Multiuser Detection for DS-CDMA Systems over ... - IEEE Xplore

2013 IEEE 9th International Colloquium on Signal Processing and its Applications, 8 - 10 Mac. 2013, Kuala Lumpur, Malaysia

Multiuser Detection for DS-CDMA Systems over Nakagami-m Fading Channels using Particle Swarm Optimization Vinay Kumar Pamula,

Srinivasa Rao Vempati,

Department of ECE, University College of Engineering, JNTUK, Kakinada, India 533003. [email protected]

Department of ECE, Anurag Engineering College, Kodad, India 508206. [email protected]

Habibulla Khan

Anil Kumar Tipparti,

Department of ECE, KL University, Vaddeswaram, India 522502. [email protected]

Department of ECE, Anurag College of Engineering, Aushapur, India 501301. [email protected] communication channels, in radar and sonar systems and in indoor wireless communication channels [9]. Hence, this paper presents the implementation and performance analysis of modified-Hampel [10] based Mestimator, which performs well in the heavy-tailed impulsive noise, using PSO algorithm. Maximal ratio combining (MRC) receive diversity is incorporated to mitigate the effects of Nakagami-m fading channel. The remaining part of the paper is organized as: Section II presents a synchronous DS-CDMA system over Nakagami-m fading channels in impulsive noise environment and modifiedHampel based M-estimator. PSO based M-decorrelating detector is presented in Section III. Section IV presents the evolutionary behavior of M-estimator and finally, conclusions are drawn in Section V.

Abstract— This paper presents a robust multiuser detection technique for direct sequence-code division multiple access (DSCDMA) systems over Nakgami-m fading channels in the presence of impulsive noise. Robust M-decorrelating detector to demodulate non-coherent DPSK signals is implemented by using the particle swarm optimization (PSO) algorithm. The PSO algorithm is used to obtain optimum M-estimates by minimizing an objective function, which is a sum of less rapidly increasing function of residuals. Simulation results are provided to show the evolutionary behavior and efficacy of the modified-Hampel based M-estimator in comparison with linear decorrelating detector, the Huber and the Hampel estimator based detectors. Index Terms—CDMA, influence function, maximal ratio combiner, M-estimator, multiuser detection, Nakagami fading, particle swarm optimization.

II. SYSTEM MODEL

I. INTRODUCTION

Consider an L-user synchronous CDMA system, where each user transmits information by modulating a PN sequence over a single-path slowly fading Nakagami-m channel. The received signal over one symbol duration can be modeled as [11]

Recent research has explored the potential benefits of evolutionary optimization algorithms and their application to multiuser detection (MUD) for direct-sequence code division multiple access (DS-CDMA) systems [1]-[4]. An adaptive robust MUD technique for CDMA by implementing Huber’s M-estimator using genetic algorithm (GA) is presented in [3] and shown that it is robust against heavy-tailed impulsive noise. Recently, particle swarm optimization (PSO) algorithm has been applied for the MUD [4] to detect received data bit by optimizing an objective function. The Nakgami-m fading channel has been received considerable attention as it can provide a good fit to different fading environments [5] and MUD for DS-CDMA systems over Nakagami-m channels has been deeply studied in the literature [6]-[8]. Experimental results have confirmed the presence of heavy-tailed impulsive noise in outdoor mobile

978-1-4673-5609-1/13/$31.00 ©2013 IEEE

⎧ L M −1 ⎫ r (t ) = ℜ ⎨ ∑ ∑ bl (i )αl (t ) e jφl (t ) sl (t − iTs - τl ) ⎬ + n(t ) (1) ⎩l=1 i =0 ⎭ where ℜ {} ⋅ denotes the real part, M is the number of data symbols per user in the data frame of interest, Ts is the symbol interval, αl (t ) is the time-varying fading gain of the lth user’s

channel, φl (t ) is the time-varying phase of the lth user’s channel, bl (i ) is the ith bit of the lth user, sl (t ) is the

275


normalized signaling waveform of the lth user and n(t ) is assumed as a zero-mean complex two-term non-Gaussian noise [12]. For synchronous case (i.e., τ1 = τ 2 = ... = τl = 0 ), assuming that the fading process for each user varies at a slower rate that the magnitude and phase can taken to be constant over the duration of a bit, the received signal can be expressed in matrix notation as [11]

r (i ) = Aθ (i ) + w (i )

where a, b and d are constants that depend on the robustness of the estimator [13]. III. PSO BASED M-DECORRELATING DETECTOR PSO is a swarm intelligence method for global optimization modeled after the social behavior of bird flocking and fish schooling [4]. In PSO algorithm the solution search is conducted using a population of individual particles, where each particle represents a candidate solution to the optimization problem (3). Each particle keeps track of the position of its individual best solution (called as pbest),

