (t)R(t)] [-1, (w(t -1) + fl(t)r(t))T ]T (3 .6a)

-Y(t)Ya (t) = 0

[_I, wT (t- 1)]R(t)[-1, WT (t _1)]T_,(t)[_1, wT(t -1)] [rT (t)g(t), rT (t)R(t)]T (3.6b)

+y(t)[±+aT(t -)a(t- 1)] = 0where

Ya(t) = aT (t - I)d(t) (3.7)After some manipulations, (3.6) can be further representedas

,8(t)[rT (t)R(t)r(t)] - Y(t)y, (t) (.a-rT(t)R(t)w(t -1) + rT(t)g(t)

-,(t)[rT (t)R(t)w(t -1)- rT (t)g(t)]

+±y(t)[1 + aT (t - I)a(t - 1)] (3.8b)[-I,w 1t-)]R(t)[-11 wT t1]

Let

all (t) = r T (t)R(t)w(t - 1) (3.9a)b1(t) = _rT (t)R(t)w(t - 1) + rT (t)g(t) (3.9b)a22 (t) 1+±aT (t - I)a(t - 1) (3.9c)

b2(t) = [1,wT(t 1)]R(t)[-1,wT(t 1)]T (3.9d)The matrix form of (3.8) is then given by

Fa, I (t) -Ya (t) P(t)l Fb (t)l (3.10)b1(t) a22 (t) ILy(t)] Lb2 (t)j

In order to compute efficiently the coefficients of (3.10), letk(t) = R(t)r(t) (3.11)bo (t) =[_I,wW(t)]R(t)[ 1,wT(t)]T (3.12)

Here the gain vector k(t) can be fast computed as shown inTable I.

TABLE IFAST SCHEME FOR COMPUTING THE GAIN VECTOR

Algorithm MAD'sInitialize BL2 (0) = 0,BL2 (0) = O, r22 (0) 0.

BL2 (t)= BL (-1) + r(t-_1)4T (t) 2L

BL2(t LBL2(t1) + r(t)p T(t) 2LL2 (t) = 322 (t-1 424()4

Fml(t) 1Qj k( t t

k(t)( t)t) -BL2(t)P2B(t) 2LTotal real MAD's: lOL + 8

Note that the fast scheme for the gain vector k(t) shownin Table I is derived by the approach similar to [15]. Itshould be emphasized that unlike the well-known Kalmangain vector that is based on the matrix-inversion lemma andcan be numerically unstable, the fast scheme for computingk(t) is independent of the matrix-inversion lemma, therebybeing numerically stable.

Using the gain vector k(t), the coefficients all(t) andb1 (t) can be efficiently computed by

all (t) = rT(t)k(t) (3.13)b1(t) =-kT(t)w(t -1) + rT (t)g(t) (3.14)

Note that with the gain vector k(t) and the coefficients

a, I(t), b1 (t) and y(t), b2 (t) and b2 (t) can be efficientlycomputed by

b2(t)= [_I,WT(t )] [yR(t-1)+ r(t)-rT(t)][-1,wT(t-_1)]T= ,ub2° (t - 1) + [y(t) - d(t)]2

(3.15)bo (t)=[ 1, wT (t _1) + P(t)r T(t)]T R(t)

1, wT(t 1) +,(t)rT (t)]T

= b2(t) + 2,8(t)[kT(t)w(t 1) gT(t)r(t)] (3.16)+ ,82 (t)kT (t)r(t)

= b2 (t)- 2,8(t)bl (t) + ,32 (t)a,1 (t)Solving (3.10), it follows that

,8(t) = [a22 (t)bl (t) + Ya (t)b2 (t)]1[all (t)a22 (t) ± Ya (t)bl (t)](3.17)

TABLE IIAIP ALGORITHM

Initialize: w(0) = [0,0,...0]T, b°(0) 0, ,= 0.99 - 1.0For t = 1,2,.. MAD's1 update the data vector r(t)2 update gain k(t) (see Table I) lOL + 8

Ya(t) = aT (t - I)d(t)3 ybt)= b (t)x(t) L

y(t) Ya (t) + Yb (t)4 g(t) ,ug(t - 1) + r(t)d(t) 2L

5 a1l(t)= rT(t)k(t) L

6 b1 (t) = _kT (t)w(t - 1) + rT (t)g(t) 2L

7 a22(t) = + aT (t - I)a(t - 1) N - 1

8 b2 (t) pubo (t - 1) + (y(t) - d(t))2 2

8 /(t) = [a22 (t)b1 (t) + Ya (t)b2 (t)] 5/[a1, (t)a22 (t) ± Ya (t)b1 (t)]

9 w(t) = w(t - 1) + /(t)r(t) L

10 bo(t)=bb (t)t)[/(t)a11 (t) - 2b1 (t)] 3Total real MAD's: 17L + N +17

4260


Notice that y(t) is not required for finding the TLS solutionof adaptive IIR filtering. The above algorithm is called theapproximate inverse-power (AIP) algorithm and issummarized in Table II.

Since the monic normalization is adopted, the above tabledoes not include the tenth and eleventh manipulations inTable 1 in [10], which saves about 2L MAD's, where1\AD's stands for the number of multiplies and divides.Moreover, since the parameter vector being tracked isreduced to L dimensions from L +1 dimensions, moremanipulations have been saved. It can be shown bycomputing the MAD's of each manipulations that theMAD's of the AIP algorithms is 17L + N +17, while theMAD's of the RTLS algorithm [10] is 19L + 3N + 74. Thisindicates that the AIP algorithm is able to achieve asubstantial reduction in the computational complexity fromthe RTLS algorithm.

