A Neural Network based Blind Multiuser Receiver ... - Semantic Scholar
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A Neural Network based Blind Multiuser Receiver for DS-CDMA Communication Systems Romano Fantacci, Mauro Forti, Mauro Marini, Daniele Tarchi, Gianluca Vannuccini Abstract Broadband wireless communications are going to be widespread by the Universal Mobile Telecommunications System (IMT/2000), which is based on Wideband-Code Division Multiple Access (W-CDMA) techniques. Adaptive Blind Multiuser Detection has been widely proposed for the mobile station receiver, which reaches the optimum solution after a certain number of bit times. In this paper, a new neural network approach is proposed in order to obtain a lower convergence time in the adaptive blind algorithm. A modified Kennedy-Chua neural network, based on the Hopfield model, is proposed in this work, and its performance in a Blind CDMA receiver have been studied by means of computer simulation. Simulation results have been discussed, highlighting the fast convergence behavior of the proposed network; its stability has also been studied in detail with a suitable analytical approach. Keywords Neural Networks, CDMA, Blind Detection.
I. Introduction
T
HE world of radiomobile communications has experienced a larger development in the last ten years, especially with the upcoming IMT/2000 third generation mo-
bile system. Among the many multiple access techniques proposed for this system, such as The authors are with the Electronic and Telecommunications Department, Florence, Italy. [email protected]
E-mail:
fan-
2
Wideband CDMA with Frequency Division Duplexing (FDD) or Time Division Duplexing (TDD), and cdma-2000 [1], [2], FDD/W-CDMA has been chosen by European Standardization groups for the future Universal Mobile Telecommunications System (UMTS); this is also known as the α Group purpose. Many research activities have been focused on the most suitable receiving techniques for W-CDMA mobile systems, and in particular the multiuser adaptive blind detection algorithm has been proposed by Verd` u [3]. This receiver has the main advantage of reaching MMSE performance without requiring any kind of training sequence to detect the received signal, and this leads to a reduced complexity. However, blind receivers need for a certain number of bit times to reach the optimal solution, since the path towards the optimum filter coefficients set is performed in a step by step approximation, based on the steepest descendent gradient technique. Neural networks are therefore widely known for their very low convergence time, being based on parallel processing algorithms. The idea proposed in this paper is to use the neural network approach in order to accelerate the convergence process of the adaptive filter coefficients set towards the optimum solution in the blind detection algorithm. Among the many types of Neural Networks, the one studied by Kennedy and Chua [4] has been selected in this work; this network is based on Hopfield neural network model [5], 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 obtain 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. II the fundamentals of adaptive blind detection are given, with respect to Verd` u’s approach. In Sec. III, Kennedy-Chua neural networks are described, starting from the Hopfield model, on which they are based. Sec. IV describes the proposed system, whose stability analysis is given in Appendix. Simulation
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results of the proposed system in a mobile CDMA communication environment are then shown in Sec. V, and, finally, conclusions are discussed in Sec. VI. II. Adaptive Blind Detection When many mobile terminals access the channel at the same time, and in the same frequency ranges, as happens in CDMA systems, many problems can arise, and one of the most important is the Multiple Access Interference (MAI). Single-user CDMA receivers, such as the conventional receiver, have one correlator for each possible spreading code in the received signal, and consider MAI as a noise to remove. In contrast, multi-user detection techniques attempt to extract useful information also from the MAI term of the received signal; a detailed description of multi-user detection models can be found in [6]. MAI is not the only critical issue in CDMA systems; if mobile terminals are at different distances from the base station, this will receive signals from the nearer terminals with a stronger power than signals from the farther terminals; this is known as the Near-Far problem, and it can be reduced with a strong power control both at the mobile terminal and at the base station ends. The optimum detector obtained in [7] is able to overcome the Near-Far problem, and to reduce MAI, at the expense of requiring the knowledge of many system parameters, such as the signature waveform and the timing of the desired user and of interfering users, and the received amplitudes of interfering users. Even when these parameters are known, to process them requires a larger complexity for the receiver end. Then, many alternative multiuser detectors have been proposed. In particular, adaptive multiuser detection techniques, described in [8], permit a good error performance detection with the only requirement of the desired user’s signature waveform and timing parameters. Such receivers are based on the minimization of mean-square error (MMSE)
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between the outputs and the input data of the receiver. MMSE detectors [9], substitute the need for interference user parameters knowledge with the requirement of training data sequences for each of the active users in the network; this knowledge is needed for the adaptive filter startup initialization. Moreover, when the channel state changes frequently, the training sequences must be updated with the same rate, thus requiring a hard processing capability. However, MMSE receivers offer a large near-far resistance, and thus they can be considered as a target model for the performance of multiuser adaptive detection techniques. In [10], a blind adaptive multiuser detection technique has been proposed, whose performance tend to the MMSE detector without requiring any training sequence. Such a technique has been considered through the rest of this paper, where the blind model has been used for a mobile terminal receiver, while for the base station more complex multi-user detection algorithms can be easily chosen. In the blind detector, the impulse response of the receiver filter is decomposed in two parts, one containing the desired user’s signature waveform and the other, adaptive and orthogonal to the first component. If the output energy of such a receiver is considered, the optimum adaptive filter’s coefficients set is the one which minimizes that 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 channel having a noise signal n(t) with variance σ 2 , assuming bit synchronism between users, we have [10]: y(t) =
N X
Ak bk sk (t) + σn(t)
(1)
k=1
where N is the maximum number of users in the communication system. In the following, the signal waveforms will be considered in the array notation, and so we will assume y(t) = y,
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y
ZM F
-
s1
-
- h + + 6
Z
-
x1
Fig. 1. Blind receiver block diagram
taking the signal values at the sampling time in the information bit interval. According to [10], the receiver filter is split into two parts, as shown in Fig. 1, where the upper filter makes the correlation between the received signal y and the desired user’s signature waveform s1 , while the lower 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 ]
(2)
which tends to the MMSE solution under the hypothesis [10]: hx1 , s1 i = 0
(3)
that is, the desired user’s signature code and the adaptive blind filter output must be two orthogonal vectors. However, the orthogonality constraint shown in (3) is not the only one applied to the output variable of the considered filter. In mobile communications, the channel state can exhibit time-varying behaviors, and so the transmitted signal coding waveform might not be the same of the spreading sequence regenerated at the receiver end. This is known as the mismatch effect, and may cause the desired signal to be canceled instead of the interference signal. So, according to [10], another constraint must be set on the output energy of the adaptive filter component. This energy bound has been chosen as a compromise between two boundary values. For
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kx1 k2 < χI the interference signal is canceled from the blind equalization, while for kx1 k2 < χS the useful information signal might be canceled by the receiver, thus leading to a faulty detection. To simplify the analytical treatment, considering that generally χI χS and thus χI is a worst-case bound, according to [10] an only value for χ has been taken, that is: χ = χI =
N −1 K − (N − 1)
(4)
where K is the number of chips per bit used in the CDMA signature sequence, and N is the number of users which are using the CDMA channel. The other constraint on the adaptive filter output signal is therefore: k x1 k2 < χ.
