A Spatial-Temporal Decorrelating Receiver for CDMA systems with ...
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A Spatial-Temporal Decorrelating Receiver for CDMA systems with Base-Station Antenna Arrays Ruifeng Wang, Student Member, IEEE and Steven D. Blosteiny, Senior Member, IEEE
Abstract We investigate multi-user signal detection with a base-station antenna array for a synchronous CDMA uplink using non-orthogonal codes in Rayleigh fading channels. We have developed a new formulation for a spatial-temporal decorrelating detector using the maximum-likelihood criteria. The detector is shown to be near-far resistant. We propose to implement the spatial-temporal decorrelating receiver iteratively by applying the space alternating generalized expectation-maximization (SAGE) algorithm. Simulation results show that the SAGE-based decorrelating receiver signi cantly outperforms the conventional single-user receiver and with performance close to that of a spatial-temporal decorrelating receiver with known channel parameters. We have observed that adding base-station antennas can actually improve convergence of the proposed iterative receiver.
I. Introduction
Code division multiple access (CDMA) systems use spreading codes to distinguish di erent mobile users. Because of the relative time delays among the active mobile users, CDMA systems su er from co-channel interference which results in the near-far problem [1]. However, the near-far problem is not inherent to CDMA systems, but due to the conventional singleuser receiver which models the interference from other users as noise. By jointly detecting all the users' signals, optimum multi-user signal detection for CDMA systems is near-far resistant and can achieve signi cant performance improvement over that of conventional single-user detection [2]. Because of the computational complexity of optimum multi-user detection, several suboptimum multi-user signal detectors have been proposed [1] [3] [4] [5] The authors are with the Department of Electrical and Computer Engineering, Queen's University, Kingston, Ontario, K7L 3N6 Canada. y denotes corresponding author. E-mail: [email protected]. This research has been supported by the Canadian Institute for Telecommunications Research under the NCE program of the Government of Canada. October 6, 1998
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for additive white Gaussian noise (AWGN) channels. In [6], multi-user signal detection is extended to fading channels. Since multi-user signal detection is near-far resistant, precise power control is not needed. Multi-access interference can also be reduced using array signal processing. By digital beamforming, a base-station antenna array can be used to improve CDMA communication system capacity and coverage (see [7] and references herein). Combined beamformer-RAKE single-user receivers have been proposed for multipath channels in [8] and [9], respectively. In [10], adaptive antenna array processing and interference cancellation approaches using the least mean squared (LMS) algorithm are analyzed and the convergence is found to be very slow, requiring several hundred training bits. The problem of integrating antenna array beamforming and multi-user signal detection is proposed for AWGN channels in [11], but channel estimation has not been addressed. In order to detect information symbols reliably, we have to estimate channel and antenna array response vectors. Feder and Weinstein apply the expectation-maximization (EM) algorithm to parameter estimation of superimposed signals [12]. Bit sequence detection with joint random parameter estimation using the EM algorithm is studied for single-user systems in [13]. Recently, applications of the EM algorithm to CDMA systems have been proposed for signal detection [14], channel estimation [15] and joint channel estimation and signal detection [16]. The space-alternating generalized EM (SAGE) algorithm has been developed to accelerate the convergence of the EM algorithm [17]. Applications of the SAGE algorithm in multi-user CDMA channels can be found in [14] for single antenna and known channels, in [18] for antenna array channel parameter estimation and in [19] for joint parameter estimation and signal detection based on the discrete wavelet transform for a single antenna system. In this paper, we investigate the integration of spatial signal processing with multi-user signal detection for the synchronous CDMA systems with non-orthogonal spreading codes over a single dominant-path Rayleigh fading channel. The motivation to focus on synchronous DRAFT
