Improved Multiuser Detection in Asynchronous Flat- Fading ... - arXiv

Improved Multiuser Detection in Asynchronous FlatFading Non-Gaussian Channels K. Vidyullatha, Student Member, IEEE, Electronics and Communication Engineering Department, Kakatiya Institute of Technology and Science, Warangal, Andhra Pradesh, India – 506 105. e-mail: [email protected]

S. V. N. L. Tejaswi, Electronics and Communication Engineering Department, Kakatiya Institute of Technology and Science, Warangal, Andhra Pradesh, India – 506 015. e-mail: [email protected]

V. Harish, Student Member, IEEE, Electronics and Communication Engineering Department, Kakatiya Institute of Technology and Science, Warangal, Andhra Pradesh, India – 506 015. e-mail: [email protected]

T. Anil Kumar, Member, IEEE Electronics and Communication Engineering Department, RRS College of Engineering and Technology, Patancheru, Andhra Pradesh, India – 502 300. e-mail: [email protected]

Abstract— In this paper, a new M-estimator based multiuser detection in asynchronous flat-fading non-Gaussian CDMA channels is considered. A new closed-form expression is derived for the characteristic function of the multiple-access interference signals. Simulation results are provided to prove the effectiveness of the derived bit-error probabilities obtained with this expression in asynchronous flat-fading non-Gaussian CDMA channels. Keywords- bit-error rate; code-division multiple-access; inter symbol interference; multiuser detection; M-estimation.

I.

INTRODUCTION

Recently, the problem of robust multiuser detection in non– Gaussian channels has been addressed in the literature [1] & [2], which were developed based on the Huber and Hampel estimators. The Huber’s estimator [3] can limit the effect of gross errors; however, the effect may still have to be large enough to reach an unacceptable level. The Hampel’s estimator [4] avoids this problem by setting up three regions to reflect the influence of gross errors with different magnitudes. It is noticed that big gross errors greater than a given threshold will have no effect on the solution as a sharp rejection point may corrupt the estimation when observations are not carefully treated due to low redundancy. Hence, a new M– estimator is proposed for multiuser detection in asynchronous flat-fading non-Guassian channels in this paper. Further, a new closed-form expression is derived for computing the average biterror rate (BER) under asynchronous transmission conditions in flat-fading non-Gaussian channels using the characteristic function method.

II.

l 1

l T N

al = [a ,......., a ]

and

,

n(i) ≅ [n1 (i),......., nN (i)]T

,

A ≡ [a1 , a2 ,....., a L ]

,

, bl (i ) denote symbol stream for the lth user,

g l (i ) is the lth channel fading coefficient (N is the

processing gain and L is the number of active users). It is assumed that the sequence of noise samples {n(i)} is a sequence of independent and identically distributed (i.i.d.) complex random variables whose in-phase and quadrature components are independent non-Gaussian random variables with a common probability density function (pdf) f. The pdf of this noise model has the form f = (1 − ε )ℵ(0, υ 2 ) + εℵ(0, κυ 2 )

(2)

with υ > 0 , 0 ≤ ε ≤ 1, and κ ≥ 1. Here ℵ(0, υ ) represents the 2 nominal background noise and the ℵ(0, κυ ) represents an impulsive component, with ε representing the probability that impulses occur. 2

III.

BER ANALYSIS

In this section, the characteristic function method is used to compute the average BER under asynchronous transmission conditions in flat-fading non-Gaussian channels. We first examine the statistics of each interferer. This analysis parallels that of [6], and so the presentation here is brief. We have the definition of interference from each interferer as I k = GkWk , where Gk is a zero-mean unit-variance guassian random

SYSTEM MODEL

Consider the signal model of [5] in matrix notation, which can be written as

y (i ) = Aθ (i ) + n(i )

y(i) ≅ [ y1 (i),......., yN (i)]T where , T θ (i) ≅ (1/ N )[b1(i)g1(i),......., bL (i)gL (i)]

(1)

variable and Wk is defined in terms of random variables

Pk , Qk , X k , Yk , S k as Wk = Pk Sk + Qk (1 − Sk ) + X k + Yk (1 − 2Sk )