(5)
The blind adaptive detector problem has then been driven to a quadratic optimization, subject to the two nonlinear constraints shown in (3) and (5). In [10], by using Lagrange multipliers to satisfy the two constraints described above, the solution to this problem has been calculated as: x1opt = ν2 (A + γIK )−1 sˆ1 − s1
(6)
where ν2 is the Lagrange multiplier relative to (3), γ is defined as: γ = ν1 + σ 2
(7)
with ν1 the Lagrange multiplier relative to (5), and σ as defined in (1). The term A is the outer-product matrix, which under AWGN assumptions becomes: A=
N X
A2k sk sTk
(8)
k=1
where Ak is the k-th user transmitted signal amplitude. The solution of (6) is however much difficult, considering that under general hypotheses the inversion of the A matrix becomes a hard topic [10]. The optimum x1 can then be calculated
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by several time steps of the steepest descendent gradient algorithm, which suggests the following adaptation rule [10]: x1 [i] = x1 [i − 1] − µZ[i](y[i] − ZM F [i]s1 )
(9)
where the subtracting term represents the output energy function gradient, µ is the adaptation rule convergence speed, which has been taken as 1i , Z[i] is defined as: Z[i] = hy[i], s1 + x1 [i − 1]i
(10)
ZM F [i] = hy[i], s1 i.
(11)
and ZM F as:
The adaptation rule in (9) has been widely accepted to solve the blind optimization problem and to reduce the system complexity. However, as its definition shows, the solution of equation (9) can lead to a high convergence time, and this can be critical when large dynamics channel models are considered, or when many users access the system simultaneously. To address this problem, neural networks parallel processing capabilities have been considered through the rest of this paper. III. Kennedy-Chua Neural Networks Neural networks have been widely studied in the last decades, and their fast processing characteristics have been largely highlighted [11]. They can be modeled as a close network of weighted connections and nodes, called artificial neurons, which are connected in a regular topology. 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 netwotks, have
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been selected in this work for their interesting capabilites in minimization problems. One of the main applications of an Hopfield neural network is the minimization of a linear function in n variables, which is obtained through n properly connected neurons with fixed weights and threshold value. During the transient process, the values of the output voltages tend to the minimum value of the function to be minimized. The analog form of an Hopfield neural network is also called gradient type network, it is characterized by a continuous time decreasing of the network total energy, and it could be entirely made of simple discrete-elements electronic devices, such as capacitors, resistors and operational amplifiers. In [12] Hopfield neural networks are used for the optimiziation of linear systems with linear constraints. In [4] 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 [13]. A generic nonlinear problem is based on the minimization of a general cost function: φ(x1 , . . . , xq )
(12)
subject to nonlinear constraints: f1 (x1 , . . . , xq ) ≥ 0 (13)
··· fp (x1 , . . . , xq ) ≥ 0
where p and q are two independent integer number. The circuit equation which solves (12) subject to (13) is: p
dvi ∂φ X ∂fj Ci =− − ij dt ∂vi j=1 ∂vi
(14)
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 Ci 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. We can
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rewrite (14) as a vector notation of the form v(t) ˙ = h (v(t)), where h(·) is a continuous function from Rq to Rp . The system in (12) and (13) tends to the equilibrium point of its Lyapunov function, also called total co-content function [4], defined as: E(v) = φ(v) +
p Z X j=1
fj (v)
gj (x)dx
(15)
0
To model non-ideal current sources imperfections, a correction term is added to (14), thus obtaining: p
dvi ∂φ X ∂fj Ci =− − ij − Gi vi dt ∂vi j=1 ∂vi
(16)
where G takes into account the above mentioned imperfections. A circuit implementation of (16) is possible using two types of amplifiers, one to realize (12) and the other to impose the constraints in (13), as shown in Fig. 2. IV. Proposed System As discussed in Section II the gradient algorithm proposed in [10] requires several bit times to reach the optimal solution. The system we have proposed has the principal objective to reach the optimal solution in almost a bit time, by using neural networks fast parallel processing capabilities. According to Section II and [10], the blind adaptive algorithm requires to minimize the following energy function: E(x1 ) = E (hy, s1 + x1 i)2
(17)
where y is the signal received from the channel, described in (1), s1 is the signature sequence of the desired user and x1 is the orthogonal adaptive filter coefficients set. In (17) h·, ·i stands for the canonical scalar product. As described in Sec. II, the coefficient set of the adaptive filter x1 must satisfy two constraints in order to reduce multiple access interference (MAI)
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and mismatch effect. Equation (17) must be then subject to the following constraints: hs1 , x1 i = 0
(18)
kx1 k2 < χ
(19)
where χ is the energy constraint in (4). To reach a higher convergence speed than (9), we have used a neural network derived from the Kennedy-Chua scheme. Since the aim of our neural network is to minimize an energy function, its dynamic equation will be a gradient-type, such as: x˙ 1 = −∇V (x1 )
(20)
where V (x1 ) is the total energy function of the considered neural network defined, like in (15), 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
(21)
0
where f (x1 ): f (x1 ) = χ − kx1 k2 ≥ 0 is the function that gives the energy constraint and 0, ρ≥0 g(ρ) = Kρ, ρ < 0
(22)
(23)
is the continuous function in [4], where K is 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
(24)
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The effectiveness of this choice can be shown if we consider the projection of x˙ 1 along s1 : hx˙ 1 , s1 i = −h∇V (x1 ), s1 i + h∇V (x1 ), s1 i
(25)