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systems is as follows: a K -user asynchronous system can be modelled as a synchronous system with K N users [1], where N is the number of bits in each transmitted block. Most importantly, the synchronous problem formulation simpli es the derivation and analysis of the new algorithm and the results obtained can be generalized to the case of higher complexity asynchronous multipath systems [20]. We rst formulate a spatial-temporal decorrelator which we show to be near-far resistant. Then, we apply the SAGE algorithm to jointly estimate channel array response vectors and detect information symbol sequences. The bit error probability and Cramer-Rao Lower Bound (CRLB) for the estimated channel are then derived. This paper is organized as follows. The discrete-time system model is developed in Section II. In Section III, we derive a spatial-temporal decorrelator using the maximum likelihood criteria and analyze its asymptotic eciency. An iterative spatial-temporal decorrelating receiver is proposed in Section IV by applying the SAGE algorithm. The bit error probability and CRLB are derived in Section V. Section VI presents simulation results. II. System Model
We consider a synchronous frequency non-selective Rayleigh fading channel for an uplink CDMA communication system with a base-station antenna array. There are K active users in the system. An N-bit transmitted signal from the kth user is
sk (t) = Ak
N
X
i=1
bk (i)ck (t ? iTb)
(1)
where Ak is the amplitude of the kth user, bk (i) 2 f?1; 1g is the ith transmitted bit of the kth user with equal probability and ck (t) represents the spreading waveform of the kth user, which is given by L?1 ck (t) = ck (l)p(t ? lTc) (2) X
l=0
where ck (l) 2 f?1; +1g (l = 1 L ? 1) is the spreading code, p(t) is the chip pulse, Tc is the chip interval, Tb is the bit interval and processing gain L is de ned as L = Tb=Tc. October 6, 1998
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We assume that the information bits from K users are independent, the spreading code sequences for K users are independent and the spreading waveform has normalized energy, i.e., Tb jck (t)j2 = 1. The transmitted signal passes through a fading channel and arrives at a base-station antenna array with M elements. The impulse response of the channel from transmitter to antenna array output can be modelled as R
0
t) = k (t)a(k (t))(t)
gk (
(3)
where k (t) represents channel attenuation for user k and a(k (t)) is the M-dimensional array response vector with direction-of-arrival (DOA) k (t) from the kth user. The channel fading attenuations for K users are assumed to be mutually independent and also independent of information bit symbols. The received composite signal at the base-station antenna array from K users is given by
t) =
x(
K
X
k=1
t) =
xk (
K
X
k=1
sk (t) gk (t) + n(t)
(4)
where n(t) is the AWGN vector with zero mean and covariance matrix 2IM , where IM is an M M identity matrix. From here on, we assume that k (t) and k (t) remain unchanged over the N-bit duration. We denote these quantities as k and k , respectively. From (1),
sk (t) gk (t) = Ak k a(k )
N
X
i=1
bk (i)ck (t ? iTb)
(5)
We denote the channel impulse response vector for user k as fk
= Ak k a(k )
(6)
where the mth component of fk is fkm = Ak k am(k ) and am(k ) is the mth component of array response vector a(k ). The received signal at the mth array element (for m = 1; ; M ) is given by K N (7) xm(t) = fkm bk (i)ck (t ? iTb) + nm(t) DRAFT
X
X
k=1
i=1
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where nm (t) is AWGN at the mth array element. The received signal at each element rst passes through a lter matched to the chip waveform, and is then sampled at the chip rate. The received discrete-time signal at the ith bit interval from the mth element can be obtained for sample g 2 f0; 1; ; L ? 1g as
xm(i; g) =
Z (g +1)T c
t=gTc
xm(t)p(t)dt
(8)
Finally, the gth sample of the ith bit for the mth antenna element is obtained in terms of the sampled chips ck (g) by substituting (7) into (8) yielding
xm(i; g) =
K
X
k=1
fkm ck (g)bk (i)=L + nm(i; g)
(9)
where nm (i; g) = t ggTc Tc nm(t)p(t)dt is Gaussian distributed with zero mean and variance 2=L. We denote the code vector for the kth user as R ( +1) =
ck
= [ck (0)=L ck (1)=L ck (L ? 1)=L]T
(10)
The matched lter output at the mth element for m 2 f1; ; M g can be written in vector form as K (11) fkmbk (i)ck + nm (i) xm (i) = X
k=1 nm(i; L ? 1)]T .
