Adaptive Multiuser Receivers for DS-CDMA Using ... - Semantic Scholar
Adaptive Multiuser Receivers for DS-CDMA Using Minimum BER Gradient-Newton Algorithms Rodrigo C. de Lamare and Raimundo Sampaio-Neto CETUC - PUC-RIO, 22453-900, Rio de Janeiro - Brazil e-mails: [email protected], [email protected]
Abstract— In this paper we investigate the use of adaptive minimum bit error rate (MBER) Gradient-Newton algorithms in the design of linear multiuser receivers (MUD) for DS-CDMA systems. The proposed algorithms approximate the bit error rate (BER) from training data using linear multiuser detection structures. A comparative analysis of linear MUDs, employing minimum mean squared error (MMSE), previously reported MBER and the proposed MBER algorithms is carried out. Computer simulation experiments show that the MBER Gradient-Newton approaches outperform other analysed algorithms and can operate with shorter training sequences.
I. I NTRODUCTION Multiple access interference (MAI) in direct-sequence code division multiple access (DS-CDMA) communications systems arise due to the non-orthogonality of the spreading codes at the receiver. The mitigation of MAI has attracted significant effort in the last years since MAI represents the main limitation on the capacity of DS-CDMA systems. The conventional receiver, which consists of a bank of matched filters, is well-known to be susceptible to the near-far problem [1] and to MAI. Several multiuser detectors (MUDs) have been shown to considerably outperform the conventional detector. Thus, superior performance is achieved at the cost of some increased complexity and often requires side-information regarding the structure of user´s signals [1]. Adaptive techniques reduce the amount of sideinformation and complexity of such MUDs, whilst improving the performance of conventional detectors [1]. Adaptive multiuser receivers employing the minimum mean square error (MMSE) [1]-[3] criterion have become rather successful, since they usually show good performance and have simple adaptive implementation [1]-[3]. However, it is well known that the MSE cost function is not optimal in digital communications applications, and the most appropriate cost function is the bit error rate [4],[5]. The approximate minimum bit error rate (AMBER) [4] and the least bit error rate (LBER) [5] are two of the most successful and suitable algorithms for adaptive implementation. However, these minimum bit error rate (MBER) algorithms usually require long training sequences to converge to lower bit error rates than those achieved by the techniques that employ the MSE cost function. In this work, we investigate MBER Gradient-Newton algorithms that can speed up the convergence of the multiuser receiver, requiring shorter training data. The proposed algorithms, denoted Gradient-Newton-AMBER and GradientNewton-LBER, are similar to the well known LMS-Newton [6] algorithm and employ the error functions used in the AMBER and the LBER, respectively. We perform a comparative analysis of linear MUDs, employing the LMS [6], the AMBER [4], the LBER [5], the LMS-Newton [6], the Gradient-Newton-AMBER and the Gradient-Newton-LBER adaptive algorithms. This paper is organised as follows. Section II briefly describes the DSCDMA system model. The adaptive linear multiuser receiver structure and the stochastic gradient algorithms are presented
0-7803-7589-0/02/$17.00 ©2002 IEEE
in Sections III and IV. Section V is dedicated to the GradientNewton based algorithms. Section VI shows and discusses the simulation results and Section VII gives the concluding remarks of this work. II. DS-CDMA S YSTEM M ODEL consider a synchronous DS-CDMA system with users, Wechips per bit and binary symbols, as shown in Fig. 1. These symbols are spread with random generated spreading sequences, modulated and transmitted through a communication channel, where denotes the bit of user , the signature sequence for user "! #$&% ')(*(*(+#$&% ,.- /10 is normalized to have a unit length, and the channel impulse response is given by
798;: 2 435 6 < ' @&A A3 : AB =?> : ' introduces a delay of one chip time in where the operator 3 the transmitted signal.
Fig. 1. Model of synchronous DS-CDMA system.
At the receiver the composite signal is demodulated, filtered by a chip-pulse matched filter and sampled at chip rate yielding the observation vector
H
FGG
C 6 ED
FGG IJ GG K
K I J
H K ...
..
O O &PQ&P@S RT .. .
O &PQUVWRT
(*(*( K
.
(*(*(
..
.
..
.
K
L MM
.. .
K IJ N
MM
L MM N VYXZ [ E\ .VYXZ
PIMRC 2002
]9
where the Gaussian noise vector XZ ! '; (*(*( ,.- / 0 7 , \ O is the noise-free signal vecwith ! XZ AX)0Z / tor, the user bit vector is given by ! B (*(*(6 I /10 , the user signature sequence matrix is described by ! 5'Y(*(*( / J , the diagonal user signal amplitude matrix is rep resented by 9 ' (*(*( , and the matrix H is expressed by
FG !#"$"%"& ' 8 ( H ) ! "$"%" ..