and we have hx˙ 1 , s1 i = 0, which means that if x(0) = 0 or if x(0) is orthogonal to s1 , then hx(t), s1 i = 0 for each t ≥ 0. To consider the real current sources imperfections, as previously discussed, we add to (24) 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
(26)
where V (x1 ) contains the energy function to be minimized by the considered network. If we suppose the channel affected by only additive white gaussian noise (AWGN), according to [10] we can simplify equation (17) in: E(x1 ) = E (hy, s1 + x1 i)2 ' (hy, s1 + x1 i)2
(27)
thus obtaining a more feasible minimization process. The stability of the proposed system can be analyzed through a suitable extension of the method discussed in [14], which defines the following: Theorem 1: Suppose that E(x) is a positive definite quadratic form. Than (26) has a unique equilibrium point xe which is Globally Asymptotically Stable (GAS). Proof of this theorem is given in Appendix. The proposed neural network system is therefore Globally Asymptotically Stable, it allows a feasible implementation in analog or digital circuitry [4], 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.
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V. Simulation Results The proposed system performance have been evaluated by means of computer simulation. We have first considered a downlink mobile CDMA system, in which all N users are synchronous; each user information sequence is spread by a Gold sequence of 31 chips per bit. The considered channel model is affected by additive white gaussian noise, and performance comparison between gradient blind algorithm [10] and proposed neural network blind receiver are shown through the rest of this section. In Fig. 3 the Bit Error Rate for several fixed SNR, under the hypotesis of 10 transmitting users, is shown. The neural network parameters have been chosen equal to G = 0.1 and K = 10. It is evident that the neural network blind receiver gives a better performance in terms of BER than classical gradient algorithm. In Fig. 4 BER in the case of 20 synchronous users is shown. Also in this result we can see how neural network blind receiver has better performance than classical gradient blind algorithm. To highlight the better behavior in fast time-varying conditions, we have considered an extremely large interference addition. In Fig. 5, we have supposed the presence of only one user until 5000 − th bit time interval, when 19 other interfering users are added to the system. In this severe dynamic condition it is more evident how the neural network blind is able to react much faster than the classical blind receiver, due to the neural network parallel processing. 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 channel, the adaptation rule of the proposed receiver is able to contrast MAI with more strength, and, moreover, the classical blind algorithm undergoes a long transient to adapt the coefficients set in the optimal configuration. In Fig. 6 we report the results for another hard time-varying environment. We have considered a window of 1000 bits (even if the same can be done for larger window sizes), in which we have computed the error rate. The number nu of transmitting users is nu = 2 for tb bit times such that 0 ≥ tb ≥ 200,
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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 experiment for users transmitting with 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. Since the proposed system reaches the optimal solution within a bit time, we have considered the extension to multipath slow fading channel models such as the Urban GSM model [15]. In Fig. 7 and Fig. 8 the bit error rate versus the transmitted SNR for an Urban GSM channel with respectively 10 and 20 synchronous users is shown, and the proposed system allows in both cases better BER performance than the classical blind detector. Finally, in Fig. 9 the same critical condition of Fig. 5 has been applied to the GSM Urbal channel case, even though the 19 users are added at the 300-th bit time, and better results in comparison to the classical blind receiver can be observed also in this situation. VI. Conclusive Remarks In this paper we have considered a DS-CDMA communication system, in which the blind detection algorithm has been proposed in literature for the receiver implementation as an effective compromise between computational complexity and convergence time. One of the main problems of the blind detector is that it reaches the optimal solution in several bit time steps, thus making it difficult to adopt such a system for the dynamic conditions in which UMTS-like communication systems work. Neural networks have been largely studied in the last decades, and their fast processing capabilities have been highlighted in many research works. In particular, Kennedy and Chua neural networks are suitable for minimization problems in which the optimization function is subject to nonlinear constraints.
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We have then adopted these networks to solve the blind algorithm with a properly designed Kennedy and Chua type neural network. A stability analysis of the proposed network is discussed, and performance evaluation of the CDMA communication system adopting the proposed network is shown by means of computer simulations. The proposed network performance have been tested first in terms of bit error rates, and then by simulating hard varying conditions of the number of users in the channel, thus highlighting the fast convergence time of the blind receiver using the proposed neural network. Simulation results are shown both for the AWGN channel model and for the well-known GSM Urban-like model, and they all show that the proposed neural network based blind receiver can be a suitable approach to reach high convergence speeds in DS-CDMA communication systems. Appendix Proposed System Stability Analysis Theorem 1: Suppose that E(x) is a positive definite quadratic form. Then, (26) has a unique equilibrium point xe which is Globally Asymptotically Stable (GAS). Proof. Let E(x) = 12 x0 Qx + x0 a, where Q ∈ Rn×n is symmetric and positive definite, and a ∈ Rn (the prime means transpose). System (26) becomes x˙ = −Gx − Qx+ < Qx, s > s − a+ < a, s > s + 2g(χ − x0 x)(x− < x, s > s).
