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CHEN et al ADAPTIVE MINIMUM BER LINEAR MULTIUSER DETECTION 1241. that is is the user signal power the CIR matrix,has the form. Fig 1 Discrete time model of synchronous CDMA downlink. and the system matrix,II SYSTEM MODEL, The discrete time baseband model of the synchronous. DS CDMA downlink system with users and chips per 9. symbol is depicted in Fig 1 where denotes the, th symbol of user the unit length signature sequence for. The channel ISI span depends on the length of the CIR. 1 related to the length of the chip sequence For, and the transfer function of the channel impulse response CIR and so on The model 3 adopted in this study is a wideband. is synchronous DS CDMA channel and it can be extended to the. case of asynchronous DS CDMA downlink systems, We will study the linear multiuser detector of the form.

sgn with 10, The baseband model for received signal sampled at chip rate is. given by 4 18 where, denotes the detector weight vector for user Let the. possible combinations or sequences of,3 and let be the th element of Define the. noise free received signal states, where denotes the noise free received signal the white. Gaussian noise vector and the set of scalars,the user symbol vector.

III MBER LINEAR MULTIUSER DETECTOR, In this section we derive the theoretical MBER solution for. the user signature sequence matrix the linear multiuser detector and present a gradient search algo. rithm for finding this MBER solution This will provide some. 6 insight into the development of our LMS style stochastic gra. dient adaptive algorithm The error probability of the linear de. the diagonal user signal amplitude matrix tector 10 is. diag 7 Prob sgn 15, 1242 IEEE TRANSACTIONS ON SIGNAL PROCESSING VOL 49 NO 6 JUNE 2001. Following 15 16 define the signed decision variable With this rescaling the gradient can be simplified as. where sgn 26, The steepest descent gradient algorithm 24 may converge. 17 slowly and a Gauss Newton algorithm is computationally. expensive The conjugate gradient method 19 offers a better. alternative A simplified conjugate gradient algorithm is. and summarized, sgn 18 Initialization Choose step size and termination scalar. given and set, Notice that can only take the values from the set de Loop If.

fined in 14 and is Gaussian with zero mean and vari goto Stop. ance Under the assumption that are, equiprobable the probability density function pdf of. Thus Stop is the solution,At a local minimum Therefore the small. positive scalar determines the accuracy of the solution ob. tained A gradient algorithm can in general find a local min. imum of To ensure a unique global solution certain. where constraints can be added to the optimization problem as shown. in 7 at a possible cost that the solution found may deviate. 21 slightly from the true MBER solution, In general the global MBER solutions are infinitely many. and and they form a half hyperplane in the dimensional weight. sgn sgn space The normalization 25 has an effect of fixing the solu. 22 tion to a unique unit length one Consider the following simplest. example with two equal power users and two chips per bit The. The gradient of with respect to is two chip codes are and respectively and the. transfer function of the CIR is,The SNR for user 1 SNR is 25 dB The BER sur. face for user 1 is plotted in Fig 2 For this example there are. only global minimum solutions and all the MBER solutions. sgn 23 form a half line half hyperplane in the two dimensional 2 D. space The point marked in the MBER solution half line is the. unit length one The MMSE solution for this example is also. The following steepest descent gradient algorithm can be depicted in Fig 2 where it can be seen that the MMSE solution. used to find the MBER solution is very different from the MBER ones In fact for the MMSE. solution whereas for an MBER one, where indicates the iteration and is an adaptive step size.