.
..
.
' 8 ( "% !"$" %" "$"* ' 8 ( ..
.
N
LM
+-, .
The multiple access interference (MAI) is originated from the non-orthogonality between the user signature sequences. The intersymbol interference (ISI) span L depends on the length of the channel response, which is related to the length ofthe chip S sequence. For (no ISI), for S ] , for ] Y and so on. Consider a one-shot MUD, depicted in Fig. 2, whose observation vector is C ;')(*(*( ,.- 0 . The detected symbols for this one shot receiver and user are expressed by:
21 3/54 06 1 /7/ 4 0 60 8 3 :9 ; %; =< > [ @? BADC 0 C [ @? BE BF5 where A + 6 ! G ')(*(*(% G ,.- /10 foris theuserreceiver weight vector and E + is the estimated symbol and symbol in a system with
users.
Fig. 2. One-shot Linear Multiuser Receiver.
O PRQ O TS PRQ O)*U =VW TS + %? BE + [ U =VW TS ? E [ U TS + E + X1ZY[U \ where E + is given through (4) and 5 + is the desired symbol taken from the training sequence for user and symbol . B. The AMBER algorithm
Given a user transmitted training sequence , the bit error probability ) , for the linear receiver, is expressed by:
The AMBER is a stochastic gradient that attempts to approximate the exact MBER performance [5]. The algorithm is appealing due to its very low complexity, simplicity and straightforward extension to the complex signalling case. The MUD solution that minimises the BER criterion via the AMBER algorithm employs the vector function ? [5] to approximate an expression for a coefficient vector + that achieves a MBER performance with linear receiver structures, as described by:
BA A
+ A\ *d ?BA + 6 J]$^`_ + a A A 0 a A\ c fe b where + is the desired transmitted symbol for user , taken ^ from the training sequence, A( is the Gaussian error function
and \ are the received samples without noise taken from the outputs of chip-matched filters. A simple stochastic solution for can be derived by using ? and adjusting the receiver weights by:
A
BA A VWB 6 JWfA l-monq ps Wr @l-VLmonqtuKgl-mo?n BA h ^Iikj v pwWfl-mon vfxzy inside the expected Note that the quantity value operator in (7) corresponds to the conditional bit error probability given the product 5 + A\ . This W quantity can be replaced in (7) by an error indicator function j given by: j W 6 ] A L? E { where E + is the estimated symbol and 5 + is the desired signal provided by the training sequence. The AMBER algorithm, as devised for linear MUDs [5], is described by the following equalities:
III. S TOCHASTIC G RADIENT A LGORITHMS A number of adaptive algorithms can be used to adjust the receiver coefficients. In this section, we describe stochastic gradient algorithms that adjust the parameters of the receivers based on the minimisation of the mean square error (MSE), the LMS [6], and the bit error rate (BER) cost functions, the AMBER [4] and the LBER [5]. A. The LMS algorithm The adaptive solution for the linear MUD via the LMS algorithm [1] is based on the MMSE error criterion formed by the error signal B+ 6 + ) 0 C , and is described by:
H IA A VWB 6 JA @VLKMHB C
*N9
where is the desired signal for the -th user taken from the training sequence, C is the observation vector for the linear MUD and is the algorithm step size.
K
J]$^`_ + a A 0 a A\ + A\ *d A + |b A + VWB 6 JA @VLK ~}q 9 j W QR + A\ < + A\ f A VWBC 6 A + @VLK 9 j W $W + A\ f< Since \ XZ , and j W and + are statistically independent, ! j $ + AXZ / ! + / ! j W AXZ /. Y weand have thus: A + VWB 6 JA @VLK } j W $ + C A Y A
+ VWB 6 JA @VLK
The AMBER stochastic gradient update equation for the linear MUD is given by:
A
+ VWB JA + @VLK? j W $ + C
A9B
? E
jW
'
A In practice, a modified error indicator function + + ) is employed, where the threshold is responsible for increasing the algorithm rate of convergence. This algorithm updates when an error is made and also when an error is almost made, becoming a smarter choice for updating the filter coefficients.