(2)

best

T

pd

T

where r (i ) ⎡⎣ r1 (i ),...,rN (i ) ⎤⎦ ,

w (i ) ⎡⎣ w1 (i ),...,w N (i ) ⎤⎦ and

l

the

it

⎡ ⎢⎣

(

l =1

)⎤⎥⎦}

the

best L

] among pbests of all

population

it

it

Step1:

represented

it

it

Compute 0

θ =

)

the

R −1 AT

r.

decorrelating Here,

R

(

detector

AT

A/ N

output,

)

is

the

normalized cross-correlation matrix of signature waveforms of all users. Step 2: Initialization: The output of decorrelating detector is 0

0

taken as input first particle d1 = θ1 .

Step 3: Fitness evaluation: The objective function (3) is used to find the fitness vector by substituting residuals.

(3) where ρ (⋅) represents a specific penalty function that is symmetric positive-definite with a unique minimum at zero, and is chosen to be less increasing than square, rn (i ) and

best

Local best position p d is recorded by looking at the history of each particle and the particle with lowest fitness is taken as g

best

of the population.

Step 4:Update the inertia weight by using the decrement

θl (i ) are the nth and lth elements of the vectors r (i ) and

it −1

it

function w = β w , where β < 1 is the decrement constant. Step 5: Update the particle velocity by using the relations

th

θ (i ) respectively, [ A]nl is the nl element of the matrix A , ⋅ denotes imaginary part. This paper considers the and ℑ() modified-Hampel based M-estimator to implement an Mdecorrelating detector with penalty function [10] ⎧ ⎪ x2 for x ≤ a ⎪ 2 ⎪ ⎪⎪ a 2 for a < x ≤ b ρ MH ( x ) = ⎨ −a x ⎪ 2 ⎪ ⎛ x2 ⎞ ab ⎪ − exp⎜1− 2 ⎟+d for x >b ⎪ ⎜ b ⎟ 2 ⎝ ⎠ ⎩⎪

,..., g

M-decorrelating detector’s implementation are [2], [3], [4]:

L ⎡ ⎤ θ = F (θl (i ) ) = arg min ∑ ρ ℜ rn (i ) − ∑ [ A]nl θ l (i ) ⎢ ⎥⎦ l =1 θ∈C L n =1 ⎣

+ ρ ℑ rn (i ) − ∑ [ A]nl θ l (i )

in

best 1

vd = [vd1 , ..., vd L ] [2], [4]. The steps involved in the PSO based

In M-estimates, unknown parameters θ 1 , θ 2 ,...θ L are solved by minimizing a sum of function, ρ (⋅) of the residuals [11]

L

it

= [g

iteration, d = 1, 2, …, Np, it = 1, 2,…, Nit, Np is the number of particles, and Nit is the maximum number of iterations. Corresponding to each position, the particle velocity is

l

{ (

particles it

best

as p d = [ pd1 ,..., pd L ] , where p d is the dth particle in the itth

A [ a1 , a 2 , ..., a L ] with a l [ a1 , a2 ,..., a N ] .

N

best

(called as gbest), g

1 T ⎡ b (i ) g1 (i ),...,bL (i ) g L (i ) ⎤⎦ . Here, wn (i ) is a θ (i ) N ⎣1 sequence of independent and identically distributed (i.i.d.) complex random variables whose in-phase and quadrature components are independent non-Gaussian random variables, gl (i ) is the lth channel fading coefficient and l

best

= [ pd1 ,..., pd L ] and the overall global best solution

privatethinking of the particle

it +1 vd

=w +

( −p ) c × μ × (g

it +1

it × vd

+

2

2

best

c1 × μ1 × p d

best

it d

it

− pd

)

(5)

socialthinking of the particle

(4)

it +1

pd

it

it +1

= p d + vd

(6)

where c1 and c2 are the acceleration constants

representing the weighting of the stochastic acceleration terms to pull the particle to pbest and

276


gbest.

μ1 and μ 2 are random numbers that are

Ω=

uniformly distributed between 0 and 1. Particle position is updated according to (6). Particle velocity is limited by the maximum velocity v

max

max

= [v1

max

,..., v2

(

it

following rule:

then Step 7: g

best

best

best pd

=

it pd

}

−1 2 2 ν ⎡ R* ⎤ + 2 SNRl σ (1 − ρl )

⎣

⎦ ll

2

2

2

(8)

2

(9)

2

with υ and κυ , respectively, as the variance of in-phase and quadrature components of noise samples),

Step 6: The individual best particle position is updated by

( )

N

{

σ = (1 − ε )υ + εκυ

].

if F p d ≤ F p d

1

).