(3)

where, S k is a uniform random variable over [0,1), Pk and

The S k ’s now appear in the numerators of the exponential

Qk are symmetric Bernoulli random variables, X k and Yk are

function arguments and averaging can be carried out to give

the discrete random variables that represent the sums of different sets of independent symmetric Bernoulli random variables. Thus, given Wk , I k is a Gaussian random variable with zero-mean and conditional variance σ I2 W = Wk2 . This k

k

implies through (3) that I k given Pk , Qk , X k , Yk , S k , B (where B equals the number of chip boundaries with transitions in target user’s signature waveform) is Gaussian and the conditional pdf for I k follows as

1 2π Pk S k + Q k (1 − S k ) + X k + Yk (1 − 2 S k )

⎧⎪ ⎫⎪ i k2 × exp ⎨− 2 ⎬ ⎪⎩ 2[Pk S k + Q (1 − S k ) + X k + Yk (1 − 2 S k )] ⎪⎭

0

⎛ A ⎞⎛ B ⎞ ⎟⎜ ⎟ i+ A ⎟⎜ j +B ⎟ i∈A j∈B ⎝ 2 ⎠⎝ 2 ⎠ ×[J (i +1, j) + J (i, j −1) + J (i, j +1) + J (i −1, j)] −( N −1)

=2

4

(4)

since Wk may take negative values, a modulus operation is

(11)

where,

(12)

then, using the fact that the I k ’s given B are independent, we have the characteristic function for the total interference term I , given B , as

required in (3). Averaging over Pk , Qk , X k , Yk (which is equivalent to averaging over all interferers, spreading sequences and data sequences), yields

K

Φ I B (ω) = ∏ Φ I k B (ω)

(13)

k =2

Let , ξ B = I B + n , where n1 is a Gaussian random variable

f I k | S k , B (i k ) =

∑∑⎜⎜

⎧ π / 2 {Q( ω (i − j)) −Q( ω (i + j))}, ( j,ω ≠ 0) ⎪ jω J(i, j) = ⎨exp(− 1 i2ω2), j =0 2 ⎪ =0 ω 1, ⎩

f I k | Pk ,Qk , X k ,Yk , S k , B (i k ) =

1

ΦIk |B (ω) = ∫ ΦIk |Sk ,B (ω) dSk

1

⎛ A⎞ 2 − ( N −1) ∑ ∑ ⎜⎜ i + A ⎟⎟ 4 2π i∈A j∈B ⎝ 2 ⎠ 1

representing the background noise, I B is the total other-user

⎛ B ⎞ ⎜⎜ j + B ⎟⎟ ⎝ 2 ⎠

interference given B , and ξ B is the total disturbance given

⎧⎪ ⎧ ⎫ ⎫⎪ i 1 exp ⎨− ×⎨ ∑ ⎬⎬ 2 ⎪⎩l =1, 2 , 3, 4 σ l (i , j , S k ) ⎩ 2σ l (i , j , S k ) ⎭ ⎪⎭ 2 k

(5)

B . Since the other-user interference and background noise are independent, we have Φ ξ B (ω ) = Φ I B (ω )Φ n1 (ω )

where,

σ (i, j, Sk ) = [1+ i + j(1− 2S k )] 2 1

2

= Φ n1 (ω ) − [1 − Φ I B (ω )]Φ n1 (ω )

(6)

(14)

σ 22 (i, j , S k ) = [2 S k − 1 + i + j (1 − 2 S k )]2

(7)

We use the Fourier inversion formula for the real integral to find the distribution function of ξ B , Fξ B (⋅) which is to be

σ 32 (i, j , S k ) = [1 − 2S k + i + j (1 − 2 S k )]2

(8)

used to calculate the BER, as

σ 42 (i, j , S k ) = [−1 + i + j (1 − 2 S k )]2

(9) Fξ | B (ξ ) =

note that to obtain (6), it was considered that X k and Yk given B are independent. It is clear from (6) that the pdf of I k given S k and B is not gaussian, though the functional form is a weighted summation of “Gaussian-like” terms. We postpone averaging S k here since they appear in the denominators of the exponential function arguments giving an intractable integral. The characteristic function of I k , given