where We de ne the overall system impulse response vector for user k, including the fading channel, array response vector and spreading code vector de ned above as nm (i) = [nm(i; 0)
nm(i; 1)
i) = fkm ck
hm k(
(12)
Equation (11) can be written in terms of (12) as xm (
i) =
K
X
hm k k(
k=1 (xM (i))T ]T , hk
b i) + nm(i)
(13)
Denote x(i) = [(x1(i))T = [(h1k )T (hMk )T ]T and n(i) = [(n1(i))T (nM (i))T ]T . The received discrete-time signal from the antenna array is given by
i) =
x(
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K
X
k=1
i) =
xk (
K
X
k=1
b i) + n(i)
hk k (
(14) DRAFT
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where xk (i) is the received signal from the kth user and n(i) is AWGN vector with zero mean and covariance matrix L IML, where IML is an ML ML identity matrix. Vector hk models the spatial and temporal channel characteristics of our system. The spatial-temporal channel vector hk can also be decomposed as 2
2
hk
=
6 6 6 6 6 6 6 6 6 6 4
ck
0
0
ck
32
0
... 0 ... . . . . . . ...
76 76 76 76 76 76 76 76 76 76 54
fk1 fk2 ...
3 7 7 7 7 7 7 7 7 7 7 5
= C k fk
(15)
fkM where Ck is an ML M spreading code sequence matrix of the kth user. Denote matrix H = [h1 ... ... hK ] and vector b(i) = [b1(i) bK (i)]T , (14) can be expressed as x(i) = H b(i) + n(i) (16) 0
0
ck
A necessary condition that K users are identi able is that H be of full column rank, i.e., ML > K . III. Spatial-Temporal Decorrelator
The log-likelihood of the received signal x(i) conditioned on the bit vector b(i) and the spatial-temporal channel matrix H is given by (for the sake of simplicity, we omit the index i in this section)
(b; H ) = ? 21 (x ? H b)H (x ? H b) (17) =L
where the superscript H denotes conjugate transpose. The unknown parameters are the information bit vector b and channel and array response matrix H. If H is available, the bit vector decision variable b^ can be obtained by maximizing (17) ^d b
= arg max
(b; H ) b
(18)
Equating the derivative of (17) with respect to b to zero, we obtain ^ b DRAFT
= signf[H H H ]?1H H xg
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where signfag = 1 if a 0 or -1 if a < 0. We de ne spatial-temporal cross-correlation matrix as H
R = H H = C1f1 C2f2 CK fK H
C1f1 C2f2 CK fK
(20)
Denoting
fij = fiH fj and
ij = cHi cj
8 > >
> :
2
3
6 6 6 6 6 6 6 6 6 6 4
7 7 7 7 7 7 7 7 7 7 5
2
z
=H
H
x
=
6 6 6 6 6 6 6 6 6 6 4
fH
1
0T
0T
f2H
0T
0T
. . . 0T ... . . . . . . ...
3
32
0T
H fK
76 76 76 76 76 76 76 76 76 76 54
y1 y2
...
yK
7 7 7 7 7 7 7 7 7 7 5
= F Hy
(24)
using (15), yk = CkH x for k = 1; 2; ; K . Thus, yk actually represents the despread output for the kth user's decision variable. In (24), the despread outputs for all K users are beamformed by a maximum SNR criterion. This detector structure is a conventional October 6, 1998
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single-user detector with maximum SNR beamforming, which we refer to as the conventional single-user detector. Using (19), (20) and (24), the detected bit vector can be written in the form ^ = signfR?1 F H yg b (25) Therefore, the nal data decision vector is obtained by decorrelating the conventional singleuser detector output. Eq. (25) can also be viewed as a maximum SINR beamforming. Matrix R includes both temporal cross-correlation due to non-orthogonal spreading codes and instantaneous spatial correlation because of the spatial distribution of the active users in the system. The detector structure is illustrated in Figure 1. This structure di ers from that of [11] since the matched- lter (despreader) is at the front end. Substituting (16) into (19) and using de nition of R (c.f. (20)), the spatial-temporal decorrelator output can be simpli ed to ^ b
= signfb + R?1 ndg = signfb + wg
(26)
where w is the spatial-temporal decorrelator output AWGN vector with zero mean and variance L R?1 . If there is no base-station antenna array, the spatial-temporal detector reduces to the classic decorrelating detector proposed in [1]. 2
A. Near-far Resistance
The asymptotic eciency is a performance measure for multi-user signal detection in the limit as the background noise goes to zero. The kth user's asymptotic eciency is de ned as [2] prw e k ( ) (27) Pk ()=Q( k ) < +1g k = lim ! ! wk = supf0 r 1 : lim 0
0
where ek() is the kth user's e ective energy required to achieve the same error probability in the absence of interferers, wk is the received energy for user k, Pk () represents bit-error-rate (BER) of the kth user when the variance of AWGN is 2 and Q(x) = x1 exp(p?2y =2) dy. R
