Proceedings of the 19th International Conference on Digital Signal Processing
20-23 August 2014
Multiuser Detection for MIMO DS-CDMA System in Impulsive Noise over Nakagami-m Fading Channels Srinivasa Rao Vempati
Vinay Kumar Pamula
Department of ECE, KITS for Women, Kodad, India 508206. e-mail:
[email protected]
Department of ECE, UCEK, JNTUK, Kakinada, India 533003. e-mail:
[email protected]
Habibulla Khan
Anil Kumar Tipparti
Department of ECE, KL University, Vaddeswaram, India 522502. e-mail:
[email protected]
Department of ECE, SR Engineering College, Warangal, India 506 371. e-mail:
[email protected]
a number of different approaches to the robust estimation problem, and the M-estimator is one of the most sophisticated approaches to this problem. The problem of robust multiuser detection in non-Gaussian channels has been addressed in the literature [6], which was developed based on the Huber and the Hampel M-estimators, respectively. This paper presents the performance of a MIMO direct-sequence CDMA (DS-CDMA) system with maximal ratio combining (MRC) receive-diversity over Rayleigh fading channels in an impulsive noise environment. An expression for average bit error rate (BER) of a decorrelating detector is derived. This expression is used to compute the probability of error of linear decorrelating detector, the Huber and the Hampel estimator based detectors, and the proposed [7] M-estimator.
Abstract—This paper presents a new M-estimator based multiuser detection technique for multiple-input multiple-output (MIMO) direct-sequence code division multiple access (DSCDMA) system including maximal ratio combining (MRC) receive diversity to combat multi access interference (MAI), multipath fading and impulsive noise. An expression for average bit error rate (BER) of M-decorrelator is derived to demodulate coherent BPSK signals over Nakagami-m fading channels. Simulation results shows that the proposed M-estimator based detector outperforms the linear decorrelating detector, the Huber and Hampel estimator based detectors. Keywords— Average BER; BPSK; CDMA; diversity; fading channel; impulsive noise; M-estimator; MIMO; MRC; Nakagami.
I.
INTRODUCTION
The remaining part of the paper is organized as follows: Downlink MIMO CDMA system over multipath fading channel in non-Gaussian environment is presented in Section II. Influence function of the proposed M-estimator is presented in Section III. An approximate closed-form expression for average BER of an M-decorrelator with MRC combining is derived in Section IV. In Section V, simulation results are presented to study the performance of proposed M-estimator based decorrelating detector in impulsive noise. Finally, conclusions are drawn in Section VI.
The multiple-input and multiple-output (MIMO) systems use antenna arrays both at transmitting and receiving ends. Capacity of MIMO systems, which use spatial diversity and spatial multiplexing, is more compared to single-input and single-output (SISO) systems [1]. As the code division multiple access (CDMA) systems provide high data rates and high spectral efficiency, the combination of MIMO and CDMA technique is well suited for supporting high data rates [2]. Multiuser detection (MUD) for CDMA presented in the pioneering work of Verdu [3] and has been extensively studied in the literature. Experimental results confirmed that the ambient noise in many wireless communication systems is impulsive in nature [4].
II.
SYSTEM MODEL
In this paper, a downlink MIMO DS-CDMA system with known pseudo noise (PN) sequences at the receiving end is considered. It is assumed that the system has L users with NR receive antennas and NT transmit antennas. The received signal at the kth receiving antenna which represents the kth diversity reception is given by [8]
Recently, [1] analyzed the performance of robust MIMOCDMA decorrelating detector in impulsive noise by using a generalized clipper decision function. Robust decorrelating detector in multipath-fading under impulsive noise is analyzed in [5] by implementing it using a non-linear clipper to implement impulsive components. There exist in the literature
M
NT
L
rp (t ) = ∑∑∑ cn , p al ,n sl bl ,n (m) + n p (t ) m =1 n =1 l =1
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(1)
Proceedings of the 19th International Conference on Digital Signal Processing where
III.
cn , p is the fading coefficient of the nth transmitting
θ = Ab ) are solved by minimizing a sum of function ρ (⋅) of the residuals [6]
th
l
user
user.
