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Applications of maximum likelihood algorithm in asynchronous CDMA systems

Applications of Maximum Likelihood Algorithm in Asynchronous DS-CDMA Systems

Pei Xiao,Erik Str¨o m

Communication Systems Group,Dept.of Signals and Systems

Chalmers University of Technology,SE-41296,G¨o teborg,Sweden E-mail:pei.xiao@s2.chalmers.se,erik.strom@s2.chalmers.se

Abstract—In this paper,we treat the problems of propagation delay and channel estimation as well as data detection of orthogonally modulated signals in asynchronous DS-CDMA system over fading chan-nels using the maximum likelihood(ML)approach. The overwhelming computational complexity of the ML algorithm makes it unfeasible for implementa-tion.The emphasis of this paper is to reduce its com-plexity by some approximation methods.The de-rived approximative ML schemes are compared with conventional algorithms as well as some others,e.g., the PIC for data detection and the subspace-based algorithm for acquisition.

I.I NTRODUCTION

The considered system in this paper is an asyn-chronous DS-CDMA system with orthogonal modula-tion.The transmitted chip sequence from a particular user is the concatenation of one of possible Walsh codes(representing the transmitted symbol)and a long scrambling code used for user separation.The rationale for using orthogonal modulation is to avoid having to track the phase in a rapid fading environment[1]. Iterative methods for demodulating orthogonal sig-nalling formats in DS-CDMA systems were proposed in[2],[3].Perfect synchronization of the chip timing is required to achieve the low bit error rate by all the demodulation schemes.The impact of synchronization errors on data demodulation was studied in[3].It was shown that errors in the delay estimates would drasti-cally degrade the system performance.

Several synchronization schemes were proposed in[4]for orthogonally modulated signals, e.g., subspace-based estimator,MMSE delay estimator, whitened sliding correlator,etc..They all require a long periodic training sequence to derive an accurate esti-mate of the covariance matrix.

To provide substantial capacity gains,multiuser de-tectors also require estimation of the fading channels. Coherent detection always outperforms its non-coherent counterpart provided that the channel gains are accu-rately estimated[3].

In this paper,we treat all the above issues in an uni-form manner,i.e.,under the maximum likelihood(ML) framework,and compare the performance with the al-gorithms mentioned earlier.In section II,the system model is presented,and the general ML algorithm is formulated.The applications of the approximative ML algorithms with reduced complexity on delay estima-tion,channel estimation and data detection are intro-duced and evaluated with numerical results in section III and IV.Finally,the proposed algorithms are summa-rized and conclusions are drawn in section V.

II.S YSTEM M ODEL

The passband received signal due to the user is denoted by.The user’s symbol is denoted by,and the or-thogonal signal alternatives are de?ned by Walsh sequences of length. The Walsh chips are randomized by a scrambling code

.Hence,every symbol which repre-sents binary bits is spread by chips and each bit by chips.

The baseband signal is formed by pulse ampli-tude modulation with the unit-energy rectangular chip waveform,and the baseband signal is multiplied with a carrier with frequency and transmitted over the chan-nel,which is represented by the complex channel gain and assumed to be a slowly time-varying Rayleigh ?at fading channel with delay.

The total received signal is the sum of the users’signals plus additive white complex Gaussian noise with power spectral density.The complex enve-lope of the received signal is

where stands for the propagation delay of the th user.The vector is de?ned as

.Let us denote

(is one chip interval)such that

stands for the chip delay,

stands for the interchip delay,and denote, the transmitted chip sequence due to the th user as

(3) where is a diagonal matrix de?ned by the th user’s scrambling code,and

is the th column()of the Hadamard matrix.Then,each column of the matrix in(2)can be expressed as

(4) where,stand for the up-shift and down-shift operators respectively:

,

.All the delay estimators intro-duced in this paper need a periodic training sequence, which means remains unchanged at the delay es-timation stage.

