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A Neural Network Approach for Blind Multiuser Detection in DS-CDMA Communication Systems D. Tarchi, R. Fantacci, M. Forti, M. Marini, G. Vannuccini Electronics and Telecommunications Department, University of Firenze. Via di Santa Marta 3, 50139 Firenze, Italy Ph. & Fax +39-055-4796485 Email: {tarchi,fantacci,vannucci}@lenst.det.unifi.it A BSTRACT In order to enhance data rate for multimedia services, new advanced receivers for next generation mobile communications are developed. Adaptive blind multiuser detection has been widely proposed for applications in CDMA (Code Division Multiple Access) wireless communication systems for its principal advantage of eliminating training sequence to set-up receiver filter coefficients. Main drawback of this technique is that it reaches the optimum behavior after a certain number of bit times, which precludes its use in typical time-varying environments. In this paper, a new neural network approach is proposed in order to solve this drawback. In particular, this paper considers the use of a modified Kennedy-Chua neural network, based on the Hopfield model. Simulation results are given to demonstrate the effectiveness of the proposed approach in different time-varying application scenarios. 1

I NTRODUCTION Code Division Multiple Access (CDMA) has recently gained a large development due to third generation mobile communication systems [1]. This technique permits an higher number of users communicating simultaneously with high bit rate. Many research activities have been focused on the most suitable receiving techniques for Direct Sequence CDMA (DS-CDMA) mobile systems, and in particular the multiuser adaptive blind detection algorithm has been proposed by Verdù [2]. This receiver has the main advantage of reaching Minimum Mean Square Error (MMSE) performance without requiring any kind of training sequence to detect the received signal. However, blind receivers need for a certain number of bit times to reach the optimal solution, since the path toward the optimum filter coefficients set is performed in a step by step approximation, based on the steepest descendant gradient technique [2]. Neural networks are therefore widely known for their very low convergence time, being based on parallel processing architectures. The idea proposed in this paper is to use a neural network approach in order to accelerate the convergence process of the adaptive filter coefficients set toward the optimum solution in the blind detection al-

gorithm. Among the many types of neural networks, the one studied by Kennedy and Chua [3] has been selected in this paper. Such a network derives from the Hopfield neural network model [4], to which the capability to solve non linear programming problems is added. The Kennedy and Chua model has been modified in this work, in order to perform a nonlinear optimization subject to two nonlinear constraints, one for the energy and one for the orthogonality of the solving variable. The rest of this paper is organized as follows. In Sec. 2, Kennedy-Chua neural networks are described. Section 3 describes the adaptive blind algorithm and the proposed neural approach. Simulation results of the proposed system in a mobile CDMA communication environment are then shown in Sec. 4, and, finally, our conclusions are discussed. 2

K ENNEDY-C HUA N EURAL N ETWORKS Neural networks have been widely studied in the last decades, and their fast processing capabilities have been largely highlighted [5]. Many neural network types can be classified, depending on the presence of feedback between output and input nodes, on the multi-layer structure with which they are modeled, or on the analog or digital domain implementation. Among these types, Hopfield networks, which are single-layer feedback neural networks, have been selected in this work for their interesting capabilities in minimization problems. One of the main applications of an Hopfield neural network is the minimization of a suitable cost function, which is obtained through n properly connected neurons with fixed weights and threshold value. Since the analog form of an Hopfield neural network is represented through a gradient type network, the values of the output voltages tend in a few time constants to the minimum value of the energy function to be minimized, and such a network could be entirely made of simple electronic devices, such as capacitors, resistors, and operational amplifiers, also suitable for the implementation in VLSI (Very Large Scale of Integration) technology. In [3] Kennedy and Chua proposed an extension of an Hopfield neural network for nonlinear programming with nonlinear constraints; this network was a neural implementation of the canonical nonlinear circuit proposed by Chua and Lin [6]. A generic non-

linear problem is based on the minimization of a general cost function: φ(x1 , . . . , xq ) (1) subject to nonlinear constraints: f1 (x1 , . . . , xq ) ≥ 0 ··· fp (x1 , . . . , xq ) ≥ 0

(2)

where p and q are two independent integer numbers. The circuit equations which minimize (1) subject to (2) are: p

