2014 Fifth International Conference on Intelligent Systems, Modelling and Simulation
Robust Multiuser Detection in Asynchronous DS-CDMA Systems over Nakagami-m Fading Channels
Srinivasa Rao Vempati
Umamaheshwar Soma
Department of ECE, Anurag Engineering College, Kodad, India 508206. e-mail:
[email protected]
Department of ECE, VR Engineering College, Warangal, India 506371. e-mail:
[email protected]
Srinivasa Rao Kunupalli
Habibulla Khan
Department of ECE, TRR College of Engineering, Patancheru, India 502319. e-mail:
[email protected]
Department of ECE, KL University, Guntur, India 522502. e-mail:
[email protected]
Anil Kumar Tipparti Department of ECE, SR Engineering College, Warangal, India 506 371. e-mail:
[email protected] throughput [1 and 4]. Subspace-based blind multiuser detection technique is known for its better performance among all the other blind methods. In [1], an improved subspace-based robust blind multiuser detection technique for synchronous and asynchronous DS-CDMA systems over non-Gaussian channels is presented. A new adaptive algorithm for blind multiuser detection of coherent BPSK signals in DS-CDMA systems operating under non-Gaussian impulsive noise is also presented in [1]. Recently, in [6] the robust blind multiuser detection for synchronous DS-CDMA systems over Nakagami-m fading channels is considered. In this paper, multiuser detection in asynchronous DSCDMA systems over Nakagami-m channels in the presence of heavy-tailed impulsive noise is considered and a new Mestimator is proposed for its robustification. Further, subpace-based blind multiuser detection technique is also applied for asynchronous DS-CDMA systems over Nakagami-m channels in impulsive channels. An expression for average probability of error is derived and used to compute the probability of error for linear decorrelating detector, the Huber and the Hampel estimator based detectors along with the proposed M-estimator based detector. Simulation results shows that the proposed Mestimator based detector outperforms the linear decorrelating detector, Huber and Hampel estimator based detectors.
Abstract—This paper presents a new M-estimator based robust multiuser detection technique in asynchronous direct sequence-code division multiple access systems over Nakagamim fading channels with impulsive noise. A new M-estimator, which performs well in the heavy-tailed impulsive noise, is proposed and analyzed for robustifying the detector. Average probability of error is derived and the performance of the new M-estimator based multiuser detector is evaluated in comparison with the linear decorrelating detector, Huber and Hampel estimator based detectors. Simulation results show that the proposed M-estimator based detector performs well with little attendant increase in computational-complexity. The proposed robust multiuser detection technique is also extended to blind version of asynchronous CDMA channels. Keywords- multiuser detection; CDMA; impulsive noise; M-estimator; Nakagami-m distribution; probability of error
I. INTRODUCTION The problem of robust multiuser detection in nonGaussian channels has been addressed in the literature [1-2], which were developed based on the Huber and the Hampel M-estimators, respectively. Differential non-coherent data detection for direct-sequence code-division multiple-access (DS-CDMA) over flat-fading non-Gaussian channels with impulsive noise is presented in [3-4]. Robust multiuser detection in synchronous DS-CDMA system with MRC receive diversity over Nakagami-m fading channel is presented in [5] by assuming that the modulation is binay PSK (BPSK). Blind multiuser detection can be used to reduce the intersymbol interference (ISI) and also to improve system 2166-0662/14 $31.00 © 2014 IEEE DOI 10.1109/ISMS.2014.50
II.
SYSTEM MODEL
In this section, an asynchronous DS-CDMA system shared by L-users, over a single-path Nakagami-m fading channel, is considered. The received signal over one symbol duration can be modeled (when the users are symbol and chip asynchronous) as [2]: 255
⎧⎪ L M −1 ⎫⎪ jθ (t ) r (t ) = ℜ ⎨ bl (i)αl (t )e l sl (t − iTs - τl )⎬ + n(t ) , (1) ⎪⎩ l=1 i =0 ⎪⎭
r k (i ) = H k (i )θ k (i ) + w k ( i )
∑∑
where ⎡ H (1) ⎢ ⎢ 0 ⎢ ⋅ H k (i ) ≅ ⎢⎢ ⋅ ⎢ ⎢ ⋅ ⎢ ⎢⎣ 0
⋅ denotes the real part, M is the number of data where ℜ {} 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, θl (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 normalized signaling waveform of the lth user, τl is the propagation delay and n(t ) is assumed as a zero-mean complex non-Gaussian noise given by [2]. At the receiver, the received signal r ( t ) is first filtered by a chip-matched
III. (3)
iTs + ( j +1)Tc
Tc
iTs + jTc
∫
n(t )e
− jωct
j > ml
dt
1 T ⎡⎣b1 (i ) g1 (i ),...,bL (i ) g L ( i ) ⎤⎦ N
.