In our case,the noise is complex Gaussian.Given observations of the received vector for

,the log-likelihood function of the received vector can be expressed as1

We denote the2-norm of a vector by Downmixing,

Chip-Matched

Filtering,

Sampling

Delay

Estimation

reformed as

Since is irrelevant to the choice of,so the decision is solely decided by,i.e.,the projection of onto the the subspace spanned by the vector. Therefore,the ML single user delay estimator(8)can be expressed alternatively as

The projection of onto the the subspace spanned by the vector is the same as the normalized correla-tion of the vectors and,therefore,in single user case,the ML delay estimator becomes identical to the conventional sliding correlator(SC)which is the stan-dard approach to delay estimation.The merit function for the SC is formed by correlating the received signal with time delayed versions of the known code se-quence

is the sample mean of the re-ceived vector.The estimated delay is the delay candi-date value that maximizes the correlation.

Like the standard receiver(i.e.,the single user matched?lter),the SC or the ML single user delay esti-mator is only optimal in the AWGN single user channel or in strict orthogonal synchronous channel.It is highly unreliable in the presence of MAI.The ML approach to delay estimation suitable for multiuser environment is presented below.

B.ML multiuser delay estimator

It is shown in[2]that the ML estimation of the fading channel vector is

(9) where denotes the left pseudoin-verse of,which is the data matrix with timing uncertainty denoted by.Substituting into(6),we yield the ML estimate of as

(10)

where is the orthogonal projection matrix onto the orthogonal complement to the subspace spanned by the columns of.

The ML multiuser delay estimator?nds the delays of

all the users simultaneously.The disadvantage of this algorithm is the unaffordable computational complex-ity because it has to search over points(is the number of users,is spreading factor,and we assume the delays are within one symbol interval).While the

ML single user estimator or the subspace-based estima-tor(see[4],[6])estimate the delay for one user at a time, the number of points to be searched goes down to. However,some approximations can be made to sim-plify the ML algorithm.The sub-optimum solutions with reduced complexity are discussed below.

B.1Hierarchic ML delay estimator

In[7],a hierarchic way to construct an ML approx-imation for delay estimation was proposed.A revised version of this algorithm adopted to the system model in question is as follows:

1.Evaluate for

.Choose corresponding to the minimum trace value,which means we?x the delay es-timate for the user.

2.Form the matrices for

https://www.doczj.com/doc/8612397128.html,pute

,evaluate for

and.Choose cor-responding to the minimum trace value.

3.Form the matrices

for https://www.doczj.com/doc/8612397128.html,pute

,evaluate. Choose corresponding to the minimum trace value.

4.Repeat step3until all the users’delays are esti-mated and?xed.

The idea of this scheme is similar to that of successive interference cancellation(SIC).We?x the delay of one user at each step,that user’s signal is reconstructed and used for detecting the next user’s delay.

B.2Multistage parallel ML delay estimator

The complexity of ML algorithm can also be reduced in an iterative multistage manner like parallel interfer-ence cancellation(PIC).We use the ML single user de-lay estimator to get initial estimates of delays for all the users and enter the iteration loop.The subsequent stages differ from the hierarchic estimator introduced earlier. Instead of?xing one user’s delay at a time,we?x the delays of all the interfering users simultaneously(in par-allel),using the estimates derived from last iteration in order to estimate the delay for the user of interest,e.g., the th user.This multistage parallel ML delay estima-

tor can be expressed as

(11)

The matrix

is de?ned as

where are the estimated delays for the interfering user at the

iteration stage.

In contrast to the original ML algorithm which jointly detects all the users,these two approximative ML algo-rithms detect the delay of one user at a time,thus greatly reduce the computational complexity compared to a full search of the ML criterion function.Like the SC,they have the property of fast convergence,require much shorter training sequence than the subspace-based algo-rithm,thus reduce the overhead induced by the training.The results of the -stage parallel ML delay estima-tion are demonstrated in Fig.2.Clearly,the perfor-mance is improved at each iteration.However,it tends to get saturated at the th stage.The simulated system for this and all the following experiments is a -user system with chips per symbol and

signal alternatives.The signal to noise ratio is set to

,and perfect power control is

assumed in the sense that the average power is equal to all users.The channels are independent Rayleigh fad-ing channels.The channel gain

is complex cir-cular Gaussian process with auto-correlation function

where is the max-imum Doppler frequency and is the zeroth order

Bessel function of the ?rst kind.Simulation results are averaged over or Monte-Carlo runs with each MCrun representing a particular realization of the data sequences,noise,and fading processes,as well as prop-agation delays.