(28)
Consider the change of variables x = Rw, where R ∈ Rn×n is such that R−1 = R0 , and R(1, 0, · · · , 0)0 = s. Also let w = (y, z), where y ∈ R is the component of w along s and z ∈ Rn−1 is the component of w on the space π orthogonal to s. After straightforward algebraic manipulations, it is verified that in the coordinate system w, (28) becomes y˙ =
f1 (y)
= −Gy
(29) 2
z˙ = f2 (y, z) = −Gz − P22 z − P21 y + 2g(χ − z z − y )z − b. 0
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Vector b ∈ Rn−1 is the component of R0 a on π. Moreover, P22 ∈ R(n−1)×(n−1) is symmetric and positive definite, P21 ∈ Rn−1 , and they are defined by the decomposition 0 P11 P21 . R QR = P21 P22 0
(30)
System (29) is the cascade of two systems. From Theorem 5 in [16], it follows that (28) is GAS if z˙ = f2 (ψ, z) is GAS for each fixed parameter ψ ∈ R. Using classical results on GAS, see e.g. [16], it is seen that GAS of z˙ = f2 (ψ, z) is an easy consequence of the following observations. a) We have f2 (ψ, z) = −∇z W (z, ψ), where 1 1 W (z, ψ) = Gz 0 z + z 0 P22 z + z 0 (P21 ψ + b) + 2 2
Z
fψ (z)
g(ρ)dρ
(31)
0
and fψ (z) = χ − ψ 2 − z 0 z. b) W is radially unbounded (W → ∞ as kzk → ∞) and strictly convex in z, being the sum of a positive definite quadratic form, and the convex function R fψ (z) 0
g(ρ)dρ. References
[1]
T. Ojanper¨ a and R. Prasad, “An overview of air interface multiple access for IMT-2000/UMTS,” IEEE Communications Magazine, vol. 36, pp. 82–95, Sept. 1998.
[2]
E. Dahlman, B. Gudmundson, M. Nilsson, and J. Sk¨ old, “UMTS/IMT-2000 based on wideband CDMA,” IEEE Communications Magazine, vol. 36, pp. 70–80, Sept. 1998.
[3]
S. Verd` u, “Blind adaptive receivers,” IEEE Transactions on Information Theory, 19...
[4]
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.
[5]
Hopfield, “Neural networks,” IEEE Trans. on ......, Apr. 1993.
[6]
S. Verd` u, Multiuser Detection, Cambridge University Press, New York, USA, 1998.
[7]
S. Verd` u, “Minimum probability of error for asynchronous gaussian multiple access channels,” IEEE Trans. on Information Theory, Jan. 1986.
[8]
S.Verd` u, “Adaptive multiuser detection,” in Proceedings of the IEEE Third International Symposium on Spread Spectrum Techniques and Applications, 1994, vol. 1, pp. 43–50.
[9]
M. Honig and U. Madhow, “MMSE interference suppression for direct-sequence spread-spectrum CDMA,” IEEE Transactions on Communications, vol. 42, pp. 3178–3188, Dec. 1994.
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[10] M. Honig, U. Madhow, and S. Verd` u, “Blind adaptive multiuser detection,” IEEE Transaction. on Information Theory, vol. 41, pp. 944–960, July 1995. [11] J. M. Zurada, Introduction to Artificial Neural Systems, PWS Publishing Company, Boston, 1995. [12] D. W. Tank and J. J. Hopfield, “Simple ”neural” optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit,” IEEE Transactions on Circuits and Systems, vol. CAS-33, pp. 533–541, May 1986. [13] 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. [14] 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. [15] GSM 05.05 version 8.5.0 Release 1999, Digital cellular telecommunications system (Phase 2+); Radio Transmission and Reception, 07 2000. [16] M. Hirsh, “Convergent activation dynamics in continous time networks,” Neural Networks, vol. 2, pp. 331–349, 1989.
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Fig. 2. An example of a Kennedy-Chua neural network
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1
Blind with Neural Network Blind with Gradient Algorithm Single-User Bound
BER
0.1
0.01
0.001
0.0001 0
2
4
6 SNR (dB)
8
10
Fig. 3. BER in downlink AWGN channel for 10 synchronous users
12
19
Fig. 4. BER in downlink AWGN channel for 20 synchronous users
20
80 Blind Receiver with Gradient Algorithm Blind Receiver with Neural Algorithm 70
60
Errors
50
40
30
20
10
0 0
2000
4000
6000
8000
10000
12000
14000
Time (bit)
Fig. 5. Total Errors in bit time for a variable environment (1+19) for AWGN downlink channel
21
1e+00 Blind Receiver with Gradient Algorithm Blind Receiver with Neural Algorithm
BER
1e-01
1e-02
1e-03 0
2
4
6 SNR (dB)
8
10
Fig. 6. BER in downlink AWGN channel with variable number of users
12
22
BER
1
Blind with Neural Network Blind with Gradient Algorithm
0.1
0.01 0
2
4
6 SNR (dB)
8
10
Fig. 7. BER vs SNR in a downlink GSM-Urban Channel with 10 users
12
23
BER
1
Blind with Neural Network Blind with Gradient Algorithm
0.1
0.01 0
2
4
6 SNR (dB)
8
10
Fig. 8. BER vs SNR in a downlink GSM-Urban Channel with 20 users
12
24
Blind with Neural Network Blind with Gradient Algorithm
140
Total Errors
120 100 80 60 40 20 0 0
200
400 600 Time (in bit intervals)
800
1000
Fig. 9. Total Errors in bit time for a variable environment (1+19) for AWGN downlink channel
A Neural Network based Blind Multiuser Receiver for DS-CDMA Communication Systems Romano Fantacci, Mauro Forti, Mauro Marini, Daniele Tarchi, Gianluca Vannuccini Abstract Broadband wireless communications are going to be widespread by the Universal Mobile Telecommunications System (IMT/2000), which is based on Wideband-Code Division Multiple Access (W-CDMA) techniques. Adaptive Blind Multiuser Detection has been widely proposed for the mobile station receiver, which reaches the optimum solution after a certain number of bit times. In this paper, a new neural network approach is proposed in order to obtain a lower convergence time in the adaptive blind algorithm. A modified Kennedy-Chua neural network, based on the Hopfield model, is proposed in this work, and its performance in a Blind CDMA receiver have been studied by means of computer simulation. Simulation results have been discussed, highlighting the fast convergence behavior of the proposed network; its stability has also been studied in detail with a suitable analytical approach. Keywords Neural Networks, CDMA, Blind Detection.