∂φ X ∂fj dvi =− − ij , i = 1, ..., q C dt ∂vi j=1 ∂vi

(3)

where vi is the output voltage of the generic i − th neuron, ij = gj (fj (v)) is the output current of j − th constraint, and C is the output capacitor of i − th neuron. We have also indicated with gj (·) the nonlinear continuous function used to impose the j − th constraint. The system in (3) tends to an equilibrium point, which is a minimum of its Lyapunov function, also called total cocontent function [3], defined as: p Z fj (v) X E(v) = φ(v) + gj (x)dx (4) j=1

0

where v = (v1 , ..., vq ). To model non-ideal current sources imperfections, a correction term is added to (3), thus obtaining: p

C

∂φ X ∂fj dvi =− − ij − Gvi , i = 1, ..., q dt ∂vi j=1 ∂vi

(5)

where G takes into account the above-mentioned imperfections. 3

P ROPOSED S YSTEM In [7], a blind adaptive multiuser detection technique has been proposed, whose performance tend to the MMSE detector without requiring any training sequence. In the blind detector, the impulse response of the receiver filter is decomposed in two parts, one containing the desired user signature waveform, and the other, which is adaptive and orthogonal to the first component. If the output energy of such a receiver is considered, the optimum adaptive filter coefficients set is the one which minimizes the mean output energy. If we denote with y(t) the signal at the receiver input, with sk (t) the signature waveform of the k − th user, which transmits an information bit bk with a signal amplitude of Ak , and we assume the AWGN (Additive White Gaussian Noise) channel having a noise signal n(t) with variance σ 2 , assuming bit synchronism between users, we have [7]: y(t) =

N X

Ak bk sk (t) + σn(t)

will be considered in the array notation, so that the signal from the channel y(t) becomes y, taking the signal values at the sampling time in the information chip interval. According to [7], the receiver filter is split into two parts, where the matched filter makes the correlation between the received signal y and the desired user signature waveform s1 , while the other filter represents an orthogonal component which changes in an adaptive way. The adaptive filter coefficients x1 change their value in order to minimize the output energy. The Minimum Output Energy (MOE) is then defined as: MOE(x1 ) = E[(hy, x1 + s1 i)2 ]

which tends to the MMSE solution under the hypothesis [7]: hx1 , s1 i = 0 (8) that is, the desired user signature code and the adaptive blind filter output must be two orthogonal vectors. In (8) h·, ·i stands for the canonical scalar product. However, the orthogonality constraint shown in (8) is not the only one applied to the output variable of the considered filter. According to [7], another constraint must be set on the output energy of the adaptive filter component, in order to reduce the so-called mismatch effect. This is given by: kx1 k2 < χ

k=1

where N is the maximum number of users in the communication system. In the following, the signal wave-forms

(9)

where χ is a constant parameter defined in [7]. The blind adaptive detector problem has then been brought back to a quadratic optimization, subject to the two nonlinear constraints shown in (8) and (9). The steepest descendant rule proposed in [2, 7] to solve this problem can lead to a high convergence time, and this can be critical when large dynamic channel models are considered, or when many users access the system simultaneously. To address this problem, neural networks parallel processing capabilities have been considered. The system we have proposed has the main objective to reach the optimal solution in a few bit times, by using neural networks fast parallel processing capabilities. According to [7], the blind adaptive algorithm requires the minimization of the following energy function: h i 2 E(x1 ) = E (hy, s1 + x1 i) (10) where y is the signal received from the channel, described in (6), s1 is the signature sequence of the desired user and x1 is the orthogonal adaptive filter coefficients set. As previously described, the coefficient set of the adaptive filter x1 must satisfy two constraints in order to reduce multiple access interference (MAI) and mismatch effect. Namely, (10) must be subject to the following constraints: hs1 , x1 i = 0 2

(6)

(7)

kx1 k < χ

(11) (12)

where χ is the energy constraint mentioned in (9). To reach a higher convergence speed than the classical blind algorithm, we have used a neural network derived

x˙ 1 = −∇V (x1 )

(13)

where V (x1 ) is the total energy function. As in (4), this is defined as the sum of the energy function to be minimized and an additive term used to impose the energy constraint on x1 : V (x1 ) = E(x1 ) +

Z

f (x1 )

g(ρ)dρ

(14)

0

where: f (x1 ) = χ − kx1 k2 ≥ 0 is the function defining the energy constraint, ( 0, ρ≥0 g(ρ) = Kρ, ρ < 0

(15)

(16)

is the continuous function in [3], and K a positive constant parameter. To impose the orthogonality constraint we have modified the classical Kennedy-Chua neural network dynamic equation by subtracting the orthogonal component of x1 : x˙ 1 = −∇V (x1 ) + h∇V (x1 ), s1 is1 .

