Multiuser Detection of Synchronous MC-CDMA in ... - Springer Link

Neural Processing Letters 18: 49–63, 2003. # 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Multiuser Detection of Synchronous MC-CDMA in Multipath Fading Channels Using Hopfield Neural Networks EBRAHIM SOUJERI1 and HUSEYIN BILGEKUL2 1

Department of Computer Engineering, Cyprus International University, Nicosia, North Cyprus, (via Mersin 10, Turkey). e-mail: [email protected] 2 Department of Electrical and Electronic Engineering, Eastern Mediterranean University, Famagusta, North Cyprus, (via Mersin 10, Turkey). e-mail: [email protected] Abstract. Sub-optimum multiuser reception using Hopfield Neural Network for synchronous Multicarrier Code-Division Multiple Access signals in a multipath fading channel is studied with respect to near-far ratio. We have shown that by the appropriate choice of Hopfield Neural Network parameters from the channel state information, the Hopfield network can collectively resolve the multipath fading effects and the multiple-access interference in the system. Moreover, the Hopfield Neural Network demonstrates multiple-access interference resilient performance regardless of the number of paths resolved at the receiver. We have also investigated the bit-error rate performance of the system with respect to channel estimation errors. Results show that performance of the proposed detection scheme is influenced by the correctness of the estimated channel state information. Key words. Hopfield neural network, multi-carrier code-division multiple-access, multipleaccess interference and multiuser detection

1. Introduction In a Code Division Multiple Access (CDMA) system used for wireless communications many users share the same radio band in the code domain by using different pre-assigned unique code sequences for different users. By the use of unique codes, users in CDMA can transmit at the same time and over the entire bandwidth. CDMA codes are generated using logical shift registers. CDMA codes are in fact, a sequence of þ1s and 1s queued next to each other. Each of these þ1s and 1s is called a chip. The length of the sequence, which is defined by the number of chips in one code, is the gain of the CDMA system. A general block diagram of CDMA system for a single (kth) user is shown in Figure 1. User data is being spread using the CDMA code and then assigned to different orthogonal subcarriers across the frequency domain by using the Multicarrier CDMA (MC-CDMA) [1] modulation technique. MC-CDMA is an accepted technique for future mobile communication systems. The idea is to transmit information not over a single carrier, which may be strongly fading, but to spread it over several independently

50

EBRAHIM SOUJERI AND HUSEYIN BILGEKUL

Figure 1. A general block diagram of the MC-CDMA system for the kth user only.

fading subcarriers. Thus, diversity gain and system performance improvement can be achieved. The conventional signal detection technique in multiuser communications employs matched filters (MF) because of its simplicity. A key factor that restricts the performance of the conventional detector is that every user interferes with all other users and the interference degrades the performance. This interference is known in CDMA communications as Multiple-Access Interference (MAI). The problem becomes increasingly significant if the received energy levels of the users are dissimilar; this is known in CDMA as the near-far problem. An alternative to the conventional receiver is to use a receiver designed to take the multiple-access interference into consideration, i.e., multiuser detection. A multiuser detector is a detector that can make a joint detection of the data of all users. Optimum multiuser detection (OMD) obtained through the maximum likelihood function has superior performance but with the price of computational complexity which grows exponentially with the number of users. Lower complexity multiuser detectors that provide near-optimum performance are of great interest. The work on multiuser receivers has demonstrated that even suboptimal detectors with a significantly lower implementation complexity than the optimal detector can greatly improve the detection performance and capacity of multiuser communication systems. That is why we consider the Hopfield neural network detector as an alternative for detecting the multiuser MC-CDMA signals. Multiuser reception using Hopfield Neural Network (HNN) [2] detector in an Additive White Gaussian Noise (AWGN) channel has earlier been studied and shown to have sub-optimum performance by Miyajima and Kechriotis [3, 4]. Even though HNN achieved excellent sub-optimum results in an AWGN channel, it failed to exhibit Multiple Access Interference (MAI) resilient performance in a Multipath Fading (MPF) channel. The reason for this failure is that HNN parameters were set without taking into consideration the channel parameters. The main contribution of this letter is to design a HNN detector capable of jointly canceling MPF effects and MAI from the received signal in the system. This is achieved by deriving the Hopfield receiver parameters using the Channel State Information (CSI) provided by a channel estimator. We will also track the performance of the proposed HNN detector under erroneous or inaccurate CSI conditions.