C. The LBER algorithm
E
The MUD BER depends on the distribution of the decision variable + , which is a function of the weights of the receiver. The sign-adjusted decision variable for the linear receiver + + is drawn from a Gaussian mixture, described by:
E W
E E W [ ? + S A 0 \ .VLA 0 XZ %U E W 6 @? E @V
AB]9
where the first term of (12) is the noise free sign-adjusted MUD output. Consider that samples of the transmitted symbols *+ and samples of the estimated symbols + are available from the samples + Z of a training sequence. A kernel density estimate [5] is given by:
E
AMBER, a non-zero defines a region boundary where the algorithm will continue to update, in the LBER, the effect of the distance from the decision boundary is controlled by an exponential term [5]. Indeed, this can be viewed as a soft distance metric, the size of an update is a continuous and decreasing function of the distance from boundary and both algorithms the with two parameters that require have a complexity of tuning.
b
IV. G RADIENT-N EWTON A LGORITHMS Gradient-Newton algorithms [6] incorporate second-order statistics of input signals, increasing their convergence rate. They usually have a faster convergence rate than gradient techniques, although they require a higher computational complexity. In practice, estimates of the autocorrelation matrix and the gradient vector are used to converge to the desired solution. In addition, to avoid the inversion of the autocorrelation matrix, the matrix inversion lemma is also employed. The update equation of Newton´s method is given by
A
VWB 6 A + ]` c : ' p R
A{
where p is the autocorrelation matrix of the observation vector c R is the gradient vector. W + W . < "!$# + W #&%( ' ) ++* 1,f++ -R... ,++-R..0/ 2 C and In practice, only estimates of the autocorrelation matrix / p c R and the gradient vector
are available. These estimates
, . can be applied to Newton´s formula + to devise an update rule where 3 is the radius parameter of the kernel density estimate give by: [5]. Substituting the expected value of expression in (13) with a single point estimate, we have: A + VWB 6 JA IK c > : ' > p R ] Y 4 W + W . 5!$# + W #6%( ' ) ++* , ++ R- ... , ++R- ..0/ 2 The convergence factor K is introduced to protect the algorithm from divergence, which is originated by the use of noisy esti/
(7 . mates of c and p R . + The probability of error for user is estimated by: To obtain an unbiased estimate of the observation matrix ced , we employ the following weighted sum: 8:9 W 8 + W= . @? 4 W + W .C*D W @E _ %(' ) ++* f, ++ R- .. GF, ++R- . > [ gf C C 0 @VEAh f > SB ]TB (BA + . /b c
IH . where f is c a small factor chosen in the range + w V Y if 4 Y ( and 1 The gradient terms of are: C is the observation vector. WLK > J 8:9 W i /L( O N( P QRB MP N R y %(' ) ++* , ++R- .. !S# T ++-R. GF ,f++-R. F To avoid the required inversion of , we use the matrix c J , lemma, described by: + . /VU +
Abstract— In this paper we investigate the use of adaptive minimum bit error rate (MBER) Gradient-Newton algorithms in the design of linear multiuser receivers (MUD) for DS-CDMA systems. The proposed algorithms approximate the bit error rate (BER) from training data using linear multiuser detection structures. A comparative analysis of linear MUDs, employing minimum mean squared error (MMSE), previously reported MBER and the proposed MBER algorithms is carried out. Computer simulation experiments show that the MBER Gradient-Newton approaches outperform other analysed algorithms and can operate with shorter training sequences.
I. I NTRODUCTION Multiple access interference (MAI) in direct-sequence code division multiple access (DS-CDMA) communications systems arise due to the non-orthogonality of the spreading codes at the receiver. The mitigation of MAI has attracted significant effort in the last years since MAI represents the main limitation on the capacity of DS-CDMA systems. The conventional receiver, which consists of a bank of matched filters, is well-known to be susceptible to the near-far problem [1] and to MAI. Several multiuser detectors (MUDs) have been shown to considerably outperform the conventional detector. Thus, superior performance is achieved at the cost of some increased complexity and often requires side-information regarding the structure of user´s signals [1]. Adaptive techniques reduce the amount of sideinformation and complexity of such MUDs, whilst improving the performance of conventional detectors [1]. Adaptive multiuser receivers employing the minimum mean square error (MMSE) [1]-[3] criterion have become rather successful, since they usually show good performance and have simple adaptive implementation [1]-[3]. However, it is well known that the MSE cost function is not optimal in digital communications applications, and the most appropriate cost function is the bit error rate [4],[5]. The approximate minimum bit error rate (AMBER) [4] and the least bit error rate (LBER) [5] are two of the most successful and suitable algorithms for adaptive implementation. However, these minimum bit error rate (MBER) algorithms usually require long training sequences to converge to lower bit error rates than those achieved by the techniques that employ the MSE cost function. In this work, we investigate MBER Gradient-Newton algorithms that can speed up the convergence of the multiuser receiver, requiring shorter training data. The proposed algorithms, denoted Gradient-Newton-AMBER and GradientNewton-LBER, are similar to the well known LMS-Newton [6] algorithm and employ the error functions used in the AMBER and the LBER, respectively. We perform a comparative analysis of linear MUDs, employing the LMS [6], the AMBER [4], the LBER [5], the LMS-Newton [6], the Gradient-Newton-AMBER and the Gradient-Newton-LBER adaptive algorithms. This paper is organised as follows. Section II briefly describes the DSCDMA system model. The adaptive linear multiuser receiver structure and the stochastic gradient algorithms are presented
0-7803-7589-0/02/$17.00 ©2002 IEEE
in Sections III and IV. Section V is dedicated to the GradientNewton based algorithms. Section VI shows and discusses the simulation results and Section VII gives the concluding remarks of this work. II. DS-CDMA S YSTEM M ODEL consider a synchronous DS-CDMA system with users, Wechips per bit and binary symbols, as shown in Fig. 1. These symbols are spread with random generated spreading sequences, modulated and transmitted through a communication channel, where denotes the bit of user , the signature sequence for user "! #$&% ')(*(*(+#$&% ,.- /10 is normalized to have a unit length, and the channel impulse response is given by
798;: 2 435 6 < ' @&A A3 : AB =?> : ' introduces a delay of one chip time in where the operator 3 the transmitted signal.
