Adaptive maximum likelihood algorithms for the blind tracking of time-varying multipath channels

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 12, 207—222 (1998)

Int. J. Adapt. Control Signal Process., 12, 207—222 (1998)

ADAPTIVE MAXIMUM LIKELIHOOD ALGORITHMSFOR THE BLIND TRACKING OF TIME-VARYING

MULTIPATH CHANNELS

JOE® L GROUFFAUD1,*, PASCAL LARZABAL1, ANNE FERREOL2 AND HENRI CLERGEOT3

1 LESiR, ENS de CACHAN, URA CNRS D1375 61, avenue du President Wilson, 94235 Cachan, France2 THOMSON CSF Communications 66, rue du Fosse blanc, 92231 Gennevilliers, France

3 C.R.S.T.G. avenue d’Estree, BP 792, 97337 Cayenne, Guyane, France

SUMMARY

Transmissions through multipath channels suffer from Rayleigh fading and intersymbol interference. Thiscan be overcome by sending a (known) training sequence and identifying the channel (active identification).However, in a non-stationary context, the channel model has to be updated by periodically sending thetraining sequence, thus reducing the transmission rate. We address herein the problem of blind identification,which does not require such a sequence and allows a higher transmission rate. We have first proposeda two-stage algorithm (see Reference 2) for the blind identification of multipath channel. We investigate herethe maximum-likelihood approach for the blind estimation of channel parameters. In order to tracknon-stationary channels, we have derived an adaptive (Kalman) algorithm which directly estimates theentire set of characteristic parameters. An original adaptive estimation of the noise model has been proposedfor this investigation. Furthermore, the proposed method can easily cope with a model including Dopplershift, which is not directly possible with more common methods. Monte-Carlo simulations confirm theexpected results and demonstrate the performance. ( 1998 John Wiley & Sons, Ltd.

Key words: Array signal processing in wireless communications; blind identification of multipath channel;tracking of nonstationary channels.

1. INTRODUCTION

1.1. Background

Mobile communications in urban areas are subjected to multipath propagation. This causesRayleigh fading and intersymbol interference, thereby deteriorating the transmission. Knowledgeof the propagation channel is then crucial in order to perform an effective equalization.

The active identification consists of sending a known training sequence, whose correspondingresponse through the channel leads, after a parametric estimation, to the propagation conditions.1Propagation channels are almost always time-varying: estimated parameters have to be updatedby periodically sending the training sequence, reducing thereby the effective transmission rate.

*Correspondence to: J. Grouffaud. LESiR, ENS de CACHAN, URA CNRS D1375, 61, avenue du President Wilson,94235 Cachan, France.

CCC 0890—6327/98/020207—16$17.50( 1998 John Wiley & Sons, Ltd.

A two-stage algorithm for blind identification has been proposed in Reference 2. It needs notraining sequence, thus, allowing a higher transmission rate. The first stage uses recent results onblind identification of multichannel,3,4 which can be naturally connected to the conditionalmaximum-likelihood method (CML).2,5 The second stage is a parametric identification usinga spatio-temporal high-resolution method.2

Generally, the channel is time-varying, due either to the movement of the mobile or tovariations in the environment itself (wind in trees, interference with other vehicles, etc.). In sucha context, these algorithms have to be made adaptive. Rather than working on doubly adaptivealgorithms (due to the two stages), we wish to perform a direct estimation using a parametricmodel of propagation, from which we can express the log-likelihood to be maximized.

Moreover, the movement of the mobile induces Doppler shift on the emitted signal, whichcannot be taken into account by a simple convolution with the impulse response. For all thesereasons, high-performance receivers need accurate instantaneous propagation conditions. Thispaper focuses on the adaptive ML estimation of these parameters.

This paper describes the CML method applied to parametric estimation and providesa Gauss—Newton procedure for implementation. In addition, an original adaptive scheme isproposed for tracking non-stationary channels. Analysis, numerical stability and comparativestudy are performed by simulations.