MULTIUSER DETECTION OF SYNCHRONOUS MC-CDMA

51

2. Channel Model We are considering an uplink scenario of multiuser communication where the base station receives the signals of all K active synchronous users with different energies and over L different paths simultaneously. In this study, the reception is assumed to be performed without power control so that the received signal energies of different users distinct, i.e. if we consider two CDMA users k and q, then under the absence of power control, Ek is not necessarily equal to Eq at the receiver. The detailed transmitter block diagram of the MC-CDMA system employed is shown in Figure 2a. The kth user input binary data bk is serial to parallel converted so that every P consecutive bits are transmitted in parallel as symbols. Parallel transmission is necessary for high bit rate data applications. Each parallel data is multiplied by the spreading sequence ak of length L so that L copies (chips) of the pth information bit bk;p are transmitted on L different orthogonal and overlapping subcarriers. The chips are Binary Phase Shift Keying (BPSK) modulated to different subcarriers spaced 1=Ts Hz apart, where Ts ¼ PTb ¼ PLTsmp with Ts , Tb and Tsmp representing symbol, bit and chip sampling intervals, respectively. Bit duration Tb and symbol duration Ts after serial-to-parallel (S/P) conversion are illustrated in Figure 2b. Notice that chip durations are also equal to Ts . The mth frequency fm of the orthogonal subcarriers is given by fm ¼ f1 þ ðm 1Þ=Ts , m ¼ ðp 1ÞL þ l for l ¼ 1; . . . ; L and p ¼ 1; . . . ; P. Here, f1 is the first subcarrier frequency. The transmitted MC-CDMA signal Sk ðtÞ for user k, can be written in terms of the energy of kth user Ek as follows Sk ðtÞ ¼

P L X pffiffiffiffiffiffi X Ek bk;p ak;l cosðwm tÞ 0 4 t < Ts p¼1

ð1Þ

l¼1

where the radian frequency wm ¼ 2pfm . The transmitted signal of the kth user Sk ðtÞ carries P bits using L chips for each bit such that a total of PL subcarriers are modulated. A fading channel is a channel in which the transmitted signal reaches the receiver after experiencing many reflections and refractions through obstacles in the communication medium. Reflections may cause the received signal to add destructively. The destructive addition of signals is known as fading. Many multipath channel models have been used in the literature. In this Letter, the multipath fading channel is modeled as consisting of L resolvable Rayleigh faded paths. The low-pass impulse response gk ðtÞ of the channel for user k is expressed as [5] gk ðtÞ ¼

L X

bk;l e jgk;l dðt tk;l Þ

ð2Þ

l¼1

where bk;l e jgk;l is a complex Gaussian random variable with zero mean and variance s2l and tk;l represents the channel delay of the lth path of the kth user. The path gains bk;l are independent for different k and l and are Rayleigh distributed, while the angle gk;l is a uniform random variable in the interval [0, 2p). The complete

52


Figure 2. Block diagram of the MC-CDMA transmitter of the kth user. a) The overall MC-CDMA transmitter structure for P parallel bits. b) Representation of the time durations of the serial and parallel data. c) The MC-CDMA MF receiver for the kth user.


53

Figure 3. The complete multiuser communication channel employing L multipath components.

multiuser communication channel employed in this work is illustrated in Figure 3. The block diagram in Figure 3 shows the multipath fading channel effect together with AWGN and the interference from other users affecting the kth user. The MPF channel response in the frequency domain Gð f Þ results in different channel gains Gðmf1 Þ and phase angles ffGðmf1 Þ at the subcarrier frequency mf1 . Thus, the complex channel frequency response in general is given by Gð f Þ ¼ Gð f ÞffGð f Þ. The composite synchronous MC-CDMA [6] received signal in MPF channel with AWGN n(t) having spectral density N0 =2 is given below rðtÞ ¼ nðtÞ þ

K pffiffiffiffiffiffi X P L X X Ek bk;p ak;l Gðmf1 Þ cosðwm t þ ffGðmf1 ÞÞ k¼1

p¼1

ð3Þ

l¼1

where ak;l is the lth chip in the spreading sequence of user k and m is the subcarrier index of the MC-CDMA signal.