S k and B , is 2 − ( N −1) Φ I k |S k , B (ω ) = 4

(10)

⎛ A ⎞⎛ B ⎞ ⎜⎜ i + A ⎟⎟⎜⎜ j + B ⎟⎟ ∑∑ i∈ A j∈B ⎝ 2 ⎠⎝ 2 ⎠

{

}

∫

+∞

0

Φ I | B (ω )

ω

sin(ξω ) dω

(15)

The conditional BER for our target user can be expressed, by symmetry, as (16)

Pe| A1 , B = P{ξ < − A1 N } = 1 − Fξ ( A1 N ) 1 1 +∞ sin( A1 N ) − Φ ξ | B (ω ) dω 2 π ∫0 ω +∞ sin( A Nω ) 1 = Q( σA1nN1 ) + π1 ∫ × [1 − Φ I |B (ω)] Φ n1 (ω) dω =

0

⎫ ⎧ × ⎨ ∑ exp − 12 σ l2 (i, j , S k )ω 2 ⎬ ⎭ ⎩l =1, 2 ,3, 4

1 1 + 2 π

ω

Averaging over the pdf of A1 , and using the integral identity

N

n =1

∫

+∞

0

sin( kx) xe − x

2

/2

π

dx =

2

ke − k

2

/2

⎧⎪ ⎛ ⎧

∑ ⎨⎪ψ ⎜⎜ ℜ⎨⎩ y ⎩ ⎝

n

L ⎫⎞ (i) − ∑[ A]nk θ k (i)⎬ ⎟⎟ k =1 ⎭⎠

L ⎛ ⎧ ⎫ ⎞⎫⎪ +ψ ⎜⎜ ℑ⎨ y n (i) − ∑[ A]nk θ k (i)⎬ ⎟⎟⎬[ A]nk = 0, k = 1,…, L k =1 ⎭ ⎠⎪⎭ ⎝ ⎩

(17)

(22) we have, Pe| B =

⎤ N N 1⎡ ⎢1 − ⎥+ 2 2 2⎢ 2π σ n1 + N ⎥⎦ ⎣ ⎫ ⎧ 1 × exp ⎨− ω 2 N 2 ⎬ dω 2 ⎭ ⎩

∫

+∞

0

In an impulsive noise environment, a more efficient estimator can be obtained by considering a less sensitive function ρ (.) of the residuals. Hence, the following penalty function ρ (.) and the corresponding influence functions ψ (.) are proposed (also see Figure 1)

[1 − Φ I |B (ω )] Φ n1 (ω )

(18)

which finally yields, 1 N Pe| B = − 2 2π

∫

+∞

0

Φ I | B (ω )Φ n1 (ω ) exp {− ω N 1 2

2

2

} dω

ρ PROPOSED (19)

⎧ x2 , for x ≤ a ⎪ 2 ⎪ 2 a ⎪ ( x) = ⎨ − a x , for a < x ≤ b 2 ⎪ 2 ⎪ − ab exp ⎛⎜ 1 − x ⎞⎟ + d , for x > b 2 ⎜ ⎪ 2 b ⎟⎠ ⎝ ⎩ (23)

where the characteristic function of noise is given by

Φ n1 (ω ) = e −{v

2

(1−ε ) 2 + k 3v 2ε 2 }ω 2

(20)

and

ROBUST MULTIUSER DETECTION

IV.

The basic idea of M–estimator based multiuser detection is to detect the symbols in (1) by first estimating the vector θ (i) , and then extracting symbol estimates from these continuous estimates [1] & [2]. The required estimates of θ (i) are obtained by using M–estimators proposed by Huber [3]. M– estimators minimize a sum of function ρ(.) of the residuals ⎧⎪ ⎛ ⎧ n =1 ⎪ ⎩ ⎝ ⎩ N

L

⎫⎞

k =1

⎭⎠

ψ PROPOSED

⎧ ⎪ x , for ⎪⎪ ( x ) = ⎨ a sgn( x ), for ⎪a ⎛ x2 ⎪ x exp ⎜⎜ 1 − 2 b ⎝ ⎩⎪ b

θ ( i )∈C

⎞ ⎟⎟ , for ⎠

x >b (24)

where d is a constant.