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Because the asymptotic eciency of the temporal decorrelator in a single-path Rayleigh fading channel is the same as that of an AWGN channel [6], we only consider AWGN channel here. In this case, fk = Akak (k ). The kth user error probability for the spatial-temporal decorrelator can be obtained as
Pkd = Q(
p L
1 ) (R?1 )kk
(28)
q
where (R?1 )kk represents the kth diagonal element of matrix R?1 [3]. Thus, the asymptotic eciency of the spatial-temporal decorrelator for the kth user is given by
kd = max2f0;
1 1 g = Rkk (R?1 )kk Rkk (R?1 )kk
(29)
q
By formulating the bit error probability for the conventional signal-user detector [21], the kth user's asymptotic eciency of the conventional single-user detector can be obtained
c k
= max2f0;
pR ? jR j=pR kk kk pi6 Rk ik g = max2f0; 1 ? P
=
kk
X
6
i=k
jRik j g Rkk
(30)
The kth user's near-far resistance is de ned as its worst case asymptotic eciency over all possible energies of the interferers and given by k = inf wi 0;i6=k k [1]. Example 1: To numerically compare the asymptotic eciency for the spatial-temporal decorrelator and the conventional single-user detector, we consider a four-user system based on a set of spreading codes from Gold sequences of length seven. The corresponding crosscorrelation matrix of the spreading codes is taken from [5]. We assume a uniform linear array (ULA) with half-wavelength spacing at the base-station. The direction-of-arrivals (DOA) for four active users with known array responses are [25; 5; ?15; ?35] with respect the array boresight. We consider two cases: a single antenna and a three-element antenna array. We refer the rst user as the desired user and de ne the power ratio as A2k =A21, for k = 2; ; K . In this example, we assume that all interferers have equal transmitted power. Figure 2 shows that performance of the spatial-temporal decorrelator is independent of the received energies from interferers and is near-far resistant. Also note that the asymptotic eciencies October 6, 1998
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for both conventional single-user detectors tends to zero as the interferers become stronger: the conventional single-user detector is not near-far resistant even when an antenna array is used. From Figure 2, it can be observed that using an antenna array can potentially improve the asymptotic eciency for either the spatial-temporal decorrelator and the conventional single-user detector. The above analysis is based on the assumption that matrix R is available, implying that channel array response vectors, fk (for k = 1; 2; ; K ), are known at the receiver. This is not the case in a practical application. We now investigate the problem of joint channel estimation and signal detection. IV. Decorrelating Receiver via SAGE Algorithm
In this section, we address the key implementation issue of the proposed spatial-temporal decorrelating receiver, namely the joint parameter estimation and spatial-temporal signal decorrelation. We derive a sequential decision-feedback receiver structure based on the space alternating generalized EM (SAGE) algorithm. The available observed data at the receiver is a data vector set fx(i); 1; ; N g. The SAGE algorithm can achieve improved performance, such as faster convergence, over the standard EM algorithm [17]. The key idea for the SAGE algorithm is to choose a less informative hidden-data space to improve convergence rate. Starting with an initial parameter vector estimate at iteration j = 0, iterate the following: 1. Choose new index set S = S j , a subset of the parameters, which de nes an admissible hidden-data space xSj ; 2. Compute the following conditional expectation of xSj given observed data x and the previous estimate of parameter vector ^j
(Sj j^j ) = E [ln f (xSj jxSj ; ^Sj )jx; ^j ] +1
(31)
3. Obtain the next estimate by maximizing over the chosen subset while keeping the other DRAFT
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parameters xed
^Sj+1 = arg maxSj (Sj j^j ) j ^Sj+1 = ^Sjj j where the index set Sj is the complement of S j . 4. Increment j and go to step 1. +1
+1
(32)
+1
+1
+1
A. Application to CDMA
For the CDMA system model considered, we would like to estimate each user's channel parameter and detect the information symbol individually. In step 1, we choose user index k as the index set. Thus, the admissible hidden-data space for index k, for k = 1; ; K and i = 1; ; N , is given by (14) 2 i) N(hk bk (i); L IML)
xSk (
(33)
The log-likelihood function after removing the terms independent of hk and bk (i) is straightforward N (34)
(xSk (1); ; xSk (N )) = ? 21=L (xSk ? hk bk )H (xSk ? hk bk ) X
i=1
Given the estimation results at the jth iteration, the conditional expectation of xSk (i) (for k = 1; ; K and i = 1; ; N ) in step 2 is given by
i) = h^ jk^bjk (i) + (x(i) ?