3. Interference Analysis In order to derive HNN parameters from the CSI, we must proceed in the following manner. We express the channel induced spreading sequence of the kth user code ck in terms of the kth spreading sequence ak , and the frequency response of the channel at the mth subcarrier Gðmf1 Þ as follows

54


ck ¼

L pffiffiffiffiffiffi X Ek Gðmf1 Þak;l

ð4Þ

l¼1

Having done this, we can express the cross-correlation matrix H in terms of its (q,k)th element hq;k as given in (5) below, where ()* stands for complex conjugate operation. Note that hq;k represents the correlation between the spreading sequences of the kth and the qth users, which results in MAI between these users. Z Ts hq;k ¼ cq ck dt ð5Þ 0

The pth MF output of the kth user, yk;p is the sum of the MF outputs of all L chips, and can be expressed as below yk;p ¼

L X l¼1

yk;p;l ¼

L Z X l¼1

Ts

rðtÞak;p;l cosðwm tÞ dt

ð6Þ

0

The parameters H and yk ¼ ½yk;1 ; . . . ; yk;P T will be used in realizing the HNN detector, as explained in the next section. The conventional matched filter detection is based on the sign of Re yk in obtaining an estimate of the binary output as ðb^k ÞMF ¼ sgn Re yk . A MF receiver for MC-CDMA is shown in Figure 2c. The chips are collected by MF receivers as yk;p;l and combined to form the decision variables yk;p . It is reminded that in a synchronous CDMA scenario, one could perform symbol-by-symbol detection without caring about left- or right-overlapping bits. This is because no overlapping occurs in synchronous transmission. At this point, we must emphasize that the presence of accurate and flawless CSI to the receiver is very crucial. Classically, one could find an estimate of the channel coefficients by sending training sequences or using a pilot channel. These approaches rely on periodic transmission of training sequences [7], making the identification of CSI feasible since both input and output signals are known during the transmission of these sequences. Recently, some training-based channel estimation algorithms were proposed for CDMA systems [7]. One could use either of the methods mentioned here to deliver the required CSI to the receiver.

4. Optimum and Hopfield Detection 4.1.

MECHANISM OF OPTIMUM DETECTION

The optimum multiuser detection considers the optimum multiuser demodulation of PK bits, which includes data from the 1st bit to the Pth bit of all the K active users in the system, where P is the packet size. The reason to consider the optimum data demodulation is to derive a corresponding suboptimum receiver with lower complexity. Since the noise is white and Gaussian and all the transmitted data sequences are assumed to be equiprobable, the maximum-likelihood sequence detector selects the estimates of the data sequence expressed in vector form b^p ¼ ½b^1;p ; . . . ; b^K;p T ,


55

for p ¼ f1; 2; . . . ; Pg, where ½ T stands for the transpose operation. The estimation is done by minimizing the Euclidean distance dE between the received signal rðtÞ and ^ ^ can be respectively the estimate of the transmitted signal SðtÞ. The terms dE and SðtÞ expressed as Z Ts ^ 2 dt dE ¼ jrðtÞ SðtÞj ð7Þ 0

S^k ðtÞ ¼

P L X pffiffiffiffiffiffi X b^k;p Ek ak;l cosðwm tÞ p¼1

ð8Þ

l¼1

Minimization of the Euclidean distance given in (7) is expressed as in (9) in terms of the likelihood function L that is also given by (10) [3, 4]. ^bOMD ¼ Arg Min ðRefLgÞ ð9Þ b2f1gPK

L ¼ yT b þ 12bT Hb

ð10Þ

Where y 2 C1PK in (10) is the MF output with its kth element as given in (6) and b 2 f1gPK1 is a pool in which the optimum detector searches for the estimate bÔMD 2 f1gPK1 of the originally transmitted sequence b. The optimum detection algorithm finds an estimate bÔMD by substituting a stream b into L such that the real part of L is minimized. The stream b begins with b ¼ ½1 1 1 T 2 f1gPK1 and ends up with b ¼ ½1 1 1 T 2 fþ1gPK1 ; for every stream b, the algorithm stores the value of RefLg in a stock and after finishing with all streams, the algorithm checks which stream resulted in a minimum value for RefLg and chooses that stream as bÔMD . The OMD detector has a complexity in the order of 2PK. This complexity is considered to be high in CDMA since the product of number of users into the packet size is very high, i.e. PK 1. Inapplicability of the optimum detector for practical CDMA communications has led people in the field to search for alternative tools to carry out detection of multiuser CDMA signals [3, 4].

4.2.

MECHANISM OF HNN DETECTION

In applying the HNN to our multiuser multicarrier system, we consider K users who transmit P bits in parallel, so, the relevant number of inputs to HNN detector is PK. The bias vector I ¼ ½I1 I2 IPK T and the interconnection matrix T of HNN [3, 4] are taken from the communication system parameters. HNN output vector is denoted by V ¼ ½V1 ; . . . ; VPK T . A simple 5-neuron model of the HNN detector that has been deployed as a CDMA detector in this work is shown in Figure 4. Hopfield [2] has shown that if interconnection weights are symmetric ðTqi ¼ Tiq Þ, the state of the network always converges to a stable state. The stable states of the network are the minima of the Energy Function of HNN described by

56


Figure 4. Interconnection diagram of Hopfield neural network receiver with 5 neurons.