By
stacking
⋅
H (1)
(7)
M-ESTIMATOR
and
(5)
⎧ ⎪ x, for x ≤ a ⎪ ⎪ ψ PROPOSED ( x) = ⎨ a sgn( x), for a < x ≤ b (10) ⎪ 2 ⎪ a x exp ⎜⎛ 1 − x ⎟⎞ , for x > b 2 ⎪⎩ b ⎝ b ⎠
T T Let r (i ) ⎡⎣ r0 ( i ),..., rN −1 ( i ) ⎤⎦ , w (i ) ⎡⎣ w0 (i ),...,w N −1 (i ) ⎤⎦ and
θ (i)
⋅
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ H (0)⎥⎦
⋅ ⋅ ⋅
2 ⎧ x , for x ≤ a ⎪ 2 ⎪ 2 a ⎪ ρ PROPOSED ( x ) = ⎨ (9) − a x , for a < x ≤ b 2 ⎪ 2 ⎪− ab exp ⎛1 − x ⎞ + d , for x > b ⎜ 2⎟ ⎪ 2 ⎝ b ⎠ ⎩
(4)
⎢τl ⎥ τl − ml and ⎥ , πl ≡ ⎢⎣ Tc ⎥⎦ Tc
2
⋅
⋅
⋅ ⋅ ⋅
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 (⋅) (that exponentially suppresses the large values of noise) are proposed [4]
with ml ≡ ⎢
w j (i) =
⋅
0 ⎤ ⎥ 0 ⎥
where k is smoothing factor and is chosen such that H k has full column rank.
j < ml j = ml
⋅ ⋅
(8)
where
l ⎧ π al + (1−π l ) a , j + N − ml ⎪ l j + N − ml −1 ⎪ l h j (1) ⎨ π l alj − m + N −1, l ⎪ ⎪0, ⎩
⋅
⎡ hl (0) ⋅ ⋅ ⋅ hL(0)⎤ ⎡ hl (1) ⋅ ⋅ ⋅ hL (1)⎤ 1 1 ⎥ ⎢ 1 ⎥ ⎢ 1 ⎢ ⋅ ⎢ ⋅ ⋅ ⋅ ⎥ ⋅ ⋅ ⎥ ⎢ ⎥ ⎢ ⎥ ⋅ ⋅ ⎥ and H(1) ≅ ⎢ ⋅ ⋅ ⋅ ⎥ H(0) ≅ ⎢ ⋅ ⎢ ⎥ ⎢ ⎥ ⋅ ⋅ ⎥ ⋅ ⋅ ⎥ ⎢ ⋅ ⎢ ⋅ ⎢ l ⎥ ⎢ l ⎥ L L ⎣⎢hN (0) ⋅ ⋅ ⋅ hN (0)⎦⎥ ⎣⎢hN (1) ⋅ ⋅ ⋅ hN (1)⎦⎥
1 L ⎡ ∑ gl (i )bl (i ) hlj (0)+ g l ( i −1)bl ( i −1) hlj (1) ⎥⎤ + w j (i ) (2) ⎦ N l =1 ⎢⎣
⎧ ⎪0, j < ml ⎪⎪ l l h j (0) ⎨ (1−π l ) a j − m , j = ml l ⎪ l ⎪ π al + (1−π l ) a , j > ml j − ml ⎪⎩ l j − ml −1
H (0) 0 ⋅
with
filter and then sampled at the chip rate, 1 / Tc . The resulting discrete-time signal sample corresponding to the nth chip of the ith symbol (assuming that the fading process for each user varies at a slower rate that the phase and amplitude can taken to be constant over the duration of a bit) is given by r j (i ) =
(6)
k
successive data samples, r k (i ) can be written as [2-3]
256
2
over the joint probability distribution function (PDF) of the random variables x and y defined as
where d is a constant. The choice of the constants a (= κυ ) 2
and b (= 2 κυ ) depends on the robustness measures derived from the influence function. IV. ROBUST BLIND MULTIUSER DETECTION In robust blind multiuser detection for asynchronous channels the signal subspace components of the covariance matrix of the signal r k (i ) are computed as [1]
and
1
y
1
N
N
T
2
σ θˆ l
Here, the signal subspace has dimension of L(m+1). Then, the robust estimation in the signal subspace follows as
(
k
r (i) ≅ ψ r − U sξ (i) ξ
k +1
k
(i ) = ξ ( i ) +
1
μ
)
T k U s r (i)
where μ is a step-size parameter chosen as
θ l (i ) =
∑
λj
j =1
gl (i ) + gl (i − 1)
(18)
N
−1 2 ν ⎡ R* ⎤
⎣
(19)
⎦ ll
2
∫ ψ (u ) f (u ) du
2
ν =
1
(20)
(12)
with
(13)
and R* is the cross-correlation matrix of the random infinite-
[ ∫ψ ′(u ) f (u ) du ]
2
length signature waveforms of the L users, QM (⋅, ⋅) is the
1