I. Introduction
T
HE world of radiomobile communications has experienced a larger development in the last ten years, especially with the upcoming IMT/2000 third generation mo-
bile system. Among the many multiple access techniques proposed for this system, such as The authors are with the Electronic and Telecommunications Department, Florence, Italy. [email protected]
E-mail:
fan-
2
Wideband CDMA with Frequency Division Duplexing (FDD) or Time Division Duplexing (TDD), and cdma-2000 [1], [2], FDD/W-CDMA has been chosen by European Standardization groups for the future Universal Mobile Telecommunications System (UMTS); this is also known as the α Group purpose. Many research activities have been focused on the most suitable receiving techniques for W-CDMA mobile systems, and in particular the multiuser adaptive blind detection algorithm has been proposed by Verd` u [3]. This receiver has the main advantage of reaching MMSE performance without requiring any kind of training sequence to detect the received signal, and this leads to a reduced complexity. However, blind receivers need for a certain number of bit times to reach the optimal solution, since the path towards the optimum filter coefficients set is performed in a step by step approximation, based on the steepest descendent gradient technique. Neural networks are therefore widely known for their very low convergence time, being based on parallel processing algorithms. The idea proposed in this paper is to use the neural network approach in order to accelerate the convergence process of the adaptive filter coefficients set towards the optimum solution in the blind detection algorithm. Among the many types of Neural Networks, the one studied by Kennedy and Chua [4] has been selected in this work; this network is based on Hopfield neural network model [5], 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 obtain 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. II the fundamentals of adaptive blind detection are given, with respect to Verd` u’s approach. In Sec. III, Kennedy-Chua neural networks are described, starting from the Hopfield model, on which they are based. Sec. IV describes the proposed system, whose stability analysis is given in Appendix. Simulation
3
results of the proposed system in a mobile CDMA communication environment are then shown in Sec. V, and, finally, conclusions are discussed in Sec. VI. II. Adaptive Blind Detection When many mobile terminals access the channel at the same time, and in the same frequency ranges, as happens in CDMA systems, many problems can arise, and one of the most important is the Multiple Access Interference (MAI). Single-user CDMA receivers, such as the conventional receiver, have one correlator for each possible spreading code in the received signal, and consider MAI as a noise to remove. In contrast, multi-user detection techniques attempt to extract useful information also from the MAI term of the received signal; a detailed description of multi-user detection models can be found in [6]. MAI is not the only critical issue in CDMA systems; if mobile terminals are at different distances from the base station, this will receive signals from the nearer terminals with a stronger power than signals from the farther terminals; this is known as the Near-Far problem, and it can be reduced with a strong power control both at the mobile terminal and at the base station ends. The optimum detector obtained in [7] is able to overcome the Near-Far problem, and to reduce MAI, at the expense of requiring the knowledge of many system parameters, such as the signature waveform and the timing of the desired user and of interfering users, and the received amplitudes of interfering users. Even when these parameters are known, to process them requires a larger complexity for the receiver end. Then, many alternative multiuser detectors have been proposed. In particular, adaptive multiuser detection techniques, described in [8], permit a good error performance detection with the only requirement of the desired user’s signature waveform and timing parameters. Such receivers are based on the minimization of mean-square error (MMSE)
4
between the outputs and the input data of the receiver. MMSE detectors [9], substitute the need for interference user parameters knowledge with the requirement of training data sequences for each of the active users in the network; this knowledge is needed for the adaptive filter startup initialization. Moreover, when the channel state changes frequently, the training sequences must be updated with the same rate, thus requiring a hard processing capability. However, MMSE receivers offer a large near-far resistance, and thus they can be considered as a target model for the performance of multiuser adaptive detection techniques. In [10], a blind adaptive multiuser detection technique has been proposed, whose performance tend to the MMSE detector without requiring any training sequence. Such a technique has been considered through the rest of this paper, where the blind model has been used for a mobile terminal receiver, while for the base station more complex multi-user detection algorithms can be easily chosen. In the blind detector, the impulse response of the receiver filter is decomposed in two parts, one containing the desired user’s signature waveform and the other, adaptive and orthogonal to the first component. If the output energy of such a receiver is considered, the optimum adaptive filter’s coefficients set is the one which minimizes that 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 channel having a noise signal n(t) with variance σ 2 , assuming bit synchronism between users, we have [10]: y(t) =
N X
Ak bk sk (t) + σn(t)
(1)
k=1
where N is the maximum number of users in the communication system. In the following, the signal waveforms will be considered in the array notation, and so we will assume y(t) = y,
5
y
ZM F
-
s1
-
- h + + 6
Z
-
x1
Fig. 1. Blind receiver block diagram
taking the signal values at the sampling time in the information bit interval. According to [10], the receiver filter is split into two parts, as shown in Fig. 1, where the upper filter makes the correlation between the received signal y and the desired user’s signature waveform s1 , while the lower 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 ]
(2)
which tends to the MMSE solution under the hypothesis [10]: hx1 , s1 i = 0
(3)
that is, the desired user’s signature code and the adaptive blind filter output must be two orthogonal vectors. However, the orthogonality constraint shown in (3) is not the only one applied to the output variable of the considered filter. In mobile communications, the channel state can exhibit time-varying behaviors, and so the transmitted signal coding waveform might not be the same of the spreading sequence regenerated at the receiver end. This is known as the mismatch effect, and may cause the desired signal to be canceled instead of the interference signal. So, according to [10], another constraint must be set on the output energy of the adaptive filter component. This energy bound has been chosen as a compromise between two boundary values. For
6
kx1 k2 < χI the interference signal is canceled from the blind equalization, while for kx1 k2 < χS the useful information signal might be canceled by the receiver, thus leading to a faulty detection. To simplify the analytical treatment, considering that generally χI χS and thus χI is a worst-case bound, according to [10] an only value for χ has been taken, that is: χ = χI =
N −1 K − (N − 1)
(4)
where K is the number of chips per bit used in the CDMA signature sequence, and N is the number of users which are using the CDMA channel. The other constraint on the adaptive filter output signal is therefore: k x1 k2 < χ.