1.2. Outline of paper

The organization of this paper is as follows. In Section 2, we propose a parametric model ofthe channel propagation including group delay, azimuth and elevation angle, and Doppler shift.This model will be demonstrated by means of the impulse response of this channel. Problemformulation and feasibility are exposed in Section 3. Section 4 introduces the correspondinglog-likelihood and provides an ML estimate of the parameters. In Section 5, a recursiveimplementation of ML estimation, based on estimation and covariance update, is proposed forcomputational consideration. Computer simulations exhibit convergence behaviour. A generaliz-ation for time-varying parameters is proposed in Section 6 such as an adaptive Kalman filtering.For this, an original adaptive estimator of noise modelling is introduced. Tracking properties areanalysed by computer simulations. Finally, the conclusion and future works are presented inSection 7.

1.3. Notations and definitions

A lower-case boldface letter denotes a vector while an upper-case boldface letter denotesa matrix. We use furthermore the following notations:

(.)T transpose(.)H hermitian transpose(.)d pseudo-inversetr( .) tracexi

ith element of vector xX

ij(i, j) entry of matrix X

I identity matrixE.E euclidean norms( . ) continuous-time signal

208 J. GROUFFAUD E¹ A¸.

Int. J. Adapt. Control Signal Process., 12, 207—222 (1998) ( 1998 John Wiley & Sons, Ltd.

s[.] discrete-time signald(t) dirac delta functionsinc(.) function defined by

Gsinc(t)"sin(nt)

ntif tO0

sinc(0)"1

E[.] expected valueN number of sensorsK number of samples for a single spatio-temporal observationM number of paths¸ order of the channel impulse response

For the variable x (scalar, vector or matrix), xL represents the estimate generally according to theMaximum-Likelihood (ML) criterion.

2. A PARAMETRIC MODEL OF CHANNEL PROPAGATION

A radio channel can be depicted by M paths each one characterized by qm, h

m, *

m(m"1,2 , M),

which are the group delay, azimuth and elevation angles, respectively, while am

and /m

are thecorresponding attenuation and phase. The emitted signal is supposed to be received by anN-sensor array. The complex, low-pass impulse response between the emitter and the ith sensor,noted h

i(t), is2,6,7

hi(t)"

M+

m/1

amai(h

m, *

m) d (t!q

m)e+(m for i"1,2 , N, (1)

where ai(h, *) is the response of the ith sensor in the bearings (h, *) relative to a reference sensor.8

It has been assumed in this work that the incoming wave fronts are plane.In this paper, we assume that the channel, described in Figure 1, is ideal; its response is uniform

within the passband and zero elsewhere. The numerical impulse response (finite and causal)is then:2

hi[l]"GFs

M+

m/1

amai(h

m)e+(m sinc(l!F

sqm), l3M0,1,2 ,¸N

0, l(0 or l'¸

(2)

A more complete model could be used, including Doppler shift lm

due to motion of the tranmitter,receiver or scatterers.9 However, the input/output relationship is no longer a convolution:

xi[k]"

L+l/0

hi[k,l] s[k!l]#n[k] (3)

with

hi[k,l]"

M+

m/1

h (m)i

[l]ei(2nklmTs`(0,m) for l3M0, 1,2 ,¸N (4)

h(m)i

[)] is the contribution of the mth path in the impulse response of the ith channel (see (2)), andthe initial phase /

0,mwill be attached, in the sequel, to /

m.

ADAPTIVE MAXIMUM LIKELIHOOD ALGORITHMS 209

Int. J. Adapt. Control Signal Process., 12, 207—222 (1998)( 1998 John Wiley & Sons, Ltd.

Figure 1. A digital transmission through a multipath channel

From a theoretical point of view, it seems that Doppler shifts could be estimated rather simply.However, these parameters are strongly connected to propagation conditions, and this delicateproblem is not investigated in this first approach.