Fig. 1. Model of synchronous DS-CDMA system.
At the receiver the composite signal is demodulated, filtered by a chip-pulse matched filter and sampled at chip rate yielding the observation vector
H
FGG
C 6 ED
FGG IJ GG K
K I J
H K ...
..
O O &PQ&P@S RT .. .
O &PQUVWRT
(*(*( K
.
(*(*(
..
.
..
.
K
L MM
.. .
K IJ N
MM
L MM N VYXZ [ E\ .VYXZ
PIMRC 2002
]9
where the Gaussian noise vector XZ ! '; (*(*( ,.- / 0 7 , \ O is the noise-free signal vecwith ! XZ AX)0Z / tor, the user bit vector is given by ! B (*(*(6 I /10 , the user signature sequence matrix is described by ! 5'Y(*(*( / J , the diagonal user signal amplitude matrix is rep resented by 9 ' (*(*( , and the matrix H is expressed by
FG !#"$"%"& ' 8 ( H ) ! "$"%" ..
.
..
.
' 8 ( "% !"$" %" "$"* ' 8 ( ..
.
N
LM
+-, .
The multiple access interference (MAI) is originated from the non-orthogonality between the user signature sequences. The intersymbol interference (ISI) span L depends on the length of the channel response, which is related to the length ofthe chip S sequence. For (no ISI), for S ] , for ] Y and so on. Consider a one-shot MUD, depicted in Fig. 2, whose observation vector is C ;')(*(*( ,.- 0 . The detected symbols for this one shot receiver and user are expressed by:
21 3/54 06 1 /7/ 4 0 60 8 3 :9 ; %; =< > [ @? BADC 0 C [ @? BE BF5 where A + 6 ! G ')(*(*(% G ,.- /10 foris theuserreceiver weight vector and E + is the estimated symbol and symbol in a system with
users.
Fig. 2. One-shot Linear Multiuser Receiver.
O PRQ O TS PRQ O)*U =VW TS + %? BE + [ U =VW TS ? E [ U TS + E + X1ZY[U \ where E + is given through (4) and 5 + is the desired symbol taken from the training sequence for user and symbol . B. The AMBER algorithm
Given a user transmitted training sequence , the bit error probability ) , for the linear receiver, is expressed by:
The AMBER is a stochastic gradient that attempts to approximate the exact MBER performance [5]. The algorithm is appealing due to its very low complexity, simplicity and straightforward extension to the complex signalling case. The MUD solution that minimises the BER criterion via the AMBER algorithm employs the vector function ? [5] to approximate an expression for a coefficient vector + that achieves a MBER performance with linear receiver structures, as described by:
BA A
+ A\ *d ?BA + 6 J]$^`_ + a A A 0 a A\ c fe b where + is the desired transmitted symbol for user , taken ^ from the training sequence, A( is the Gaussian error function
and \ are the received samples without noise taken from the outputs of chip-matched filters. A simple stochastic solution for can be derived by using ? and adjusting the receiver weights by:
A
BA A VWB 6 JWfA l-monq ps Wr @l-VLmonqtuKgl-mo?n BA h ^Iikj v pwWfl-mon vfxzy inside the expected Note that the quantity value operator in (7) corresponds to the conditional bit error probability given the product 5 + A\ . This W quantity can be replaced in (7) by an error indicator function j given by: j W 6 ] A L? E { where E + is the estimated symbol and 5 + is the desired signal provided by the training sequence. The AMBER algorithm, as devised for linear MUDs [5], is described by the following equalities:
III. S TOCHASTIC G RADIENT A LGORITHMS A number of adaptive algorithms can be used to adjust the receiver coefficients. In this section, we describe stochastic gradient algorithms that adjust the parameters of the receivers based on the minimisation of the mean square error (MSE), the LMS [6], and the bit error rate (BER) cost functions, the AMBER [4] and the LBER [5]. A. The LMS algorithm The adaptive solution for the linear MUD via the LMS algorithm [1] is based on the MMSE error criterion formed by the error signal B+ 6 + ) 0 C , and is described by:
H IA A VWB 6 JA @VLKMHB C
*N9
where is the desired signal for the -th user taken from the training sequence, C is the observation vector for the linear MUD and is the algorithm step size.