(5)
The blind adaptive detector problem has then been driven to a quadratic optimization, subject to the two nonlinear constraints shown in (3) and (5). In [10], by using Lagrange multipliers to satisfy the two constraints described above, the solution to this problem has been calculated as: x1opt = ν2 (A + γIK )−1 sˆ1 − s1
(6)
where ν2 is the Lagrange multiplier relative to (3), γ is defined as: γ = ν1 + σ 2
(7)
with ν1 the Lagrange multiplier relative to (5), and σ as defined in (1). The term A is the outer-product matrix, which under AWGN assumptions becomes: A=
N X
A2k sk sTk
(8)
k=1
where Ak is the k-th user transmitted signal amplitude. The solution of (6) is however much difficult, considering that under general hypotheses the inversion of the A matrix becomes a hard topic [10]. The optimum x1 can then be calculated
7
by several time steps of the steepest descendent gradient algorithm, which suggests the following adaptation rule [10]: x1 [i] = x1 [i − 1] − µZ[i](y[i] − ZM F [i]s1 )
(9)
where the subtracting term represents the output energy function gradient, µ is the adaptation rule convergence speed, which has been taken as 1i , Z[i] is defined as: Z[i] = hy[i], s1 + x1 [i − 1]i
(10)
ZM F [i] = hy[i], s1 i.
(11)
and ZM F as:
The adaptation rule in (9) has been widely accepted to solve the blind optimization problem and to reduce the system complexity. However, as its definition shows, the solution of equation (9) can lead to a high convergence time, and this can be critical when large dynamics channel models are considered, or when many users access the system simultaneously. To address this problem, neural networks parallel processing capabilities have been considered through the rest of this paper. III. Kennedy-Chua Neural Networks Neural networks have been widely studied in the last decades, and their fast processing characteristics have been largely highlighted [11]. They can be modeled as a close network of weighted connections and nodes, called artificial neurons, which are connected in a regular topology. 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 netwotks, have
8
been selected in this work for their interesting capabilites in minimization problems. One of the main applications of an Hopfield neural network is the minimization of a linear function in n variables, which is obtained through n properly connected neurons with fixed weights and threshold value. During the transient process, the values of the output voltages tend to the minimum value of the function to be minimized. The analog form of an Hopfield neural network is also called gradient type network, it is characterized by a continuous time decreasing of the network total energy, and it could be entirely made of simple discrete-elements electronic devices, such as capacitors, resistors and operational amplifiers. In [12] Hopfield neural networks are used for the optimiziation of linear systems with linear constraints. In [4] 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 [13]. A generic nonlinear problem is based on the minimization of a general cost function: φ(x1 , . . . , xq )
(12)
subject to nonlinear constraints: f1 (x1 , . . . , xq ) ≥ 0 (13)
··· fp (x1 , . . . , xq ) ≥ 0
where p and q are two independent integer number. The circuit equation which solves (12) subject to (13) is: p
dvi ∂φ X ∂fj Ci =− − ij dt ∂vi j=1 ∂vi
(14)
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 Ci 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. We can
9
rewrite (14) as a vector notation of the form v(t) ˙ = h (v(t)), where h(·) is a continuous function from Rq to Rp . The system in (12) and (13) tends to the equilibrium point of its Lyapunov function, also called total co-content function [4], defined as: E(v) = φ(v) +
p Z X j=1
fj (v)
gj (x)dx
(15)
0
To model non-ideal current sources imperfections, a correction term is added to (14), thus obtaining: p
dvi ∂φ X ∂fj Ci =− − ij − Gi vi dt ∂vi j=1 ∂vi
(16)
where G takes into account the above mentioned imperfections. A circuit implementation of (16) is possible using two types of amplifiers, one to realize (12) and the other to impose the constraints in (13), as shown in Fig. 2. IV. Proposed System As discussed in Section II the gradient algorithm proposed in [10] requires several bit times to reach the optimal solution. The system we have proposed has the principal objective to reach the optimal solution in almost a bit time, by using neural networks fast parallel processing capabilities. According to Section II and [10], the blind adaptive algorithm requires to minimize the following energy function: E(x1 ) = E (hy, s1 + x1 i)2
(17)
where y is the signal received from the channel, described in (1), s1 is the signature sequence of the desired user and x1 is the orthogonal adaptive filter coefficients set. In (17) h·, ·i stands for the canonical scalar product. As described in Sec. II, the coefficient set of the adaptive filter x1 must satisfy two constraints in order to reduce multiple access interference (MAI)
10
and mismatch effect. Equation (17) must be then subject to the following constraints: hs1 , x1 i = 0
(18)
kx1 k2 < χ
(19)
where χ is the energy constraint in (4). To reach a higher convergence speed than (9), we have used a neural network derived from the Kennedy-Chua scheme. Since the aim of our neural network is to minimize an energy function, its dynamic equation will be a gradient-type, such as: x˙ 1 = −∇V (x1 )
(20)
where V (x1 ) is the total energy function of the considered neural network defined, like in (15), 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
(21)
0
where f (x1 ): f (x1 ) = χ − kx1 k2 ≥ 0 is the function that gives the energy constraint and 0, ρ≥0 g(ρ) = Kρ, ρ < 0
(22)
(23)
is the continuous function in [4], where K is 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
(24)
11
The effectiveness of this choice can be shown if we consider the projection of x˙ 1 along s1 : hx˙ 1 , s1 i = −h∇V (x1 ), s1 i + h∇V (x1 ), s1 i
(25)