3. FEASIBILITY AND PROBLEM FORMULATION

3.1. Feasibility study

The blind deconvolution is aimed at estimating the N impulse responses hi[k], without

knowing the emitted sequence. The question is whether it is possible to find N filtersgi[k] allowing the inversion of the input/output relationship, as explained in Figure 2, in the

noiseless case.By taking the z transform, it follows that

Xi(z)"H

i(z)S (z) for i"1,2 , N (5)

If S(z) is a polynomial and if H1(z),2 ,H

N(z) (which are polynomials of degree ¸ since they

correspond to FIR filters) are coprime, S (z) is the highest common factor of X1(z),2 ,X

N(z).

According to the generalized Bezout theorem, there exist N polynomials G1(z),2 , G

N(z)

such that

N+i/1

Gi(z)H

i(z)"1 (6)

Figure 2. Framework of the multichannel deconvolution



The blind deconvolution of FIR multichannel is realized by

S (z)"N+i/1

Gi(z)X

i(z) (7)

3.2. Representation of spatio-temporal data

The output of the ith sensor at the time k is given by the convolution product

xi[k]"

L+l/0

hi[l] s[k!l]#b

i[k] (8)

The data can be written under vectorial shape as the set of the outputs of the ith sensor forK successive snapshots

xi"[x

i[K] 2 x

i[1]]T (9)

It can be written, according to the emitted signal values

xi"H

is[K,K#¸]#n

i, (10)

with Hibeing a K](K#¸) Sylvester matrix

Hi"

hi[0] 2 h

i[¸] 0 2 0

0 hi[0] 2 h

i[¸] } F

F } } } 0

0 2 0 hi[0] 2 h

i[¸]

(11)

and

s[K,K#¸]"[s[K] 2 s[1!¸]]T (12)

Finally, the whole set of data can be gathered into a spatio-temporal vector:

x"[xT12 xT

N]T (13)

which may be written as follows:x"H s[K, K#¸]#n (14)

withH"[HT

12 HT

N]T (15)

H is a Sylvester matrix made up of the N impulse responses. Including Doppler shift poses notheoretical problem: H has to be filled according to (4). The only difference will be that its rowsare no longer identical (up to a right shift).

According to (2),H is actually a functionH(#), where # is the vector of unknown parameters:

#"[a1

/1

h1

*1

q1

l1

,2 , *M

qM

lM]T (16)

3.3. Formulation of the problem

Given only the data collected on the array of sensors, we intend to identify the propagationchannel (characterized by the set of physical parameters #) in a blind way, that is without



knowing the transmitted signals, whose estimation is furthermore achieved after channel equal-ization. Tracking of the propagation channel (almost always time varying) has to be considered bythe use of an adaptive algorithm.

Remark 1

The channel model, defined by (2), is characterized by integer parameters. Some of these areknown and depend on the system (N and K), while others are unknown and depend onpropagation conditions, and may be time-varying (M and ¸). In an operational system, theirvalue has to be detected and updated periodically. However, we are assuming in this paper thatthese parameters are known. Study of the model order selection has been widely discussed (seeReference 10), and we have already highlighted algorithms for the detection in the specificproblem addressed herein.11

Remark 2

Some uncertainties lie in the problem as it stands. Indeed, some constraints have to be imposed.From the shape of the input/output relationship (14), it turns out that multiplying each gainamei(m by any complex scalar b amouts to working with the signal bs ( )). So the constraint may be

(1) a linear constraint. We shall take in the following simulations aL1ei(K 1"1 (i.e. aL

1"1) and

/K1"0.

(2) a power constraint for the gains am: +M

i/1aL 2m"A, where A is the total multipath power gain,

(3) a power constraint for the emitted signal: sHs"&, where & is the power of the signal.

Moreover, estimating the M group delays separately is not possible, because no temporalreference is available (the input signal is totally unknown). Actually, only the differential groupdelays Mq

2!q

1,2 , q

M!q

1N can be estimated. This can be easily accomplished by imposing

q1"0.