θˆ(i) = arg min ∑ ⎨ρ ⎜⎜ ℜ⎨ y n (i) − ∑ [ A] nk θ k (i)⎬ ⎟⎟ L

x ≤a a< x ≤b

L ⎛ ⎧ ⎫ ⎞⎫⎪ + ρ ⎜⎜ ℑ⎨ y n (i) − ∑ [ A] nk θ k (i)⎬ ⎟⎟⎬ k =1 ⎭ ⎠⎪⎭ ⎝ ⎩

(21) where yn(i) and θk(i) are the nth and the kth element of the vectors y(i) and θ (i) , respectively, [A] nk is the n,k th element of the matrix A, ℑ denotes imaginary part, and ρ is a symmetric, positive-definite function with a unique minimum at zero, and is chosen to be less increasing than square. Suppose that ρ has a derivative (ψ = ρ ′) , then the solution to (21) satisfies the implicit equation

(a)

epsilon=0.1, kappa=100,L=6

0

10

Least Squares Huber Hampel Proposed Influence Function

probability of error

-1

10

-2

10

-3

10

(b) Figure 1. (a) Penalty function and (b) influence functions of the proposed estimator

V.

SIMULATION RESULTS

In simulations, a CDMA system with 6 users, in which the spreading sequence of each user is a shifted version of msequence, is considered. The fading channel is modeled (having a perfect knowledge of the channel coefficients gl(i) (l=1,2,..,L)) as [5]. The performance of the proposed detector as a function of signal-to-noise ratio (SNR) in asynchronous flat-fading non-Gaussian channel (with ε = 0.1 and 0.01, κ = 100) for N=127 is shown in Fig. 2 and 3. The bit rate, the pole radius, and the spectral peak frequency have been fixed at 1/ T = 10Kb / s , rd = 0.998 , and f p = 80Hz . epsilon=0.01, kappa=100,L=6

0

10

Least Squares Huber Hampel Proposed Influence Function

-1

probability of error

10

0

5

10

15

20 snr in dB

25

30

35

40

Figure 3. Probability of error versus SNR for user 1 for the considered detectors in asynchronous flat-fading CDMA channel with non-Gaussian noise. N =127.

These Simulation results show that the proposed detector with the proposed influence function outperforms the linear decorrelating detector and minimax detectors (with Huber and Hampel estimators) in asynchronous flat-fading non-Gaussian channels. VI.

CONCLUSIONS

In this paper, a new closed-form expression is derived for computing the average BER in asynchronous flat-fading nonGaussian channels using the characteristic function method. Further a new M–estimator based multiuser detection technique is proposed that is seen to significantly outperform linear decorrelating detector and minimax detectors (with Huber and Hampel M–estimators) in asynchronous flat-fading CDMA channels with impulsive noise with little attendant increase in the computational complexity. REFERENCES

-2

[1]

10

[2] -3

10

[3] [4] -4

10

0

5

10

15

20 snr in dB

25

30

35

40

Figure 2. Probability of error versus SNR for user 1 for the considered detectors in asynchronous flat-fading CDMA channel with non-Gaussian noise. N =127.

[5]

[6]

X. Wang and H. V. Poor, “Robust multiuser detection in non-Gaussian channels,” IEEE Trans. Signal Processing, vol. 47, 289-305, Feb. 1999. T.Anil Kumar, and K.Deergha Rao, “Improved robust techniques for multiuser detection in non- Gaussian channels,” Circuits Systems and Signal Processing J., Vol. 25, No. 4, 2006. P. J. Huber, Robust Statistics. New York: Wiley, 1981. F. R. Hampel et al., Robust Statistics: The Approach Based on Influence Functions. New York: Wiley, 1986. Z. Zvonar and D. Brady, “Multiuser detection in single-path fading channels, ” IEEE Trans. Commun., vol. 42, pp. 1729-1739, Feb.-Apr. 1994. J. Cheng and N. C. Beaulieu, “Accurate DS-CDMA Bit-Error Probability Calculation in Rayleigh Fading,” IEEE Trans. Wireless Commun., vol. 1, pp. 3-15, Jan. 2002.