Notice that the orientation of the weight vector defines the IV ADAPTIVE MBER LINEAR MULTIUSER DETECTOR. decision boundary and thus the BER and not the size of It The key to developing an effective adaptive algorithm is the. is computationally advantageous to normalize to a unit length pdf of the signed decision variable Kernel density. after every iteration estimation is known to produce reliable pdf estimates with short. data records and in particular is extremely natural in dealing. with Gaussian mixtures 20 21 Given a block of training. CHEN et al ADAPTIVE MINIMUM BER LINEAR MULTIUSER DETECTION 1243. The LMS algorithm can be viewed as replacing the ensemble. average of the gradient in its related steepest descent gradient. algorithm by a single data point estimate of the gradient In a. similar manner at sample a point estimate of the pdf is simply. Using the instantaneous or stochastic gradient,and rescaling after each update to ensure gives. rise to a LMS style stochastic algorithm, Fig 2 Bit error rate surface of user one detector for a simple two user system. with two chips per bit and SNR 25 dB sgn 33,which we refer to as the LBER algorithm. samples a kernel density estimate of the pdf is, Two important issues for any stochastic gradient adaptive. MBER algorithm are the convergence speed and steady state. BER misadjustment with respect to the optimal MBER A. theoretical analysis of these two properties for the LBER 33. is extremely complex and is still under investigation We will. sgn use computer simulation to study these two properties. B Comparison with the DMBER and AMBER Adaptive, where the radius parameter is related to the noise standard Algorithms.

deviation A lower bound of is 20, The motivation of the above LBER multiuser detector is dif. ferent from those of the existing DMBER and AMBER detec. 28 tors 8 9 For the purpose of a comparison we present a mod. ified version of the DMBER adaptive algorithm reported in 8. From the estimated error probability Define the one sample decision distortion measure3. Notice that this distortion measure is a function of as. can be calculated as,sgn Obviously is, the one sample error probability The following difference. approximation for the gradient of is adopted as, Thus block adaptive gradient algorithms can readily be devel. oped by substituting with in the steepest de,scent or conjugate gradient updating mechanisms. A LBER Stochastic Adaptive Algorithm 3In 8 the distortion measure is defined in terms of two samples one for. Our aim is however to develop a LMS style adaptive algo. b k 0 1 and the other for b k 1 1 which lead to some complica. tions in sample by sample adaptation Our modification has a nature and simple. rithm with sample by sample adjustment as in 15 and 16 adaptive implementation and is equivalent to the original formulation. 1244 IEEE TRANSACTIONS ON SIGNAL PROCESSING VOL 49 NO 6 JUNE 2001. where for some and denotes the th, coordinate unit vector The DMBER adaptive algorithm takes.

where is an adaptive step size, In theory the DMBER should work as it attempts to min. imize the BER directly As a difference approximation of the. stochastic gradient is used the algorithm does not rely on the. assumption of a Gaussian channel noise at a cost of increased. complexity The DMBER has a complexity of whereas,the AMBER and LBER has a complexity of However. in practice its rate of convergence is very slow after the weight. Fig 3 Linear detector BERs for user 1 of Example 1 SNR SNR. vector has reached the region of small error rate This is obvious. since will be zero most of the time, A more efficient stochastic gradient adaptive algorithm is the. AMBER 9 which can be expressed as,with the error signal. and the indicator function, sgn 40 Fig 4 Linear detector BERs for user 2 of Example 1 SNR SNR.

