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Minimum Probability of Error Based Methods for Adaptive Multiuser Detection in Multipath DS-CDMA Channels Aditya Dua ? , U. B. Desai †
Ranjan K. Mallik †
Department of Electrical Engineering
Department of Electrical Engineering
Indian Institute of Technology, Bombay
Indian Institute of Technology, Delhi
Powai, Mumbai - 400076
Hauz Khas, New Delhi - 110016
India
India
{dua,ubdesai}@ee.iitb.ac.in
[email protected]
?
Student Member, IEEE
†
Member, IEEE
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Abstract DS-CDMA is a popular multiple access technology for wireless communications. However, its performance is limited by multiple access interference and multipath distortion. Multiuser detection and space time processing are two signal processing techniques employed to improve the performance of DSCDMA. Two minimum probability of error based space-time multiuser detection algorithms are proposed in this paper. The first algorithm, MJPOE, aims to minimize the joint probability of error for all users. The second algorithm, MCPOE, minimizes the probability of error of each user conditioned on the transmitted bit vector, for each user individually. In both the algorithms the optimal filter weights are computed adaptively using a gradient descent approach. The MJPOE algorithm is blind and offers a BER performance better than the non-adaptive MMSE algorithm, at the cost of higher computational complexity. An approach for reducing the computational overheads of MJPOE using Gram-Schmidt orthogonalization is suggested. The BER performance of the MCPOE algorithm is slightly inferior to MMSE, however it has a computational complexity linear in the number of users. Both blind and training based implementations for MCPOE are proposed. Both MJPOE and MCPOE have a convergence rate much faster than earlier known adaptive implementations of the MMSE detector, viz. LMS and RLS. Simulation results are presented for synchronous single path channels as well as asynchronous multipath channels, with multiple antennas employed at the receiver. Keywords DS-CDMA, Multiuser Detection, Space-Time Processing, MPOE, MCPOE, MJPOE, MMSE
I. I NTRODUCTION Direct Sequence Code Division Multiple Access (DS-CDMA) is widely used for multiplexing users in a wireless scenario [1]. However, its performance is limited by multiple access interference (MAI) and multipath channel distortion. The conventional DS-CDMA matched filter detector fails to combat these problems. Many advanced signal processing techniques have been proposed to enhance the performance of DS-CDMA systems, and these techniques fall into two broad categories: multiuser detection [2], [3], [4] and space-time processing [5], [6]. The former exploits the underlying statistical structure of MAI for interference cancellation, while the latter employs an antenna array at the receiver to optimally combine the different multipath signals
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of users. Combined multiuser detection and space-time processing has also been addressed in literature. The initial focus was on systems with a single transmit antenna and multiple receive antennas [7], [8]. Recently, much research has focused on systems with multiple transmit and receive antennas, employing space-time coding at the transmitter to achieve higher diversity gain [9], [10]. Blind multiuser detection for systems employing Alamouti space-time block codes at the transmitter has been investigated in [11]. Adaptive multiuser detectors are especially attractive because they can potentially adapt to unknown and time varying channel parameters [12]. Amongst these, blind adaptive multiuser detectors have the advantage that they eliminate the need for a training sequence in adaptation mode, which translates to savings in bandwidth [13], [14], [15]. Although not many wireless standards today use blind algorithms, it shall definitely be an advantage, in terms of lesser bandwidth consumption, if we can develop blind multiuser detection algorithms which have a performance comparable to training sequence based algorithms. A space-time multiuser detector exploits the signal structure in both time domain and spatial domain for interference cancellation. Two adaptive space-time multiuser detectors based on the criterion of minimum probability of error (MPOE) are proposed in this paper, for multipath asynchronous DS-CDMA channels, with multiple antennas at the receiver. The first algorithm, minimum joint probability of error (MJPOE) minimizes the joint probability of error for all users. The algorithm is blind, i.e., no training sequence is required in adaptation mode. MJPOE offers a BER performance better than the non-adaptive MMSE detector [16], [?]. It, however, has a computational complexity which is exponential in the number of users. A scheme based on the Gram-Schmidt orthogonalization procedure is proposed to reduce the computational burden of MJPOE. In order to further reduce the computational complexity we propose the minimum conditional probability of error (MCPOE) algorithm, which minimizes the probability of error conditioned on the transmitted bit vector, for each user individually [19], [20]. It has a BER performance that is slightly inferior to the MMSE detector, however at a
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convergence rate comparable to MJPOE and a computational complexity linear in the number of users. Both MJPOE and MCPOE have a convergence rate much faster than adaptive implementations of MMSE, viz. LMS and RLS. In fact, MCPOE offers a convergence rate faster than RLS with computational complexity comparable to LMS. We also propose a blind MCPOE algorithm, whose performance is comparable to the training based MCPOE algorithm. The rest of this paper is organized as follows. Section II provides a mathematical description of an asynchronous multipath DS-CDMA channel with multiple antennas at the receiver. An expression for the received signal vector in terms of the system parameters is obtained. In Section III, the receiver structure is explained, probability distribution for the decision statistic vector is derived, and expressions for conditional and joint probability of error are also derived. The adaptive algorithms MJPOE and MCPOE are presented in Section IV. Adaptation equations for filter weights based on gradient descent are derived in this section. Section V presents simulation results, and comparisons of MJPOE and MCPOE with other multiuser detection algorithms proposed in literature. The paper concludes in Section VI. II. S IGNAL M ODEL Consider a DS-CDMA channel with K users sharing the same bandwidth. A schematic of the channel model described in this section is depicted in Fig. 1. The signaling interval of each user is T seconds, and the input alphabet is antipodal binary: {−1, 1}. During the ith signaling interval, the input bit vector 1 b(i) = [b1 (i), . . . , bK (i)]T , where bk (i) denotes the ith input symbol of the k th user and (·)T denotes the transpose. User k is assigned a spreading waveform ck which is supported on [0, T ] and is normalized to 1. Let sk = [sk1 , . . . , skN ]T (N × 1 vector) denote the corresponding spreading chip sequence, so that the spreading waveform can be expressed as N 1 X skn rect(t − (n − 1)Tc ) , ck (t) = √ N n=1 1
All vectors and matrices are displayed in boldface
(1)
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where rect(t) is a rectangular waveform with unit amplitude in [0, Tc ], and skn ∈ {−1, +1} (1 ≤ k ≤ K, 1 ≤ n ≤ N). The transmission amplitude of the k th user is denoted by Ak , which is real. The processing gain N is defined as the number of chips in a signaling period. The chip period Tc is thus equal to T /N. The baseband signal of the k th user in the ith bit interval can now be expressed as xk (t) = Ak bk (i)ck (t − iT ),
iT ≤ t < (i + 1)T .