4. MAXIMUM-LIKELIHOOD ESTIMATION

4.1. Conditional Maximum-Likelihood method (CML)

The input s[k] is considered as an unknown, with the parameters to be estimated being thephysical parameters of the impulse responses. The only random data element is the additive noise.Under the assumption of white (both spatially and temporally) Gaussian noise, of power p2, thelikelihood of observations on sensors is

p (x DH,s)"1

(J2np2)NK

expG!1

2p2Ex!HsE2H (17)

The CML criterion leads to the minimization of

Q(s,H)"Ex!HsE2"Ex!xE2 (18)

In order to eliminate s, we must search for the minimum of Q relative to s, which will depend onlyon H.



Let us note that (23) can be written as

Q(s,H)"tr((x!x)(x!x)H) (19)

which leads to

dQ"tr(G ) dsH#GH ) ds) (20)

with

G"HHHs!HHx. (21)

Subject to H being of full column rank,3 we can easily deduce, from (20) and (21), the CMLestimation of s:

s"(HHH)~1HHx (22)

and of x:

x"H(HHH)~1HHx (23)

The criterion to be minimized becomes

Q(H)"Ex!%HxE2"E%MHxE2 (24)

where %H"H(HHH)~1HH is the projector on the subspace spanned by the columns ofH (‘signal subspace’), and %MH is the projector on the orthogonal subspace (‘noise subspace’). Thiscriterion depends no longer on s, and the minimization is thus first performed with respect to #.Afterwards, the equalization can simply be achieved by using (22).

Remark 3

If many independent observations xp(p"1,2 , P) are available, the likelihood takes the form:

p (x1,2 , x

PDH, s)"

1

(J2np2)NKPexpG!

1

2p2

P+

p/1

Exp!HsE2H (25)

while the criterion to be minimized can be expressed in terms of their covariance matrix

Q(H)"tr(%MHRx) (26)

with

Rx"

1

P

P+p/1

xpxHp

(27)

4.2. Optimization procedure

This problem can be related to the well-known ML search of directions-of-arrival (D.O.A.) inarray processing. Optimization of this problem can be conducted by some conventional methodslike Wax and Ziskind,12 E.M.,13 and Gauss-Newton. Usually, the final stage of optimization isa Gauss—Newton descent. Equations are derived in the appendix.



Remark 4

In order to obtain low computational cost, some suboptimal methods have been proposed in theliterature.8,14 They are based on the concept of signal and noise subspace orthogonality. Thetracking of time-varying parameters can be performed by adaptive subspace tracking algorithms.However, they exhibit from two main limitations:

(1) they do not produce perfectly orthogonal eigenvectors and thus require additionalcomputation;

(2) they are sensitive to a few outliers that are far from the bulk of the data.

For these reasons of suboptimality and inefficiency in the tracking algorithms, we proposea scheme of blind parametric tracking with an ML method.

5. RECURSIVE ML IMPLEMENTATION

The maximum-likelihood method opens the way for new tracking algorithms. Moreover, thesealgorithms can easily cope with Gaussian prior on the parameters to estimate. All the ingredientsare present for the design of a Kalman-like adaptive estimator. In the proposed algorithm, anupdate can be obtained with just a few snapshots. In the case of one snapshot, the algorithm takesa very particular form.

5.1. Taking into account prior information

(a) Combination of efficient estimates. Let us note #1

and #2

as two independent efficientestimations of the vector #; they are assumed to follow normal laws, N(#,!

1) and N(#,!

2),

respectively. For efficient estimation, the log-likelihood is quadratic, and the value #0

of # thatmaximizes p (X D#

1) ) p (X D#

2) is easily obtained by maximizing the sum of log-likelihoods

given by

L(#)"(#!#1)H !~1

1(#!#

1)#(#!#

2)H!~1

2(#!#

2). (28)

The identification with (#!#0)H!~1

0(#!#

0) leads to

#0"(!~1

1#!~1

2) (!~1

1#

1#!~1

2#

2), (29a)

!~10

"!~11

#!~12

. (29b)

(b) Combination of one estimate with a new observation. As opposed to the previous case, onlyone efficient estimate and a new observation are available. In fact, the new observation is unableto provide alone an efficient estimation for the parameters. We assume that # is normallydistributed with the density being, at time t, N(#ª

t,!