Ts is the symbol duration. τ n , p is the timing delay
from
sl ≡ sl (t − mTs − τ n, p )
the n transmit antenna. is the signature sequence of the lth
channel coefficients are zero-mean independent complex Gaussian random variables with unit variance. At the receiving end the signal is passed through a matched filter and the discrete-time version of its output is given by [8]
(2) N
j =1
(7)
(
L
l
)
k
l =1
(8)
or in vector form
NR
i ST S C ip R = ∑C p p H p
(3)
S ψ ( r − Sθ ) = 0 L T
p =1
and
(9)
T
NR
i p ST n . n = ∑C p p H
where S is the transpose of S and 0 L is an all zero vector of length L . Eq. (7) is called an M-estimator. Different influence functions yield solutions with different robustness properties. Therefore, an influence function ψ (⋅) should be chosen such that it yields a solution that is not sensitive to outlying measurements.
(4)
p =1
The noise term, in (1), is modeled as the two-term Gaussian mixture model with probability density function (PDF) [6] 2
2
f = (1 − ε )ℵ(0, ν ) + εℵ(0, κν )
An influence function is proposed (see Fig. 1), such that it yields a solution that is not sensitive to outlying measurements, as [7]
(5)
with ν > 0 , 0 ≤ ε ≤ 1 , κ ≥ 1 . Here ℵ(0, ν 2 ) represents the nominal background noise and the ℵ(0, κν 2 ) represents an impulsive component, with ε representing the probability that impulses occur. This noise model serves as an approximation to more fundamental Middleton class A noise model [9-10] with PDF
∞
f (n p ) = ∑
αm
m = 0 πσ
2
)
∑ ψ r j − ∑ s jθ l s j = 0, k = 1,..., L
where
where
L
where ρ is a symmetric, positive-definite function with a unique minimum at zero, and is chosen to be less increasing than square and N is the processing gain. Suppose that ρ has a derivative with respect to the unknown parameters θ (ψ = ρ ' ), called the influence function, since it describes the influence of measurement errors on solutions. The solution to Eq. (7) satisfies the implicit equation [6]
n p (t ) is the ambient noise. It is assumed that the
⎧ NR i H T ⎫ y = ℜ ⎨∑ C p S p rp ⎬ = RAb + n ⎩ p=1 ⎭
(
N
l θˆ = arg min ∑ ρ r j − ∑ s jθ l = 1 = 1 L j l θ∈ℜ
between the nth transmitting antenna and pth receiving antenna. bl ,n (m) is BPSK modulated data sequence. M is the frame size and
M-ESTIMATION BASED REGRESSION
In M-estimates, unknown parameters θ1 , θ 2 ,...θ L (where
al , n is the amplitude of the
antenna and the pth receiving one. th
20-23 August 2014
αm =
σ = var( n p ) .
Zm m!
e
−Z
,
2
2 m 2
−
e
np
⎧ x ⎪ ⎪ ⎪a sgn( x ) ⎪ ψ PRO ( x ) = ⎨ ⎪ ⎪ a ⎛ x2 ⎞ ⎪ x exp⎜1− 2 ⎟ ⎜ b ⎟ ⎪ b ⎝ ⎠ ⎩
2
σ m2
σ m = σ ( m / Z + T ) / (T + 1)
(6)
x ≤a
for a< x ≤b for
(10)
x >b
where the choice of the constants a and b depends on the robustness measures. IV. AVERAGE BER OF M-DECORRELATOR In this section, an approximate expression of average BER is derived for a decorrelating detector over Rayleigh fading channels.
and
Here, T represents the power ratio of the
background Gaussian noise and the impulsive component, and Z is the impulsive index [11].
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for
The asymptotic BER for the class of decorrelating detectors, for large processing gain N, is given by [6]
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Proceedings of the 19th International Conference on Digital Signal Processing
⎛ ⎞ Al ⎜ ⎟ l Pe = Q ⎜ ⎡ −1 ⎤ ⎟ ⎜ ν ⎣R ⎦ ⎟ ll ⎠ ⎝
20-23 August 2014 l
Pe
(11)
(19) where m is the fading severity parameter and D is the diversity order equivalent to NR for MRC diversity. Now, using the relation [15]
where Q ( x ) is the Gaussian Q - function defined by ∞
Q( x) = ∫
x
1 2π
(
2
∞
)
exp −ξ /2 d ξ , x ≥ 0 ,
α −1 − β x dx = ∫ x e
(12)
Γ( mD ) 2 β mD
(20)
and following the steps as in [11-12], (19) can be simplified to 2
ν =
2
0
and 2
( )
∞ 2 1 ⎧⎪∞ ⎫⎪ m mD ∑ αq ⎨ ∫ x2mD−1e−β x dx⎬ Ω ⎪⎭ Γ(mD) q=0 ⎪⎩ 0
∫ ψ (u ) f (u ) du 2
[ ∫ψ ′(u ) f (u )du ]
l
.