(17)

The effectiveness of this choice can be shown by considering the projection of x˙ 1 along s1 : hx˙ 1 , s1 i = −h∇V (x1 ), s1 i + h∇V (x1 ), s1 i = 0. (18) Thus (18) implies that hx(t), s1 i = 0 for each t ≥ 0, when x(0) = 0 or x(0) is orthogonal to s1 . The real current sources imperfections, as previously claimed, are considered by adding to (17) a positive constant term G. The final form of the neural network equation used to solve the blind algorithm, with the above-mentioned corrective term G, is therefore: x˙ 1 = −Gx1 − ∇V (x1 ) + h∇V (x1 ), s1 is1

(19)

where V (x1 ) is defined in (14). For practical implementation of neural algorithm on mean energy function, we can simplify (10) in: h i 2 E(x1 ) = E (hy, s1 + x1 i) L

'

1X 2 (hy[i], s1 + x1 i) L i=1

(20)

where L is the averaging window in bit time. The stability of the proposed system can be analyzed through a suitable extension of the method discussed in [8]. In fact it is possible to prove the following: Theorem 1 The Neural Network in (19), where the energy function V (x) is given by (14), has a unique equilibrium point, which is Globally Asymptotically Stable.

The proof of this theorem is not reported here due to space limitations; the interested reader may find it in [9]. The proposed neural network system is therefore Globally Asymptotically Stable, it allows a feasible implementation in analog or digital circuitry [3], and represents a possible way to solve the blind algorithm problem in a very short convergence time, as it will be shown in the next Section. 4

S IMULATION R ESULTS The proposed system performance have been evaluated by means of computer simulations. We have first considered a mobile CDMA system, in which N users access in a synchronous way for the downlink channel, where the Base Station operation can be easily modeled as a synchronous system, and then in an asynchronous way for what concerns the uplink access. Bit synchronization is however assumed between the sender and the receiver of each message. Each user information sequence is spread by a Gold sequence of 31 chips per bit. We suppose also a perfect power control for all transmitting users. The neural network parameters have been chosen equal to G = 0.01, K = 100, and L = 10; these values have been selected after an optimization process [7]. The environment considered in computer simulations is a non selective multipath-fading channel with second and third path attenuated of, respectively, -10dB and -15dB with respect to the main path. Figure 1 shows the Bit Error Rate (BER) performance of the proposed receiver in the case of 20 synchronous users (downlink case). It is evident that the neural network blind receiver gives at least better performance in terms of BER than the classical gradient algorithm, as defined in [7]. In order to highlight the better behavior in fast time-varying conditions, we have considered the following scenario. In particular, in Fig. 2 we have supposed the presence of only one user until the 5000−th bit time interval, then 19 other interfering users are added to the system. Such severe dynamic conditions highlight the good neural network blind capability to react to variations of the application scenario. It can be noted that the neural network blind has a much shorter initial transient, and it drastically reduces the adaptation time. When a large number of interfering users enters the 1

Blind with Neural Network Blind with Gradient Algorithm Single-User Bound

0.1

BER

from the Kennedy-Chua scheme [3]. Since the aim of our neural network is to minimize an energy function, its dynamic equation will be a gradient-type, such as:

0.01

0.001

0.0001 0

2

4

6 SNR (dB)

8

10

12

Figure 1: Bit Error Rate for downlink transmission with 20 synchronous users

80

1E+00

Blind with Neural Network Blind with Gradient Algorithm

70

Blind with Gradient Algorithm Blind with Neural Algorithm

50 BER

Total Errors

60

40 30 20 10

1E-01

0 0

2000

4000

0

6000 8000 10000 12000 14000 Time (in bit intervals)

2

4

6 SNR (dB)

8

10

12

Figure 2: Total Errors vs bit time for a variable environment (1+19 users) in downlink transmission