t). Furthermore, the log-likelihood at time

t can be written according to its second-order expansion (quadratic approximation) using thegradient and the Hessian of the log-likelihood (24):

L(X D#)"+Ht(#!#ª

t)#1

2(#!#ª

t)HH

t(#!#ª

t). (30)

It is assumed that the resulting log-likelihood function (after integration of the new measure-ments) is sufficiently sharply peaked around #ª

t, so that the second-order approximation remains



Figure 3. Maximum likelihood and recursive implementation

valid over the interval [#ªt, #ª

t`1]. The resulting log-likelihood (up to a constant term) is obtained

from Bayes integration formula:

L(#DX)"L(X D#)#12(#!#ª

t)H !~1

t(#!#ª

t) (31)

The identification with the a posteriori Gaussian distribution N(#ªt`1

, !t`1

) leads to themaximum a posteriori (MAP) estimate

#ªt`1

"#ªt!!

t)+

t!~1

t`1"!~1

t#H

t

(32)

Figure 3 provides an overview of the recursive implementation of the ML method for the updateof estimation and covariance. Both gradient and Hessian are calculated from the new data alone(see the appendix), and are combined with a priori estimate (at time t) to give the a posterioriestimate (at time t#1), according to (32).

5.2. Simulations

Computer simulations were carried out to demonstrate the applicability and the performanceof the proposed algorithm. It is not the aim of this paper to conduct a comparative study withsubspace tracking methods. The large number of subspace tracking algorithms available in theliterature and the extreme difficulty in comparing their convergence rate, steady-state perfor-mance, accuracy and robustness from a theoretical point of view are two reasons why a specificpublication is necessary. However, estimated values obtained after convergence of the recursivealgorithm will be compared to the one provided by a subspace algorithm,3 applied to the wholeset of data used by the former.

All of the following simulations have been conducted for a 5-element linear array withequispaced sensors. A spatio-temporal observation has been built from K"20 samples (it is thusa 100-dimensional vector). Five snapshots were used to compute the covariance matrix (27), andthe gradient and Hessian (see the appendix). The input signal type is FSK, and the received dataare sampled at four times the baud rate. The channel is represented by two paths, whose mainparameters are given in Table I.



Table I

m am

/m

hm

*m

qmF4

lm¹

4

1 1 0 90° 30° 0)2 02 0)7 20° 90° 55° 2)9 0)012

Figure 4. Root-mean-square error versus time for different SNR

The order of the impulse response is assumed to be known in this study (see Remark 1). Wehave determined this on line by applying the minimum description length (MDL) criterion10 onthe covariance matrix, and the selected value is ¸"4.

Root-mean-square error (RMSE) is used to measure the accuracy of parameter estimates:

RMSE"S -%/'5)(#)+i/1

A#

i!#ª

i#

iB2

(33)

We have studied the convergence of this algorithm by plotting in Figure 4 RMSE versus iterationnumber for many values of the signal-to-noise ratio (SNR).

Results have been compared to those given by a subspace algorithm, using the covariancematrix calculated with all the spatio-temporal observations. ML algorithm exhibits betterperformance than subspace method. This can be related to a well-known result for the direction-of-arrival estimation: subspace methods (e.g. MUSIC) are only suboptimal approximations ofML method. The two approaches are asymptotically equivalent only if the emitted signal is white,which is not the case in these simulations (FSK).



6. AN ADAPTIVE ALGORITHM FOR TIME-VARYING CHANNELS

6.1. Dynamic model

For time-varying channels, a dynamic model for the evolution of parameters is assumed:

#t`1

"#t#u

t#e

t. (34)

utaccounts for the long-term average motion, and e

tfor fast decorrelated position increments

(random walk). In fact, utcan be seen as a command while e

tmay be the process noise, zero mean,

uncorrelated with # and u, and with Re"E[eteHt].