EHNN ¼

PK X i¼1

Vi Ii

PK X PK 1X 1 Tij Vi Vj ¼ IT V VT TV 2 i¼1 j¼1 2

ð11Þ

Once the output of HNN at certain iteration is determined, back propagation starts to update the states of HNN nodes for next iteration. Considering that the states of HNN change as their dynamics change, this change is best characterized by the equation of motion of the HNN neurons which is given by [3, 4] dUi ðtÞ @EHNN Ui ðtÞ ¼ dt t @Vi

ð12Þ

Considering the discrete-time approximation of, ½ðd=dtÞUi ðtÞ that is given by dUi ðtÞ Ui ðt þ DtÞ Ui ðtÞ ¼ dt Dt

ð13Þ


57

and letting Dt ¼ 1,one might express the change given in (13) as DUi ðnÞ ¼ Ui ðn þ 1Þ Ui ðnÞ

ð14aÞ

which leads to Ui ðn þ 1Þ ¼ DUi ðnÞ þ Ui ðnÞ

ð14bÞ

where n represents the time index. Note that t is replaced by the discrete time index n. P HNN Noting that @E@V ¼ i6¼j Tij Vj Ii , the expression in (14a) may be rewritten as i DUi ðnÞ ¼

X i6¼j

1 Tij Vj þ Ii Ui ðnÞ t

ð15Þ

where t ¼ RC is the time-constant of HNN neurons used in VLSI design [4]. The expression given in (15) represents the change in HNN states at the nth iteration. The HNN output after the ðn þ 1Þst iteration is represented as 2 3 31 02 V1 ðn þ 1Þ U1 ðn þ 1Þ B6 U2 ðn þ 1Þ 7C 6 V2 ðn þ 1Þ 7 B6 6 7 7C 6 V3 ðn þ 1Þ 7 ¼ jB6 U3 ðn þ 1Þ 7C ð16Þ B6 6 7 7C @4 4 5 5A g g VPK ðn þ 1Þ

UPK ðn þ 1Þ

where the sigmoid function is taken as jðÞ ¼ tanhðÞ. We may now transform the minimization of the likelihood function given in (10) into the minimization of HNN energy function that is described in (11) by setting HNN biases I as MF outputs y, and interconnections T as cross-correlations of spread sequences H as below I¼y

ð17Þ

T ¼ H

ð18Þ

Once the transformations given in (17) and (18) are done, HNN is operated for sufficient iterations, typically around 10 iterations until convergence is established. The data estimate b^ HNN , could be simply driven by hard-limiting the real part of HNN output V, as follows b^ HNN ¼ sgnfReðV Þg

Figure 5. The HNN detection strategy for multiuser CDMA.

ð19Þ

58


Here, the output vector V 2 CPK1 is obtained after the network has reached a stable state by performing the necessary number of iterations required. The block diagram of the complete CDMA receiver that utilizes HNN detector based on (17), (18) and (19) is given in Figure 5.

5. Discussion To carry out detection in an AWGN channel, the despreading sequences at the receiver ck has no frequency-selective fading characteristics, thus, by letting Gð f Þ ¼ 1 we get the spreading sequences required for AWGN channel, as given below ck ¼

L pffiffiffiffiffiffi X Ek ak;l

ð20Þ

l¼1

The generalization of HNN parameters from (20) to (4) is the contribution of this study. It must be mentioned that the availability of the CSI to the receiver is very important. The reader is reminded that the HNN parameters that are drawn from the communication system are: the bias vector I and the interconnection matrix T. Once HNN interconnections T ¼H and biases I ¼ y are estimated accurately, it is known that HNN is guaranteed to provide a sub-optimum solution [3, 4].