and we have hx˙ 1 , s1 i = 0, which means that if x(0) = 0 or if x(0) is orthogonal to s1 , then hx(t), s1 i = 0 for each t ≥ 0. To consider the real current sources imperfections, as previously discussed, we add to (24) 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
(26)
where V (x1 ) contains the energy function to be minimized by the considered network. If we suppose the channel affected by only additive white gaussian noise (AWGN), according to [10] we can simplify equation (17) in: E(x1 ) = E (hy, s1 + x1 i)2 ' (hy, s1 + x1 i)2
(27)
thus obtaining a more feasible minimization process. The stability of the proposed system can be analyzed through a suitable extension of the method discussed in [14], which defines the following: Theorem 1: Suppose that E(x) is a positive definite quadratic form. Than (26) has a unique equilibrium point xe which is Globally Asymptotically Stable (GAS). Proof of this theorem is given in Appendix. The proposed neural network system is therefore Globally Asymptotically Stable, it allows a feasible implementation in analog or digital circuitry [4], 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.
12
V. Simulation Results The proposed system performance have been evaluated by means of computer simulation. We have first considered a downlink mobile CDMA system, in which all N users are synchronous; each user information sequence is spread by a Gold sequence of 31 chips per bit. The considered channel model is affected by additive white gaussian noise, and performance comparison between gradient blind algorithm [10] and proposed neural network blind receiver are shown through the rest of this section. In Fig. 3 the Bit Error Rate for several fixed SNR, under the hypotesis of 10 transmitting users, is shown. The neural network parameters have been chosen equal to G = 0.1 and K = 10. It is evident that the neural network blind receiver gives a better performance in terms of BER than classical gradient algorithm. In Fig. 4 BER in the case of 20 synchronous users is shown. Also in this result we can see how neural network blind receiver has better performance than classical gradient blind algorithm. To highlight the better behavior in fast time-varying conditions, we have considered an extremely large interference addition. In Fig. 5, we have supposed the presence of only one user until 5000 − th bit time interval, when 19 other interfering users are added to the system. In this severe dynamic condition it is more evident how the neural network blind is able to react much faster than the classical blind receiver, due to the neural network parallel processing. 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 channel, the adaptation rule of the proposed receiver is able to contrast MAI with more strength, and, moreover, the classical blind algorithm undergoes a long transient to adapt the coefficients set in the optimal configuration. In Fig. 6 we report the results for another hard time-varying environment. We have considered a window of 1000 bits (even if the same can be done for larger window sizes), in which we have computed the error rate. The number nu of transmitting users is nu = 2 for tb bit times such that 0 ≥ tb ≥ 200,
13
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 experiment for users transmitting with 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. Since the proposed system reaches the optimal solution within a bit time, we have considered the extension to multipath slow fading channel models such as the Urban GSM model [15]. In Fig. 7 and Fig. 8 the bit error rate versus the transmitted SNR for an Urban GSM channel with respectively 10 and 20 synchronous users is shown, and the proposed system allows in both cases better BER performance than the classical blind detector. Finally, in Fig. 9 the same critical condition of Fig. 5 has been applied to the GSM Urbal channel case, even though the 19 users are added at the 300-th bit time, and better results in comparison to the classical blind receiver can be observed also in this situation. VI. Conclusive Remarks In this paper we have considered a DS-CDMA communication system, in which the blind detection algorithm has been proposed in literature for the receiver implementation as an effective compromise between computational complexity and convergence time. One of the main problems of the blind detector is that it reaches the optimal solution in several bit time steps, thus making it difficult to adopt such a system for the dynamic conditions in which UMTS-like communication systems work. Neural networks have been largely studied in the last decades, and their fast processing capabilities have been highlighted in many research works. In particular, Kennedy and Chua neural networks are suitable for minimization problems in which the optimization function is subject to nonlinear constraints.
14
We have then adopted these networks to solve the blind algorithm with a properly designed Kennedy and Chua type neural network. A stability analysis of the proposed network is discussed, and performance evaluation of the CDMA communication system adopting the proposed network is shown by means of computer simulations. The proposed network performance have been tested first in terms of bit error rates, and then by simulating hard varying conditions of the number of users in the channel, thus highlighting the fast convergence time of the blind receiver using the proposed neural network. Simulation results are shown both for the AWGN channel model and for the well-known GSM Urban-like model, and they all show that the proposed neural network based blind receiver can be a suitable approach to reach high convergence speeds in DS-CDMA communication systems. Appendix Proposed System Stability Analysis Theorem 1: Suppose that E(x) is a positive definite quadratic form. Then, (26) has a unique equilibrium point xe which is Globally Asymptotically Stable (GAS). Proof. Let E(x) = 12 x0 Qx + x0 a, where Q ∈ Rn×n is symmetric and positive definite, and a ∈ Rn (the prime means transpose). System (26) becomes x˙ = −Gx − Qx+ < Qx, s > s − a+ < a, s > s + 2g(χ − x0 x)(x− < x, s > s).