where is a non negative threshold parameter In terms of the V SIMULATION RESULTS. algorithm tuning requirements the two adaptive algorithms the. Computer simulation was conducted to investigate the. LBER 33 and AMBER 38 are similar The former requires. convergence speed and steady state BER misadjustment for. the tuning of the adaptive gain and kernel width whereas. the three stochastic adaptive algorithms For the DMBER. the latter needs the tuning of the adaptive gain and threshold. algorithm a fixed step size was used with a time varying. difference step as was used in 8 For, The following comparison of the two adaptive mechanisms. the AMBER algorithm the threshold was fixed with a. the LBER and AMBER can be made 15 16 The algorithm. time varying adaptive step size given by, 38 in its simplest form has and it only updates when a. decision error is observed When the algorithm is initialized it is 42. unlikely to separate all the noise free states correctly Thus. the indicator function will be on most of the time in The LBER algorithm had a constant width and employed. which case it is equivalent to the signed error LMS algorithm the time varying adaptive step size as given by 42 The two. 10 When the algorithm has converged to a point where it can algorithm parameters and for the DMBER and for the. separate the noise free states correctly the probability of the al AMBER and and for the LBER were chosen to give an. gorithm updating may be low because in this region errors will adequate combined result of convergence rate and steady state. be predominated by noise and hence further convergence may error for the respective algorithm. be slow Introducing the threshold essentially defines a region Example 1 A two user system with four chips per symbol. around decision boundary where the algorithm will continue to was used in the simulation The code sequences of the two users. update even when errors do not occur This region is defined by were and respectively. and the transfer function of the CIR was, In the algorithm 33 the effect of the distance from the. decision boundary is controlled by the exponential term The two users had equal signal power that is SNR was equal. This can be viewed as a soft distance to SNR Figs 3 and 4 depict the linear detector BERs for the. measure The size of an update is a continuous and decreasing two users respectively The BER formula 20 was used with the. function of the distance from the boundary The distance is detector weight vector set to the MMSE and MBER solutions. scaled by the kernel width which in turn is a function of the respectively to produce the corresponding error rate curves For. noise root mean square this example the difference between the MMSE and MBER. CHEN et al ADAPTIVE MINIMUM BER LINEAR MULTIUSER DETECTION 1245. Fig 5 Distribution of the signed decision variable for user 1 of Example 1. SNR SNR 16 5 dB the detector weight vector was set to the MBER. Fig 7 Learning curves of the three stochastic adaptive MBER algorithms and. solution and the kernel estimate was constructed from 100 data samples. the LMS algorithm for user 1 of Example 1 SNR SNR 19 dB. Fig 6 Convergence behaviors of the two block adaptive MBER algorithms. for user 1 of Example 1 Length of the data block is 100 and SNR SNR. 16 5 dB Fig 8 Linear detector BERs for user 1 of Example 2 SNR 1 4 are. solutions for user 1 is significant for the range of SNR 14 to an adaptive gain 0 01 is also depicted in Fig 7 The value of. 26 dB the learning curve at is the true BER of 20 for. The kernel density estimate 27 constructed from 100 data the given weight vector and not any approximation For. samples at SNR SNR dB is compared with the true this example it can be seen that the proposed LBER algorithm. pdf 19 in Fig 5 for user 1 where the detector weight vector is superior over the other two adaptive MBER algorithms in. was set to the MBER solution For this example the kernel esti terms of convergence speed and steady state error. mate and the true density are indistinguishable Using the con Example 2 This was a four user system with 8. structed kernel density estimate with the detector weight vector chips per symbol The code sequences for the four. the block adaptive steepest descent and conjugate gradient users were. algorithms were applied to find the MBER solutions and the. two iterative procedures are illustrated in Fig 6 It can be seen and respectively and the. that starting from the MMSE solution the block adaptive con transfer function of the CIR was. jugate gradient algorithm took fouriterations to converge to the. MBER solution 44, The three stochastic gradient adaptive algorithms the. DMBER AMBER and LBER were applied to user 1 with The four users had equal signal power Figs 8 10 depict the. SNR SNR dB and the initial weight vector set to linear detector BERs for the four users respectively For users 2. the MMSE solution For the DMBER and and 4 the MMSE and MBER solutions are indistinguishable. was used The AMBER used and whereas Fig 11 shows the kernel estimate and true pdf of the signed. the LBER had and noise variance decision variable for user 1 at SNR dB. These algorithm parameters were found in simulation to be The pdf estimate was constructed from 1500 data samples and. adequate for the respective algorithm The convergence per the detector weight vector was set to the MBER solution It. formance of these three algorithms are shown in Fig 7 where can be seen that for this example the kernel estimate approx. the results were averaged on 100 runs The standard LMS with imates the true density reasonably well Based on the kernel. 1240 IEEE TRANSACTIONS ON SIGNAL PROCESSING VOL 49 NO 6 JUNE 2001 Adaptive Minimum BER Linear Multiuser Detection for DS CDMA Signals in Multipath Channels Sheng Chen Senior Member IEEE Ahmad K Samingan Bernard Mulgrew Member IEEE and Lajos Hanzo Abstract The problem of constructing adaptive minimum bit

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