(13)
⎛ m ⎞ where G = ⎜ ⎟ ⎝ βΩ ⎠
For NR receiving diversity with MRC, the asymptotic BER is given by [12]
⎛ NR 2 ⎜ ∑ Ail l i =1 Pe = Q ⎜ ⎜ ν ⎡R −1 ⎤ ⎜ ⎣ ⎦ ll ⎝
⎞ ⎟ ⎟. ⎟ ⎟ ⎠
NR 2 ∑ Ail i =1 ⎡ R −1 ⎤ ⎢⎣ ⎥⎦ll
D D ⎛ D⎞ ∑ ⎜⎜ ⎟⎟ d =0 ⎝ d ⎠
with β =
m 1 + . Ω 2ν 2 ⎡ R −1 ⎤ ⎣
⎦ ll
SIMULATION RESULTS
In this section, simulation results for the average BER are presented by computing (21) for different values of NR and different noise parameters. Fig. 2 and Fig. 3 show the average BER versus signal-tonoise ratio (SNR) corresponding to user 1of MIMO DSCDMA system with six active users (L = 6) under the assumption of perfect power control. Performance of the decorrelator with different influence functions in nearGaussian noise (A = 1) channels and highly impulsive noise (A = 10-4) channels is depicted in Fig. 2 for NT = 1 and NR = 1, 3.
(14)
⎞ ⎟ ⎟ ⎟ . ⎟ ⎟ ⎟ ⎠
⎛ ⎞⎤ 1 ⎡ d i ⎜ 1 i −1 j⎟ ( 1) Z − ∑ ∑ ⎢ ⎜ ⎟⎟ ⎥ (21) ( ZT )d ⎣i =1 ⎜⎝ d ! j =0 ⎠⎦
mD
V.
By averaging (14) with respect to the random variable that describes fading channel, the BER of M-decorrelator under impulsive noise model (6) with MRC is given by [12] ⎛ ⎜ ∞ ⎜ l Pe = ∑ α m Q⎜ ⎜ν m=0 ⎜ ⎜ ⎝
⎛ T ⎞ ⎟ ⎝ T +1 ⎠
Pe = G ⋅ ⎜
(15)
Using the well known upper-bound of Q ( x ) given by [13]
Q( x) ≤
1 2
⎛ − x2 ⎞ ⎟, ⎟ ⎝ 2 ⎠
exp ⎜⎜
(16)
(15) can be expressed as
l Pe
⎛ − Al2 1 ∞ ⎜ ∑ α m exp⎜ 2 2 −1 2 m=0 ⎜ 2(ν =σ m ) ⎡⎢ R ⎤⎥ ⎣ ⎦ll ⎝
⎞ ⎟ ⎟ . ⎟ ⎠
(17)
Assuming that Al = x is Nakagami-m distributed with PDF given by [14] 2 mD −1 − m x 2 m mD x p( x) = e Ω Ω Γ ( mD )
( )
(18)
Fig. 1.
Influence function of proposed M-estimator.
and averaging (17) with respect to the random variable x, the average BER of the M-decorrelator can be expressed as
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Proceedings of the 19th International Conference on Digital Signal Processing
20-23 August 2014
near-Gaussian noise and highly impulsive noise over Nakagami-m fading channels. VI.
CONCLUDING REMARKS
This paper presented the multiuser detection for MIMO DSCDMA system in impulsive noise with MRC receive diversity over Nakagami-m fading channels. An approximate closedform expression for average BER of an M-decorrelated is derived to study the performance of proposed M-estimator. Simulation results shows that the proposed M-decorrelator outperforms the linear decorrelating detector, the Huber and the Hampel estimator based detectors. REFERENCES [1]
Fig. 2. Average BER versus SNR for user 1 for linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and proposed (Pro) M-estimator in MIMO DS-CDMA system with A = 1.
[2]
[3] [4]
[5]
[6]
[7]
[8]
[9] Fig. 3. Average BER versus SNR for user 1 for linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and proposed (Pro) M-estimator in MIMO DS-CDMA system with A = 10-4.
[10]
In Fig. 3, the performance of M-decorrelator with different influence functions is shown for near-Gaussian and highly impulsive noise. In this, case it is assumed that NT = 1 and NR = 1, 3.
[11]
[12]
Simulation results show that the proposed M-estimator based detector performs well in the heavy-tailed impulsive noise compared to linear multiuser detector, minimax detector with Huber and Hampel estimators. It is observed that the increase in diversity order improves system performance in
[13] [14] [15]
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