6. Simulation Results The following transmission parameters were used throughout all simulations in this Letter: K ¼ 10 users, L ¼ 15 chips / bit, P ¼ 20 bits/symbol and E1 =N0 ¼ 7:5 dB. We have used Gold sequences as spreading codes because of their low cross-correlation property. With Gold sequences of 15 chips, the CDMA system can serve a maximum number of 15 users, thus with K ¼ 10 users, we are considering a transmission case which is below full-load capacity. The MAI is expressed as the ratio of the energies of the other users Ei (assuming that they are equal) to the energy of the desired user E1 . The ratio Ei =E1 is an indication of MAI since the other users are considered as interferers. The bit-error-rate (BER) performance of HNN vs. MAI in MPF channel without using CSI and for various number of multipath components L is shown in Figure 6. The multipath performance of the AWGN-designed HNN degrades drastically compared to AWGN channel. In fact, no difference is observed between MF and HNN performance in terms of MAI-resilience, thus, the AWGN-designed HNN is rendered totally useless in a MPF channel. The BER performance of HNN versus Ei =E1 in MPF channel using perfect CSI estimate for various number of multipath components L ¼ 1; 3; 5 is shown in Figure 7. The Gaussian channel performances of MF and HNN detectors are also shown as reference. As observed in Figure 7, HNN with CSI achieves a good BER performance regardless of the number of fading paths and especially for increasing Ei =E1 . This is because HNN detection is based on MAI estimation and cancellation through the use of the cross-correlations H of the channel. When MAI is high, better


59

Figure 6. BER performance of HNN vs. MAI ðEi =E1 Þ in MPF channel without using CSI and for various number of multipath components L. The performances of MF and HNN detectors in AWGN are also shown for reference.

estimation and cancellation is done, resulting in better BER performance. Therefore, the BER performance in the region above 5 dB is better than BER performance in the region below 0 dB. Furthermore, BER performance of HNN using erroneous CSI estimation for various CSI errors in the range 2% to 12% is examined and shown in Figure 8 for a 3path channel. The percentage error mentioned here indicates equal percentage errors in both gains bk;l and phases gk;l of all of the multipath components. As we can see,

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Figure 7. BER performance of HNN vs. MAI ðEi =E1 Þ in MPF channel using perfect CSI estimate for various number of multipath components L. The Gaussian channel performances of MF and HNN detectors are also shown as reference.

the MAI-resilient property of HNN continues for acceptable error rates of few percents. Three dimensional plot of the performance of HNN versus MAI ðEi =E1 Þ versus CSI error, in a MPF channel having L ¼ 3 paths with various CSI errors in the range 0 to 12% is shown in Figure 9. Consequently, with easily achievable CSI errors of few percents, HNN detector has an acceptable (less than 103 ) BER performance when near-far ratio is high. The case of 0 % error in Figure 8 and 9 corresponds to the delivery of perfect CSI to the HNN receiver.


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Figure 8. BER performance of HNN vs. MAI ðEi =E1 Þ in MPF channel using imperfect CSI estimate with various CSI errors and L ¼ 3 paths fading channel. The Gaussian channel performances of MF and HNN detectors are also shown as reference.

7. Conclusions Unlike the AWGN-designed HNN, where the MAI-resilience property of HNN receiver was confined merely to AWGN channels, with a poor performance in MPF channels, the new design exhibits an outstanding MAI-resilient performance in MPF channels. We have shown that, if the HNN is to remove MAI from the received signal in a MPF channel, its parameters must be extracted from the communication system encountering the fading effects. The spreading sequences, if correctly attributed, build up a right measure for the cross-correlation matrix, which is used by the HNN detector to remove MAI. Detection using HNN is achieved by adjusting

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Figure 9. Three dimensional BER performance of HNN vs. MAI ðEi =E1 Þ vs. CSI error, in a MPF channel having L ¼ 3 paths with various CSI errors in the range 0 to 12%.

the bias vector and the interconnection matrix of the HNN detector by the aid of CSI. The proposed HNN detector shows a superior MAI-resilient performance regardless of the number of paths resolved at the receiver. However, the performance degrades when the proposed HNN performs the detection process under corrupt CSI conditions. Therefore, we come to the conclusion that the proposed HNN detector is guaranteed to sustain a sub-optimum performance in a MPF channel provided that the channel estimator is catering a correct CSI.

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4. Kechriotis, G. I. and Manolakos, E. S.: Hopfield neural network implementation of the optimal CDMA multiuser detector, IEEE Transactions on Neural Networks, 7(1) (1996), 131–141. 5. Proakis, J. G.: Digital Communications. (3rd Edition, ed.), McGraw-Hill, Singapore: 1995, pp. 759–762. 6. Sourour, E. A. and Nakagawa, M.: Performance of orthogonal multicarrier CDMA in a multipath fading channel, IEEE Transactions on Communications, 44(3) (1996), 356–366. 7. Stro¨m, E. G. and Malmsten, F.: A maximum likelihood approach for estimating DSCDMA multipath fading channels, IEEE Jour. on Sel. Areas of Commun., 18(1) (2000), 132–140.