Marcum’s Q-function and I o (⋅) is the modified Bessel function of the first kind with order zero. It is assumed that the random variables x and y are statistically independent Nakagami-m random variables with PDF given as
2
= ν . The
μ estimate of the product of the amplitude and the data bit θ l (i ) = Wl bl (i ) of the lth user is given by [1] L ( k +1) uT h j l
(17)
(11)
= U s Λ s U s T + υ 2 U n U nT
k
gl (i ) − gl (i − 1)
where
C ≅ E ⎡⎣r l (i )r l (i )T ⎤⎦ = H k (i )W k H k + υ I N 2
x
ξl (i ), l =1,2,..., L
⎛m⎞ ⎟ ⎝Ω⎠
pα (α l ) = 2 ⎜
(14)
mD
αl
2 mD −1
Γ(mD )
(
m exp − αl2 Ω
)
(21)
where m is the Nakagami fading parameter that determines where hl ≅
⎡ T ⎢ hl ⎣
⎤ 0TN ( k − 2) ⎥ ⎦
T
the severity of the fading, Ω = E [αl2 ] is the average channel power, E [⋅] is statistical expectation and Γ(⋅) is the complete Gamma function. The average probability of error for the lth user can be expressed as
th
and finally, the l user data bit
is demodulated as ⎪⎧ ⎡ * ⎤ ⎪⎫ b l (i + 1) = sgn ⎨ℜ ⎢θ k ,2 L + l ( i )θ k ,2 L +l (i ) ⎥ ⎬ ⎦ ⎭⎪ ⎩⎪ ⎣
V.
(15) T
1 ⎡ ⎛ ⎞⎤ 1 1 l ⎟⎥ , Pe = E ⎢QM ⎜ x y 2 ⎢ ⎜ 2σ θ2ˆ 2σ θˆ ⎟⎥ l l ⎠⎦ ⎣ ⎝
AVERAGE PROBABILITY OF ERROR FOR MDECORRELATOR
The asymptotic probability of error of non-coherent DPSK demodulator can be evaluated by averaging the conditional probability [3]
( {
}
* P ℜ θˆl (i )θˆl (i − 1) < 0 Δ φl (i ) = 0, x , y
⎛ 1 1 = QM ⎜ x, 2 2 ⎜ 2σ θˆ 2σ θˆ l l ⎝
⎞ 1 ⎛ xy y ⎟ − Io ⎜ ⎟ 2 ⎜ 2σ θ2ˆ ⎝ l ⎠
x +y ⎡ ⎛ xy ⎞ − 4σ 2 ⎤ 1 θˆ ⎥ − E ⎢ Io ⎜ 2 ⎟ e ⎢ 2 ⎜ 2σ ˆ ⎟ ⎥ ⎝ θl ⎠ ⎢ ⎥ ⎣
⎦ 2
)
⎞ − ⎟e ⎟ ⎠
2
(22)
l
x2 + y2 4σ θ2ˆ
(16)
T 2 Using approximation to Marcum’s Q-function given by [7]
l
257
Γ ( P +i ) P1− 2 i a 2 i 2−i Γ ( M +i ,b 2 /2) (23) 2 i =0 Γ (i +1) Γ( P −i +1) Γ ( M +i )exp( a /2) P
QM ( a, b) ≅ ∑
Ω=
and the relations [(6.455) and (6.631), 8], the terms T1 and T2 of (22) can be expressed, respectively, as [6]
2
⎞ ⎟ ⎟ ⎠
⎛m 1 2mD ⎜ + ⎜ Ω 4σ θ2ˆ l ⎝
2(i +mD )+ M
⎞⎫ ⎟⎪ ⎟⎪ ⎟ ⎪⎬ ⎟⎪ ⎟⎪ ⎟⎪ ⎠⎭
(24) and
⎛
T2 = 0.5 2 F1 ⎜ mD, mD;1; −
⎜ ⎝
⎛Ω⎞ 4 ⎜⎝ m ⎟⎠ 16σ ˆ
1
θl
2⎞
⎟ ⎟ ⎠
l
2
⎦ ll
(27)
2
E ⎡ gl (i )
⎣
σ
2
2
⎤ ⎦
(28)
(25) VII. CONCLUDING REMARKS Asynchronous multiuser detection and its blind version of DS-CDMA systems over Nakagami-m fading channels in an impulsive noise environment is presented and analyzed in this paper. A new M-estimator is proposed for robustification and its performance is analyzed by computing the average probability of error. Simulation results show that the proposed M-estimator based multiuser detector offers significant performance gains over the linear multiuser detector and the minimax decorrelating detectors with Huber and Hampel M-estimators in impulsive noise. Further, effect of fading parameter and diversity order on the performance of the decorrelator is also studied.