6.2. Algorithm derivation

With the ‘‘command’’ utbeing assumed to be known, the best estimate of # at time t#1,

without any new measurement, is

#ªt`1D t

"#ªt D t#u

t. (35)

The error of estimation is et, and the computation of its covariance matrix, associated with

equation (39), leads to

!ªt`1D t

"!ªt D t#Re (36)

Assuming that the resulting log-likelihood is sufficiently sharply peaked to be well approximatedby its second-order expansion on the interval [#ª

t`1D t, #ª

t`1 D t`1], and considering the set of

equations (32), (35) and (36), it is possible to propose the prediction—estimation filter

#ªt`1D t

"#ªt D t#u

tComputation of H(#ª

t`1D t) and + (#ª

t`1D t)

!ªt`1D t

"!ªt D t#Re

!ª ~1t`1D t`1

"!ª ~1t`1D t

#H (#ªt`1D t

)

#ªt`1D t`1/

#ªt`1D t

!!ªt`1D t`1

)+(#ªt`1D t

)

(37)

However, the previous filter does not remain efficient in the most general situation, becauseneither u

tnor Re are known, and they are generally time-varying.

6.3. Recursive noise covariance computation

The simplest way to estimate ut, by using the filtered positions, is

ut"#ª

t D t!#ª

t~1D t~1(38)

However, this kind of estimation is very sensitive to statistical fluctuations and may degradeperformance. The proposed solution consists of estimating u through a finite time averaging(smoothed estimation) according to

ut"(1!k

1) u

t~1#k

1(#ª

t D t!#ª

t~1D t~1), (39)

The covariance matrix of the random walk is recursively estimated in the same way. At first, let usconsider the quadratic expansion of the log-likelihood, derived from the new measurements



Figure 5. Kalman filter for the tracking of a time-varying channel

obtained at time t and maximized by the value #ªML,t

L(#)"12(#!#ª

ML, t)HH

t(#!#ª

ML,t)#cte (40)

The gradient calculated for the a priori estimate is

+t"H

t) (#ª

t D t!#ª

ML,t)

"Ht) (#K

t D t!#

t##

t!#K

ML,t)

(41)

By taking the covariance of +t, we have

cov(+t)"H

t(cov(#ª

t D t!#

t)#cov(#

t!#ª

ML,t))H

t (42)"H

t(!ª

t D t#R

t#H~1

t)H

t

The covariance matrix is easily derived from the previous equation:

Rt"H~1

tcov(+

t)H~1

t!!K

t D t!H~1

t(43)

However, the covariance of the gradient is untractable, and we shall use a smoothed estimation

Rªt"(1!l

2)Rª

t~1#l

2(H~1

t+t+HtH~1

t!!ª

t D t!H~1

t)

(44)"(1!l

2)Rª

t~1#l

2Hd

tM+

t+Ht!H

t!ªt D t

Ht!H

tNHd

t

The whole Kalman filter for the adaptive estimation of # is merely derived from equations (37) byreplacing u

tand Re by u ˆ

tand Rª

t, respectively, as estimated on line according to (39) and (44)

(see Figure 5).

6.4. Robustness and real-time computation

It is well known that ML methods are not practical in terms of real-time computation andsuboptimal algorithms are generally prefered. The use of a recursive algorithm, which deals withthe flow of data, allows to reduce the computation complexity, since, for each new set of data,convergence is achieved in only one step. In fact, a promising way of investigation may concern



Figure 6. Plots of trajectories and tracking error for a sinusoidal evolution of channel parameters (solid lines representtrue trajectory while dotted lines represent estimated one)

the search of an approximate ML method, based on a subspace approach, able to provide thesame performance for a lower complexity.15

In the case of mobile communication systems, for which signal-to-noise ratio may be signifi-cantly poor, such a method may fail in identifying lowest paths. Simulations have shown thatproposed algorithms could accommodate with SNR until almost 0 dB, if at least a path is above10 dB. In fact, a main limitation in real environment seems to be the mismatch between physicalpropagation features and too much simple model (1). A more complete model should take intoaccount time-spread and space-spread of each path, rather than a single ray.