SNRl

is the global best particle position among all the

2 E ⎡ gl ( i ) ⎤

⎣

σ

⎦.

2

(10)

it

individual best particle positions p d at the itth

(

iteration such that F g

best

)

TABLE I. PSO PARAMETERS USED FOR SIMULATION

( ) it

≤ F pd .

Step 8: The above steps are repeated until the maximum number of iterations has been reached. The computed g

best

value is used to evaluate average

probability of error of non-coherent DPSK demodulator, over Nakagami-m fading channel including MRC receive-diversity, by an approximate expression given by [14]

Acceleration constants c1 and c2

2 max

for

all users Initial inertial weight, w Decrement constant, β

2 1 0.99

IV. SIMULATION RESULTS In this section, the performance of M-decorrelating detector is presented by computing (7) for dual-branch diversity (D = 2) and m = 1 with least-squares, Huber, Hampel and modifiedHampel penalty functions. The PSO parameters used for

M +l

best

Γ ( mD + M + i )

⎞ ⎟ ⎟ ⎠

Value 20 100

Maximum velocity of the particles vl

i ⎧ ⎛ ⎞ ⎪ ⎜ 1 ⎟ −i 1 − 2 i ⎪ Γ ( P +i ) P 2 ⎜ 2 ⎟ 2 2 mD ⎪ σ 2 ⎜ ⎟ P ⎛ 1 ⎞ ⎛m⎞ ⎪ l ⎝ θˆl ⎠ Pe ≅ 2 ⎜ ⎟ ×∑⎨ ⎜ ⎟ Γ i + Γ P − i +1)Γ ( M +i ) ( 1) ( ⎝Ω⎠ ⎝ Γ ( mD ) ⎠ i =0 ⎪ ⎪ ⎪ ⎪ ⎩

⎛ 1 ⎞ ⎟ (i + mD − 1)!⎜ ⎜ 4σ θ2ˆ ⎟ ⎝ l⎠ × ⎛m 1 2 mD ⎜ + ⎜ Ω 4σ θ2ˆ l ⎝

Parameter Number of particles, Np Maximum number of iterations, Nit

2(i + mD ) + M

⎛ ⎜ m ⎜ Ω × 2 F1⎜1,mD + M +i;mD +1; m 1 ⎜ + ⎜ Ω 4σ 2 ⎜ θˆl ⎝

⎞⎫ ⎟⎪ ⎟⎪ ⎟ ⎪⎬ ⎟⎪ ⎟⎪ ⎟ ⎠ ⎪⎭

⎛ ⎞ 1 ⎛ Ω ⎞2 ⎜ ⎟ + 0.5 2 F1 mD, mD;1; − 2 ⎜ ⎟ ⎜ 16σ θˆ ⎝ m ⎠ ⎟ l ⎝ ⎠

(7)

simulation are presented in Table I. The value of g is computed for specified penalty functions and is used to compute the average probability of error (7) of DPSK signals. In Fig. 1, the average probability of error versus the signal-tonoise ratio (SNR) corresponding to the user 1 under perfect power control of a synchronous DS-CDMA system with six users (L = 6) and processing gain, N = 31 is plotted for moderate impulsiveness ( ε = 0.01) of noise. Similarly, the average probability of error is plotted in Fig. 2 with highly impulsive noise ( ε = 0.1). Evolutionary behavior of the Mdecorrelator is depicted in Fig. 3 and Fig. 4. In Fig. 5, the average probability of error of PSO-based M-decorrelator is plotted with respect to number of iterations. These results show that the decorrelating detector with modified-Hampel estimator based detector performs well even in highly impulsive noise when compared to least-squares, Huber and Hampel estimator based detectors. V. CONCLUDING REMARKS

where , m is Nakagami fading severity parameter, D is diversity order, 2 F1 is the Gauss hypergeometric function [15],

Multiuser detection technique for DS-CDMA systems over Nakagami-m fading channels using PSO was presented in this paper. An objective function, which is a sum of less rapidly increasing function of residuals, was used to obtain optimum estimates. An M-decorrelator is implemented with different

277


influence functions. Simulation results shows that the modified-Hampel based M-decorrelator performs better when compared to least-squares, Huber and Hampel estimator based detectors.

Fig. 3. Evolution of optimization function of PSO based M-decorrelating detector with respect to number of iterationsfor user 1 with linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and modified-Hampel (MH) in moderate impulsive noise ( ε = 0.01) environment.