The comparison among different delay estimators is illustrated in Fig.3.The parallel ML estimator per-forms slightly better than the hierarchic one.They both achieve good acquisition performance with a short training sequence (approximately pilots).As ex-pected,the subspace-based estimator does not work with short training sequence.We anticipate the curve of the subspace-based algorithm would go down to the same level as the approximative ML algorithms when the number of pilots increases,e.g.,to around .

B.3Combined hierarchic and parallel ML delay esti-mator Since the hierarchic and parallel estimator have dif-ferent mechanisms for searching the optimum values of the propagation delays,we tend to think that when

estimator

these two are coupled together,the combined algorithm has better chance to get out of the local minimum and

converge to the global optimum attainable by the origi-

nal ML algorithm.That is indeed the case as shown in

Fig.4.In the experiment,the hierarchic scheme is used

as the?rst stage followed by-stage parallel scheme. The combined scheme performs better than each indi-

vidual one.For instance,with pilot symbols,the

acquisition error probability goes down to when these two methods are coupled together.While the ac-quisition error probability of each individual scheme is around.

IV.ML A PPROACH TO D ATA D ETECTION The ultimate goal of the receiver is to detect the sym-bols from all the users given the received signal vector ,i.e.,detect for(see Fig.1). The decision on the th user’s symbol at the th itera-tion stage(if multistage detection is applied)is denoted by and is found as

where is the soft decision function at the iteration stage.

It was shown in[8]that with proper synchronization (delay estimation)mechanism,an asynchronous system with random propagation delays approximates the per-formance of a synchronized and chip-aligned system (meaning is known and,i.e.,the received signal is aligned with chip matched?lter),for which in equation(4)reduces to

(12) In the following,we shall derive the soft decision

for the ML demodulation schemes based on this simpli-?ed setting.

A.ML single user detector

The extension of the ML single user delay estimation to data detection is straightforward.Instead of having different candidate values of delay for the th user in equation(7),we try the possible transmitted wave-forms,,,,,() when detecting the th transmitted symbol of the th user.The ML single user detector can therefore be for-mulated as

where,. The matrix is the instantaneous covariance matrix de?ned as.

Here again,the ML single user detector becomes identical to the conventional matched?lter(MF)which forms the soft decision by correlating the received sig-nal with the local reference signals,i.e.,

.This simple scheme is particularly useful in the beginning of the detection process,e.g.,at the?rst iteration stage,when the estimates of the fading chan-nels are lacking,we must therefore carry out the detec-tion in a noncoherent manner.

B.ML multiuser detector

Although optimum in the single user AWGN channel, the above mentioned detector has poor performance in multiuser environment.From equations(6)and(9),we know that the ML estimation of the data matrix

can be derived as the minimizer of

(13) This ML multiuser detection algorithm jointly detects the symbols transmitted by all the users,it requires a test of hypotheses,thus the com-plexity grows exponentially with the number of users .A sub-optimum solution that reduces the complex-ity from exponential to polynomial is the multistage ML detector.We use the ML single user detector,or equiva-lently,the conventional detector for the?rst stage to get initial estimates of the transmitted data.Then the ML algorithm switches to the decision directed mode.To detect the data transmitted by the th user,we replace in equation(13)with de?ned as

(14) where the vectors

are obtained by substituting

into equations(3),(12),and we assume the chip delay is known.

C.A comparison between PIC and ML algorithm Parallel interference cancellation(PIC)scheme de-tects all the users at the same iteration(in parallel)and then cancels the MAI at the next iteration stage.The soft decision of PIC can be formed as

(15) where and are obtained by substituting into(12),and is the estimate of the fading vector.

Let us denote as the MAI matrix formed as

The matrix is similar to except that the th user transmitted symbol is deleted.

Similarly,we delete the estimate of the th user’s channel coef?cient from the vector and form the channel vector corresponding to the MAI

Then the PIC algorithm can be reformulated as

(16) Meanwhile,the expression of the approximative ML algorithm–equations(13)and(14)can be expanded as

(17) The?rst term of the expression(17)is irrelevant to the choice of,and all the possible transmitted wave-forms for have the same energy, thus the third term is also common for the decision met-rics.Minimization of the decision function(17)is there-fore equivalent to maximization of

which is exactly the same as the PIC algorithm ex-pressed by the equation(16).We can conclude that the PIC is an approximative ML approach to data detection. We know that the channel gains are correlated in time,and the channel estimates can be improved by smoothing.A simple smoothing procedure[2]is to feed through an FIR?lter with impulse response which yields the smoothed channel vector as

.