(2)
At the receiver an array of P elements is employed. Assuming that each transmitter is equipped with a single antenna, the baseband signal between the k th user’s transmitter and the base station receiver can be modeled as a single input multiple output (SIMO) channel with the impulse response hk (t) =
M X
m=1
akm gkm δ(t − τkm ) ,
(3)
where M is number of multipaths in each user’s channel, gkm and τkm are respectively the complex gain and delay of the mth multipath of the k th user’s signal, and akm = [akm,1 , . . . , akm,P ]T is the array response vector corresponding to the mth path of the k th user’s signal. The total received signal r(t) at the receiver is a superposition of the signal from the K users plus the additive noise given by r(t) =
K XX
xk (t) ? hk (t) + σn(t)
i
=
k=1 K XX i
k=1
Ak bk (i)
M X
m=1
(4) akm gkm ck (t − iT − τkm ) + σn(t) ,
where ? denotes the convolution operation, n(t) = [n1 (t), . . . , nP (t)]T is a P × 1 vector of independent zero-mean complex Gaussian noise processes, each with unit variance, and σ 2 is the variance of the ambient noise at each antenna element. Let r p (i) denote the received signal vector at the pth antenna in the ith signaling interval, obtained by sampling rp (t) at chip rate. Since we have considered an asynchronous channel, r p (i) contains contributions from bits transmitted in the (i − 1)th interval as well as the ith interval. Let us assume that the maximum multipath delay of any user does not exceed the symbol period,
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so that contributions from (i − 2)th and all preceding signaling intervals are zero. Also, let us assume that multipath delays are resolvable upto the accuracy of a chip period, i.e, τkl is an integral multiple of Tc (∀k, l). Now define the following: (n)
skL = [skN −n+1, . . . , skN , 0, . . . , 0]T | {z } | {z }
(5)
N −n
n
(n)
skR = [0, . . . , 0 , sk1 , . . . , skN −n ]T | {z } {z } |
(6)
N −n
n
Now, the received signal vector r p (i) can be expressed as r p (i) =
K X k=1
Ak bk (i − 1)
M X
(n ) akm,p gkm skLkm
+
m=1
K X
Ak bk (i)
M X
(n
)
akm,p gkm skRkm + σnp (i) , (7)
m=1
k=1
where nkm = τkm /Tc and np (i) denotes the sampled noise vector, obtained by chip rate sampling of receiver noise in the ith bit interval. Since the multipath delays are assumed to be multiples of the chip period, nkm is an integer. In (7), the first term represents contribution from the previous bit interval or inter-symbol interference (ISI), the second term represents contribution of the bits transmitted in the ith bit interval, whereas the last term represents AWGN. For ease of representation, define the following: αkp,L =
M X
(n
akm,p gkm skLkm
)
(8)
m=1
αkp,R =
M X
(n
)
akm,p gkm skRkm .
(9)
m=1
Thus, r p (i) can be expressed as r p (i) =
K X k=1
Ak bk (i − 1)αkp,L +
K X
Ak bk (i)αkp,R + σnp (i) .
(10)
k=1
This can further be expressed using matrix notation as rp (i) = S pL Ab(i − 1) + S pR Ab(i) + σnp (i) ,
(11)
where S pL = [α1p,L . . . αKp,L] (N × K matrix), S pR = [α1p,R . . . αKp,R] (N × K matrix), and A = diag(A1 , . . . , AK ) (K × K matrix).
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III. P ROBABILITY
OF
E RROR
A. Probability Density of the Decision Statistic Let w k (NP × 1 vector) denote the linear filter used to demodulate the bits transmitted by the k th user. w k operates on the augmented NP × 1 signal vector r(i) = [r 1 (i)T , . . . , rP (i)T ]T . Let w k be represented as wk = [w Tk1 , . . . , wTkP ]T , where each w kp (1 ≤ p ≤ P ) is an N × 1 vector. The soft output of the filter yk (i) in the ith bit interval is given by yk (i) = w H k r(i) = =
P X
p=1 P X
wH kp r p (i)
(12)
wH kp S pL Ab(i
p=1
− 1) +
P X
H wH kp S pR Ab(i) + σw k n(i) ,
p=1
where (·)H denotes the Hermitian and n(i) = [n1 (i)T , . . . , nP (i)T ]T . The bit decision for the k th user is given by bbk (i) = sgn [