6.5. Simulations

This second simulation set illustrates the behaviour of the adaptive algorithm when the channelis time-varying. We consider two kinds of parameter variations:

(i) the first one involves a slowly time-varying channel, for which the evolution is determistic,according to a sine wave. The context of simulations is the same as in Section 5.2 for fixedparameters. SNR is fixed at 12 dB. The initial conditions are merely the true values ofparameters, that is we assume that convergence has already been achieved. Figure 6 showsthe tracking of such a channel.

(ii) the second involves a fast time-varying channel, for which the evolution of parameters isa random walk (u

t"0 in (34)). Results are plotted on Figure 7. These two kinds of

simulations show that the proposed algorithm can perform the tracking of slowly as well asfast time-varying channel.



Figure 7. Plots of trajectories and tracking error for a random evolution of channel parameters (solid lines represent truetrajectory while dotted lines represent estimated one)

7. CONCLUSIONS

We have proposed an algorithm for the blind parametric estimation of time-varying multipathchannels. In addition, an adaptive equalizer has easily been derived. Statistical performanceand tracking properties have been studied by numerical simulations. Future work involvesa theoretical analysis of the convergence properties and a comparative study with other blindtracking algorithms based on a subspace approach, with a complete physical model for thepropagation.

APPENDIX

The CML criterion to be minimized can be written, according to the new data x at time t and tothe filtering matrix H(#), with a quadratic shape

Q(H)"xH (I!H(HHH)~1HH )x"xH (I!%H)x. (45)

Computation of the gradient

Let us calculate the derivative of Q as a function of the parameter #i, which can be any

element in the entire set of parameters to be estimated, #. This is simply the ith element of the



gradient +:

LQ

L#i

"xHL(I!%

H)

L#i

x

"!xHGLHL#

i

Hd#H

L (HHH )~1

L#i

HH#(Hd)HLHH

L#iHx (46)

The usual laws for the derivation of matrices yield:

L (HHH)~1

L#i

"!(HHH)~1LHHH

L#i

(HHH)~1

"!(HHH)~1ALHH

L#i

H#HHLHL#

iB (HHH)~1 (47)

and, by injecting this result into (46), this produces

LQ

L#i

"!xHG(I!%H)LHL#

i

(HHH)~1HH#H(HHH)~1LHL#

i

(I!%H)Nx (48)

Let us define

Gi"(%

H!I)

LHLH

i

(HHH)~1HH (49)

We then have

+i"

LQ

L#i

"xH(Gi#GH

i)x. (50)

The derivate of H in function of the parameter #iis obtained by a derivation element per

element, according to their analytic expansion (see (2) and (4)).

Computation of the Hessian

The second-order derivative will be calculated by deriving the matrix Gias a function of the

parameter #j:

LGi

L#j

"!

L%o

HL#

j

LHL#

i

Hd!%o

H

L2H

L#jL#

i

Hd

#%o

H

LHL#

i

(HHH)~1ALHH

L#j

H#HHLHH

L#jBHd

!%o

H

LHL#

i

(HHH)~1LHH

L#j

(51)

A Gauss—Newton algorithm does not take into account the second-order derivative. We shallthus simplify the earlier equation

LGi

L#j

")i,j"!C(Gj

#GHj)LHL#

i

#GHi

LHL#

jDHd

!GiGH

j(52)



The generic element of the Hessian can then be expressed in the following manner:

Hi,j"

L2QL#

iL#

j

"xH()i,j#)H

i,j)x (53)

Remark 5

Let us point out that the gradient and approximate Hessian are real. Furthermore, the latter ispositive definite, which is a main feature of the Gauss—Newton algorithm.

Remark 6

Should the CML criterion be expressed in terms of the covariance matrix (see (26)), the gradientand Hessian would be slightly modified and written

+i"tr((G

i#GH

i)R

x) (54)

and

Hi,j"tr(()

i,j#)H

i,j)R

x) (55)

REFERENCES

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Adaptive maximum likelihood algorithms for the blind tracking of time-varying multipath channels