IV. THEORETICAL ANALYSIS

Performing the generalized eigenvalue decomposition(GEVD) of the matrix pair (R, D) gives

RV DVdiag(y/,/I ... /{1) or Rv 2=1Dvj (4. 1)

where V is the generalized eigenvector matrix, and Vj

and Ai are the j-th generalized eigenvector and eigenvalue,respectively. Note that the eigenvalues are arranged in adescending order Al > A2 > 2 AL > AL+1 This means that

D and R have the EVD: V DV = and V RV= (,where r = diag(p, P2 ... PL+1 I (D = diag(91,92... 9PL ) I

and A, = pilpi ( 1,.,L + 1).The following lemma is obvious.Lemma 4.1 If R is of full rank, then AL > AL+1By substituting w-[ 1,wT]T into (2.20), we have the

cost functionJ(w) = [-1,wT]R[_ WT]T /(1+ aTa) (4.1)

The next Theorem 4.1 guarantees that we can search theglobal minimum point of J(w) by the gradient descentmethod. Its proof is omitted here due to limited space.

Theorem 4.1 If AL > AL+, and V-L 1 0, where V1-L1 i3the (L + 1) -th element of the first row of V, then

WL±1 VL / V1 L+1 is the global minimum point of J(w).All the other stationary points are the saddle points ofJ(w).We can show that the proposed AIP algorithm is globally

convergent. To this end, a lemma is first stated.Lemma 4.2 If t is large enough so that R(t) -> R, then

it always holds that J(w(t)) - J(w(t -1)) < 0.Theorem 4.2 Assuming that t is large enough so that

R(t) -> R , then w(t) -> h with probability 1 as t X oo.

40

mS 30

0

*E 10U]

-10

0 500 1000 1500 2000Iteration number

Fig. 1. AIP, IP, RLS and RTLS are used to identify a linear time-invariant IIR system. The errors denoted by dash-dot line, dashed line,dotted line and solid line are obtained by AIP, IP, RLS and RTLS,respectively.

60

50

E 40-03° 30

20

EI

0

-10

Onf-

--- RLSPTLSAIP

-- IP

\1

~~~~-1 j~~

0 500 1000 1500 2000Iteration number

Fig. 2. AIP, IP, RLS and RTLS are used to identify a linear time-varying IIR system. The errors denoted by dash-dot line, dashed line,dotted line and solid line are obtained by AIP, IP, RLS and RTLS,respectively.

The proofs of Lemma 4.2 and Theorem 4.2 are omittedhere due to limited space.

V. SIMULATION RESULTS

Computer simulations have been conducted to verify theperformance of the proposed AIP algorithm. Here all thesimulation results are averaged over 30 independent tests.

The unknown linear IIR system is defined by [16]1+0.4z -2 - 0.5z-3 + 0.7z-4

H(z) -1 Iz1- +0.8z-2 -0.88z-3 +0.64z-4Thus, N = 5, M = 5 and L = N +M -1 = 9. The standardRLS, the RTLS in [10], the IP in [12], and the proposed AIPalgorithms are applied to the system identificationexperiment for comparison. The IIR system is excited by the

4261

--- RLSPTLS

-AIP--IP

11I11l~~~~~~~~l1

bU

'U

,A

)nZ-v-21


colored input signal x(t), which is generated by the first-order AR model x(t) -0.25x(t -1) =(t), where c(t) is awhite noise with unit variance. The IIR system output iscontaminated by additive, zero-mean and white Gaussiannoise with a unit variance, which is statistically independentof the input signal x(t). The forgetting factor ,u used insimulation is chosen to be 0.99. The estimation error isdefined by

E(t) = llw(t) - h112 /11hl 2

where h =[1.1, -0.8, 0.88, -0.64, 1.0, 0.0, 0.4, 0.5,0.7]TWhen the system is linear time-invariant, the estimation

results of the RLS, RTLS, IP and AIP algorithms are shownin Fig. 1. In order to test the tracking behavior of the relativealgorithms in a nonstationary environment, the parameterestimation experiment is repeated, but each of the numeratorparameters of the unknown IIR system will undergo a signchange at t =1001. The obtained simulation results areplotted in Fig. 2. It is interesting to see from Figs. 1 and 2that the AIP algorithm has the slightly slow convergence inthe first segment of the learning curves, but it has thesignificantly small statistical fluctuation in comparison withthe IP and RTLS algorithms. Note that the AIP algorithm isimplemented at a numerical cost much lower than the RTLSalgorithm. As expected, the standard RLS algorithm fails towork because it produces biased parameter estimates. Thenumerical stability of the AIP algorithm has also been testedvia a long-time (t=2 x105) experiment with satisfactoryresults as shown in Fig. 3.

-101 Iil 111 AIkild , II ii, I 111l1lI III .. il .IkiJ.1[.1I,11 i

Iteration number 5x 1

Fig. 3. The long-time numerical stability of the proposed AIP algorithm istested.

VI. CONCLUDING REMARKS

The problem of adaptive identification of IIR systemssubject to output noise has been studied in this paper. A fastalgorithm has been proposed to recursively compute theTLS solution for unbiased estimation of IIR systems. Themain idea is to efficiently compute the non-normalizedeigenvector associated with the smallest eigenvalue of thesample covariance matrix via approximate inverse-poweriteration and fast gain vector computation. It has beendemonstrated that in addition to its good long-term

numerical stability, the proposed AIP algorithm has asignificant computational advantage over the fast RTLSalgorithm in [10] and the IP algorithm in [12]. The globalconvergence and the estimation unbiasedness of theproposed algorithm have been established. The goodperfornance of the AIP algorithm has been verified viacomputation simulations.

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