Figure 4: Bit Error Rate for asynchronous uplink transmission with 15 mobile users

channel, the adaptation rule of the proposed receiver is able to contrast MAI with more strength, while, the classical blind algorithm undergoes a long transient to adapt the coefficients set in the optimal configuration. In Fig. 3 we report the results for the second hard time-varying environment considered. We have assumed in this case a window of 1000 bits (even if the same can be repeated for larger window sizes), in which we have evaluated the error rate. The number nu of transmitting users is nu = 2 for tb bit times such that 0 ≥ tb ≥ 200, while nu = 20 for 200 ≥ tb ≥ 400, nu = 5 for 400 ≥ tb ≥ 600, nu = 30 for 600 ≥ tb ≥ 800, and nu = 10 for 800 ≥ tb ≥ 1000. We have repeated this simulation for users transmitting with Signal-to-Noise Ratio (SNR) varying from 0 to 12 dB, with a step size of 2 dB. It is again straightforward to note how in a fast varying environment neural blind receiver allows better performance than classical blind algorithm, both in terms of BER and transient behavior, avoiding an higher number of user transmitting over the channel. Finally we have considered the performance for the proposed system and a comparison with classical blind MOE detector [7], for the uplink channel. The considered environment implies an asynchronous access to the channel by transmitting users; the cell are supposed to have a ray of 300 m approximately and we have also supposed no interference between adiacent cells. Figure 4

shows the results for 15 asynchronous users. Results for asynchronous scenario show how proposed receiver has similar performance with that of classical blind algorithm even if it is set to work in a synchronous channel. It is also to be noted that best performance for proposed system are to be expected in a time-varying system where classical blind receiver shows its weakness.

1

Blind with Neural Network Blind with Gradient Algorithm

5

C ONCLUDING R EMARKS In this paper we have considered a DS-CDMA communication system, in which a blind detector based on neural network is used to reach a fast convergence time. In particular, we have adopted a suitable designed Kennedy and Chua type neural network. Performance evaluation of the CDMA communication system adopting the proposed receiver scheme has been shown by means of computer simulations, both for a downlink and an uplink channel. The BER performance of the proposed receiver have been derived under hard time-varying environments. The obtained results demonstrate that the proposed neural network based blind receiver can be an effective approach to reach high convergence speeds in DSCDMA communication systems and clearly outperforms classical approaches. R EFERENCES [1] T. Ojanperä and R. Prasad, “An overview of air interface multiple access for IMT-2000/UMTS,” IEEE Communications Magazine, vol. 36, pp. 82–95, Sept. 1998.

BER

0.1

[2] S. Verdù, Multiuser Detection, Cambridge University Press, New York, USA, 1998. 0.01

[3] M. P. Kennedy and L. O. Chua, “Neural networks for nonlinear programming,” IEEE Transactions on Circuits and Systems, vol. 35, pp. 554–562, May 1988.

0.001

0.0001 0

2

4

6 8 SNR (dB)

10

12

14

Figure 3: Bit Error Rate for down-link transmission with variable number of users

[4] J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two state neurons,” Proceedings of the National Academy of Sciences, vol. 81, pp. 3088–3092, Apr. 1984. [5] J. M. Zurada, Introduction to Artificial Neural Systems, PWS Publishing Company, Boston, 1995.

[6] L. O. Chua and G.-N. Lin, “Nonlinear programming without computation,” IEEE Transactions on Circuits and Systems, vol. CAS-31, pp. 182–188, Feb. 1984. [7] M. Honig, U. Madhow, and S. Verdù, “Blind adaptive multiuser detection,” IEEE Transactions on Information Theory, vol. 41, pp. 944–960, July 1995. [8] M. Forti and A. Tesi, “New conditions for global stability of neural networks with application to linear and quadratic programming problems,” IEEE Transactions on Circuits and Systems-I, vol. 42, pp. 354– 366, July 1995. [9] R. Fantacci, M. Forti, M. Marini, D. Tarchi, and G. Vannuccini, “A neural network based blind multiuser receiver for DS-CDMA communication systems,” Tech. Rep. DET 0701, Department of Electronics and Telecommunications - University of Firenze, Italy, 2001.