Fig. 1. Average probability of error versus SNR for user 1 for linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and modified-Hampel (MH) M-estimator in synchronous CDMA channel with impulse noise, N = 31, ε = 0.01.

Fig. 4. Evolution of optimization function of PSO based M-decorrelating detector with respect to number of iterationsfor user 1 with linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and modified-Hampel (MH) in highly impulsive noise ( ε = 0.1) environment.

Fig. 2. Average probability of error versus SNR for user 1 for linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and modified-Hampel (MH) M-estimator in synchronous CDMA channel with impulse noise, N = 31, ε = 0.01.

278

2013 IEEE 9th International Colloquium on Signal Processing and its Applications, 8 - 10 Mac. 2013, Kuala Lumpur, Malaysia [5] V.A. Aalo, G.P. Efthymoglou, "On the MGF and BER of Linear Diversity Schemes in Nakagami Fading Channels with Arbitrary Parameters," IEEE 69th Vehicular Technology Conference, vol., no., pp.1-5, 26-29, Apr. 2009. [6] M.K. Simon, M. S. Alouini, “A unified approach to the performance analysis of digital communication over generalized fading channels,” Proc. of the IEEE, vol. 86, no. 9, pp. 18601877, Sep. 1998. [7] D. M. Novakovic, M.L. Dukic, “Multiuser detection analysis in DS-CDMA communication systems with Nakagami fading channels,” IEEE 5th International Symposium on Spread Sprectum Techniques and Applications, vol.3, no., pp. 932-935, Sep. 1998. [8] K. A. Emad and M. S. Iman, “Performance of the decorrelator receiver for DS-CDMA mobile radio system employing RAKE and diversity through Nakagami fading channel,” IEEE Trans. Commun., vol. 50, no. 10, 2002. [9] A.M. Zoubir, V. Koivunen, Y. Chakhchoukh, M. Muma, “Robust Estimation in Signal Processing: A Tutorial-Style Treatment of Fundamental Concepts,” IEEE Signal Processing Magazine, vol.29, no.4, pp.61-80, Jul. 2012. [10] T. Anil Kumar, and K. Deerga Rao, " Improved Robust techniques for multiuser detection in non-Gaussian channels”, Circuits Systems and Signal Processing J., vol. 25, no. 4, 2006. [11] H. V. Poor, and M. Tanda, “Multiuser Detection in flat fading non-Gaussian channels,” IEEE Trans. Commun., vol. 50, no. 11, pp. 1769-1777, Nov. 2002. [12] X. Wang and H. V. Poor, "Robust multiuser detection in nonGaussian channels," IEEE Trans. Signal Processing, vol.47, no.2, pp.289-305, 1999. [13] T. Anil Kumar, K. Deerga Rao, “A New M-estimator based robust multiuser detection in flat-fading non-Gaussian channels”, IEEE Trans. Commun., vol. 57, no. 7, pp. 1908-1913, Jul. 2009. [14] P. Vinay Kumar, V. Srinivasa Rao, Habibulla Khan and T. Anil Kumar, “Robust blind multiuser detection in DS-CDMA systems over Nakagami-m fading channels with impulsive noise including MRC receive diversity,” Proc. IEEE 6th International Conference on Signal Processing and Communicaiton Systems, Gold Coast, Australia, 12-14 Dec. 2012. [15] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Boston, MA, 2007.

Fig. 5. Average probability of error of PSO based M-decorrelating detector with respect to number of iterations for user 1 with linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and modifiedHampel (MH) in impulsive noise ( ε = 0.01, ε = 0.1) environment.

REFERENCES [1] Z. Guo, Y. Xiao, M. H. Lee, “Multiuser Detection Based on Particle Swarm Optimation Algorithm,” Proc. of IEEE International Symposium on Circuits and System., vol., no., pp.2582-2585, 27-30 May 2007. [2] J-C. Chang, “Robust blind multiuser detection based on PSO algorithm in the mismatch environment of receiver spreading codes,” Comput Electr Eng, 2012, http://dx.doi.org/10.1016/j.compeleceng.2012.08.002. [3] X. Wu, T.C. Chuah, B.S. Sharif and O.R. Hinton, “Adaptive robust detection for CDMA using a gnenetic algorithm,” IEE Proc.-Commun., vol. 150, no. 6, pp. 437-444, Dec. 2003. [4] K.K. Soo, Y.M. Siu, W.S. Chan, L. Yang and R.S. Chen, "Particle-Swarm-Optimization-Based Multiuser Detector for CDMA Communications," IEEE Trans., Veh. Technol., vol.56, no.5, pp.3006-3013, Sep. 2007.

279