(28)
Consider the change of variables x = Rw, where R ∈ Rn×n is such that R−1 = R0 , and R(1, 0, · · · , 0)0 = s. Also let w = (y, z), where y ∈ R is the component of w along s and z ∈ Rn−1 is the component of w on the space π orthogonal to s. After straightforward algebraic manipulations, it is verified that in the coordinate system w, (28) becomes y˙ =
f1 (y)
= −Gy
(29) 2
z˙ = f2 (y, z) = −Gz − P22 z − P21 y + 2g(χ − z z − y )z − b. 0
15
Vector b ∈ Rn−1 is the component of R0 a on π. Moreover, P22 ∈ R(n−1)×(n−1) is symmetric and positive definite, P21 ∈ Rn−1 , and they are defined by the decomposition 0 P11 P21 . R QR = P21 P22 0
(30)
System (29) is the cascade of two systems. From Theorem 5 in [16], it follows that (28) is GAS if z˙ = f2 (ψ, z) is GAS for each fixed parameter ψ ∈ R. Using classical results on GAS, see e.g. [16], it is seen that GAS of z˙ = f2 (ψ, z) is an easy consequence of the following observations. a) We have f2 (ψ, z) = −∇z W (z, ψ), where 1 1 W (z, ψ) = Gz 0 z + z 0 P22 z + z 0 (P21 ψ + b) + 2 2
Z
fψ (z)
g(ρ)dρ
(31)
0
and fψ (z) = χ − ψ 2 − z 0 z. b) W is radially unbounded (W → ∞ as kzk → ∞) and strictly convex in z, being the sum of a positive definite quadratic form, and the convex function R fψ (z) 0
g(ρ)dρ. References
[1]
T. Ojanper¨ a and R. Prasad, “An overview of air interface multiple access for IMT-2000/UMTS,” IEEE Communications Magazine, vol. 36, pp. 82–95, Sept. 1998.
[2]
E. Dahlman, B. Gudmundson, M. Nilsson, and J. Sk¨ old, “UMTS/IMT-2000 based on wideband CDMA,” IEEE Communications Magazine, vol. 36, pp. 70–80, Sept. 1998.
[3]
S. Verd` u, “Blind adaptive receivers,” IEEE Transactions on Information Theory, 19...
[4]
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.
[5]
Hopfield, “Neural networks,” IEEE Trans. on ......, Apr. 1993.
[6]
S. Verd` u, Multiuser Detection, Cambridge University Press, New York, USA, 1998.
[7]
S. Verd` u, “Minimum probability of error for asynchronous gaussian multiple access channels,” IEEE Trans. on Information Theory, Jan. 1986.
[8]
S.Verd` u, “Adaptive multiuser detection,” in Proceedings of the IEEE Third International Symposium on Spread Spectrum Techniques and Applications, 1994, vol. 1, pp. 43–50.
[9]
M. Honig and U. Madhow, “MMSE interference suppression for direct-sequence spread-spectrum CDMA,” IEEE Transactions on Communications, vol. 42, pp. 3178–3188, Dec. 1994.
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[10] M. Honig, U. Madhow, and S. Verd` u, “Blind adaptive multiuser detection,” IEEE Transaction. on Information Theory, vol. 41, pp. 944–960, July 1995. [11] J. M. Zurada, Introduction to Artificial Neural Systems, PWS Publishing Company, Boston, 1995. [12] D. W. Tank and J. J. Hopfield, “Simple ”neural” optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit,” IEEE Transactions on Circuits and Systems, vol. CAS-33, pp. 533–541, May 1986. [13] 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. [14] 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. [15] GSM 05.05 version 8.5.0 Release 1999, Digital cellular telecommunications system (Phase 2+); Radio Transmission and Reception, 07 2000. [16] M. Hirsh, “Convergent activation dynamics in continous time networks,” Neural Networks, vol. 2, pp. 331–349, 1989.
17
Fig. 2. An example of a Kennedy-Chua neural network
18
1
Blind with Neural Network Blind with Gradient Algorithm Single-User Bound
BER
0.1
0.01
0.001
0.0001 0
2
4
6 SNR (dB)
8
10
Fig. 3. BER in downlink AWGN channel for 10 synchronous users
12
19
Fig. 4. BER in downlink AWGN channel for 20 synchronous users
20
80 Blind Receiver with Gradient Algorithm Blind Receiver with Neural Algorithm 70
60
Errors
50
40
30
20
10
0 0
2000
4000
6000
8000
10000
12000
14000
Time (bit)
Fig. 5. Total Errors in bit time for a variable environment (1+19) for AWGN downlink channel
21
1e+00 Blind Receiver with Gradient Algorithm Blind Receiver with Neural Algorithm
BER
1e-01
1e-02
1e-03 0
2
4
6 SNR (dB)
8
10
Fig. 6. BER in downlink AWGN channel with variable number of users
12
22
BER
1
Blind with Neural Network Blind with Gradient Algorithm
0.1
0.01 0
2
4
6 SNR (dB)
8
10
Fig. 7. BER vs SNR in a downlink GSM-Urban Channel with 10 users
12
23
BER
1
Blind with Neural Network Blind with Gradient Algorithm
0.1
0.01 0
2
4
6 SNR (dB)
8
10
Fig. 8. BER vs SNR in a downlink GSM-Urban Channel with 20 users
12
24
Blind with Neural Network Blind with Gradient Algorithm
140
Total Errors
120 100 80 60 40 20 0 0
200
400 600 Time (in bit intervals)
800
1000
Fig. 9. Total Errors in bit time for a variable environment (1+19) for AWGN downlink channel