where 2 F1 is the Gauss hyper geometric function [8]. Then the average probability of error of M-decorrelator for the lth user can be expressed as Pe ≅ T1 − T2 .
⎣
VI. SIMULATION RESULTS In this section, the performance of M-decorrelator is presented by computing (26) for different values of Nakagami fading parameter. It is assumed that the channel model is a lightly damped second-order autoregressive (AR) process given by [3] with bit rate, pole radius, and spectral peak frequencies as Tb = 10 kbps, rd = 0.998 and fp = 80 Hz, respectively. Further, it is also assumed that M = 1 and P = 50 in evaluating average probability of error (26). Performance of decorrelating detector with different influence functions is shown in Fig. 1 and Fig. 2. In Fig. 1, the average probability of error versus the signal-to-noise ratio (SNR) corresponding to the user 1 under perfect power control of an asynchronous DS-CDMA system with six users (L = 6) and a large processing gain, N = 127 is plotted for m = 1 and D = 1. Similarly, in Fig. 2, average probability of error is plotted for m = 1 and D = 2. Performance of blind decorrelating detector with different influence functions is shown in Fig. 3 and Fig. 4. The simulation results reveal that the increase in diversity order improves the detector performance. Simulation results also reveals that the proposed M-estimator outperforms the linear decorrelating detector and minimax decorrelating detector (both with Huber and Hampel estimators), even in highly impulsive noise. Moreover, this performance gain increases as the SNR increases.
Γ( mD + M + i )
⎛ ⎜ ⎜ m/ Ω × 2 F1⎜ 1,mD + M + i;mD +1; m 1 ⎜ + ⎜ Ω 4σ 2 ⎜ θˆl ⎝
}
−1 2 2 ν ⎡ R* ⎤ + 2 SNRl σ (1 − ρl )
SNRl
M +l
⎞ ⎟ ⎟ ⎠
N
{
where σ = (1 − ε )υ + εκυ with υ 2 and κυ 2 , respectively, as the variance of in-phase and quadrature components of noise samples w n ( i ) and
i ⎧ ⎛ ⎞ ⎪ ⎜ ⎟ ⎪ Γ ( P + i ) P1− 2i ⎜ 1 ⎟ 2−i ⎪ 2 2 mD ⎜⎜ 2σ 2 ⎟⎟ ˆ ⎛ 1 ⎞ P ⎪⎪ ⎛m⎞ ⎝ θl ⎠ T1 = 2 ⎜ ⎟ ∑⎨ ⎜ ⎟ ⎝Ω⎠ ⎝ Γ (mD ) ⎠ i = 0 ⎪ Γ(i +1)Γ ( P −i +1)Γ ( M + i ) ⎪ ⎪ ⎪ ⎪⎩
⎛ 1 (i + mD − 1)!⎜ 2 ⎜ 4σ θˆ ⎝ l ×
1
(26)
Following the steps as in [3], the average channel power Ω in (26) can be expressed as
258
Figure 3. Average probability of error versus SNR for user 1 for linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and proposed (PRO) M-estimator in asynchronous DS-CDMA channel with moderate impulsive noise, mD = 1, 2.
Figure 1. Average probability of error versus SNR for user 1 for linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and proposed (PRO) M-estimator in asynchronous DS-CDMA channel with moderate impulsive noise.
Figure 2. Average probability of error versus SNR for user 1 for linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and proposed (PRO) M-estimator in asynchronous DS-CDMA channel with highly impulsive noise.
Figure 4. Average probability of error versus SNR for user 1 for linear multiuser detector (LS), minimax detector with Huber (HU), Hampel (HA) and proposed (PRO) M-estimator in asynchronous DS-CDMA channel with higly impulsive noise, mD = 1, 2.
259
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[5]
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