The performance of the discussed data detectors is il-lustrated in https://www.doczj.com/doc/8612397128.html,ing the ML single user detector (conventional receiver)as the?rst stage,the ML mul-tiuser detector or equivalently,the PIC without channel smoothing achieves considerable gains compared to the conventional receiver as indicated by the dashed curve. The performance of the PIC can be further improved by channel smoothing(replacing with in equa-tion(15)or(16)),as illustrated by the dash-dot curve.

V.S UMMARY AND CONCLUSIONS

In this paper,some approximation methods to reduce the complexity of the ML algorithm are introduced and the applications of the derived approximative ML algo-rithms on the delay estimation and data detection of the orthogonally modulated signals in asynchronous DS-CDMA system are discussed.

The general principle is to use the single user ML algorithm as the?rst stage to get an initial estimates of the delays/transmitted data.In this case,the MAI is treated as additive noise.The ML single user delay estimator/data detector turns out to be identical to the

conventional SC/MF.Clearly,the SC/MF is optimal de-lay estimator/data detector in single user AWGN chan-nel.After obtaining the initial estimates,we switch to decision directed mode and improve the results at each iteration stage,using the ideas similar to SIC or PIC. The suggested approximative ML algorithms are eval-uated by computer simulations and found to have good performance with affordable complexity.

R EFERENCES

[1]S.Vembu,A.Viterbi.Two different philosophies

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[2] E.G.Str¨o m,https://www.doczj.com/doc/8612397128.html,ler.Iterative demodulation

of orthogonal signalling formats for DS-CDMA.

Proc.ICC2000,vol.3,pages1457-1461,2000. [3]P.Xiao,E.Str¨o m.Performance of iterative DS-

CDMA M-ary demodulation in the presence of synchronization errors.Proc.VTC’2001spring, pages1703-1707,May,2001

[4]P.Xiao,E.Str¨o m.Acquisition of Orthogonal

Modulated Signals in Rayleigh-Fading Channels.

Proc.European Wireless,pages811-816,Feb.

2002.

[5] E.Str¨o m,F.Malmsten.A maximum likelihood

approach for estimating DS-CDMA multipath fad-ing channels.IEEE Journals on Selected Areas in Communications,vol.18,pages132-140,Jan.

2000.

[6] E.Str¨o m,S.Parkvall,https://www.doczj.com/doc/8612397128.html,ler, B.Ottersten.

Propagation delay estimation in asynchronous direct-sequence code-division multiple access systems.IEEE https://www.doczj.com/doc/8612397128.html,mun.,vol.44,pages 84-93,Jan.1996.

[7]J.Joutsensalo,J.Lilleberg, A.Hottinen,J.

Karkunen.A hiearchic maximum likelihood method for delay estimation in CDMA.Proc.

VTC’1996,vol.1,pages188-192,1996.

[8]P.Xiao,E.Str¨o m.Delay estimation and data de-

tection in long-code DS-CDMA system.Submit-ted to GLOBECOM’2002.

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?一、基本情况 ? 二、 课题 设计论证 课题名称 多媒体课件优化中学生英语阅读的实践研究? 主持人 姓 名 ?刘志刚 政治面貌 党员? 性别 男? 年龄 41 行政职务 教导主任 专业 职称 中一? 学科 专业 英语? 学历 学位 本科 起止时间 2014年 12 月 10 日至 2015 年 12 月 10日 工作单位 通讯地址 伊川县江左镇中 邮政 编码 471314? 固定电话 E-mail 移动电话 主 要 参 与 者 姓 名 性别 年龄 专业 职称 学科 专业 学历学位 工作单位 韩世伟 男? 48 中二? 初中英语 本科 伊川县江左镇中 程会英 女? 40 中二? 初中英语 专科 伊川县江左镇中 黄爱香 女? 37 中二? 初中英语 本科 江左教育? 杨玉温? 女? 41 中二? 初中英语 本科 伊川县电力中学 端木梦梦 女? 26 中二? 初中英语 本科 伊川县江左镇中 预 期 成 果 (在选项上打“√” 或加粗) A .专着 B.研究报告 C.论文 D.其他

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