K
J]$^`_ + a A 0 a A\ + A\ *d A + |b A + VWB 6 JA @VLK ~}q 9 j W QR + A\ < + A\ f A VWBC 6 A + @VLK 9 j W $W + A\ f< Since \ XZ , and j W and + are statistically independent, ! j $ + AXZ / ! + / ! j W AXZ /. Y weand have thus: A + VWB 6 JA @VLK } j W $ + C A Y A
+ VWB 6 JA @VLK
The AMBER stochastic gradient update equation for the linear MUD is given by:
A
+ VWB JA + @VLK? j W $ + C
A9B
? E
jW
'
A In practice, a modified error indicator function + + ) is employed, where the threshold is responsible for increasing the algorithm rate of convergence. This algorithm updates when an error is made and also when an error is almost made, becoming a smarter choice for updating the filter coefficients.
C. The LBER algorithm
E
The MUD BER depends on the distribution of the decision variable + , which is a function of the weights of the receiver. The sign-adjusted decision variable for the linear receiver + + is drawn from a Gaussian mixture, described by:
E W
E E W [ ? + S A 0 \ .VLA 0 XZ %U E W 6 @? E @V
AB]9
where the first term of (12) is the noise free sign-adjusted MUD output. Consider that samples of the transmitted symbols *+ and samples of the estimated symbols + are available from the samples + Z of a training sequence. A kernel density estimate [5] is given by:
E
AMBER, a non-zero defines a region boundary where the algorithm will continue to update, in the LBER, the effect of the distance from the decision boundary is controlled by an exponential term [5]. Indeed, this can be viewed as a soft distance metric, the size of an update is a continuous and decreasing function of the distance from boundary and both algorithms the with two parameters that require have a complexity of tuning.
b
IV. G RADIENT-N EWTON A LGORITHMS Gradient-Newton algorithms [6] incorporate second-order statistics of input signals, increasing their convergence rate. They usually have a faster convergence rate than gradient techniques, although they require a higher computational complexity. In practice, estimates of the autocorrelation matrix and the gradient vector are used to converge to the desired solution. In addition, to avoid the inversion of the autocorrelation matrix, the matrix inversion lemma is also employed. The update equation of Newton´s method is given by
A
VWB 6 A + ]` c : ' p R
A{
where p is the autocorrelation matrix of the observation vector c R is the gradient vector. W + W . < "!$# + W #&%( ' ) ++* 1,f++ -R... ,++-R..0/ 2 C and In practice, only estimates of the autocorrelation matrix / p c R and the gradient vector
are available. These estimates
, . can be applied to Newton´s formula + to devise an update rule where 3 is the radius parameter of the kernel density estimate give by: [5]. Substituting the expected value of expression in (13) with a single point estimate, we have: A + VWB 6 JA IK c > : ' > p R ] Y 4 W + W . 5!$# + W #6%( ' ) ++* , ++ R- ... , ++R- ..0/ 2 The convergence factor K is introduced to protect the algorithm from divergence, which is originated by the use of noisy esti/
(7 . mates of c and p R . + The probability of error for user is estimated by: To obtain an unbiased estimate of the observation matrix ced , we employ the following weighted sum: 8:9 W 8 + W= . @? 4 W + W .C*D W @E _ %(' ) ++* f, ++ R- .. GF, ++R- . > [ gf C C 0 @VEAh f > SB ]TB (BA + . /b c
IH . where f is c a small factor chosen in the range + w V Y if 4 Y ( and 1 The gradient terms of are: C is the observation vector. WLK > J 8:9 W i /L( O N( P QRB MP N R y %(' ) ++* , ++R- .. !S# T ++-R. GF ,f++-R. F To avoid the required inversion of , we use the matrix c J , lemma, described by: + . /VU +