^ Sk ( x
K X
i=1
^ i ^i ( h j bj
i)) = x(i) ?
X
i6=k
^ i ^i ( h j bj
i)
(35)
Finally, substituting (35) into (34), we obtain the conditional expectation of the log-likelihood function of xSk (i). The maximization results at the next iteration, (32) in step 3, is given by N
f? (^xSk (i) ? hk bk (i))H (^xSk (i) ? hk bk (i))g [h^ jk+1; ^bjk+1(i)] = arg hmax ;b (i) X
k k
i=1
(36)
Thus, SAGE-based receiver for joint array response vector estimation and information symbol detection is obtained as for j = 1; 2 October 6, 1998
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k = (j modulo K ) E-step: compute the conditional expectation of hidden-data for k = 1; ; K and i = 1; ; N ^ Sk x
j
(i) = x(i) ?
K
X
i=1;6=k
^ i ^i ( h j bj
i)
(37)
M-step: obtain the maximum-likelihood estimates h^ k for k = 1; ; K and ^bk (i) for k = 1; ; K and i = 1; ; N N ^hjk+1 = 1 x^ Sk j (i)^bjk (i) N i=1 X
(38)
^bjk+1(i) = signf(h^ jk+1 )H x^ Sk j (i)g ^ jk+1 k0 6= k h 0 = h^ jk0 ; ^bjk+1 0 (i) = ^bjk0 (i); k 0 6= k
(39)
The above steps can be interpreted as follows: substituting (37) into (39), we obtain ^bjk+1(i) = signf((h^ jk+1 )H x^ k (i) ?
K
X
k0 =1;k0 6=k
f^kkj+10 kk0 ^bji (i))g
(40)
where f^kkj+10 = (^fkj+1 )H ^fkj0 is the estimated instantaneous spatial correlation de ned in (21) at the (j+1)st iteration and kk0 is the cross-correlation de ned in (22). Equation (40) implies that information symbol detection at each iteration involves explicit interference cancellation given the current channel estimation and previous bit detection results. Therefore, the SAGE-based receiver has a sequential interference cancellation structure. B. Convergence
From (17), it can be easily veri ed that the log-likelihood function is continuous and differentiable with respect to the unknown parameters. Since the likelihood function increases monotonically with each iteration, it is bounded above and the proposed receiver converges to a xed stationary point or local/global maxima depending on the initial guess of the unknown parameters [22]. The convergence rate of the SAGE-based receiver can be shown to DRAFT
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be the largest eigenvalue of [FxSk ? Fx](FxSk )?1 [23], where FxSk and Fx are Fisher information matrices of xSk and x, respectively. To investigate the e ect of the antenna array on receiver convergence, for simplicity, we assume that the channel array response vectors are given. From (17) and (35), Fx = 4L R and FxSk = 4L Rkk IK . 2
2
Example 2: We again consider a system with a uniform linear array at the base-station over a AWGN channel. The inter-element spacing of antenna array is half-wavelength and the DOAs of the active users are uniformly distributed in [?60; 60]. Gold sequences of length 31 from [24] are assigned to mobile users. All active users have equal transmitted power. The convergence rate curves are plotted in terms of number of users and number of antennas and averaged over 5,000 trials. From Figure 3, it can be observed that the convergence becomes slower as the number of active users increases in the system. An interesting remark is that using a base-station antenna array can accelerate the convergence of the proposed receiver. V. Receiver Performance
In this section, we assess the performance of the proposed spatial-temporal decorrelating receiver. Since we propose to jointly estimate the channel array response vectors and detect the transmitted information symbols, we determine the bit-error rate (BER) for symbol detection and the Cramer-Rao lower bound (CRLB) for channel estimation. A. Bit Error Rate (BER)
From (26), it is observed that the bit decision vector is an unbiased estimator containing a zero-mean noise vector w whose covariance matrix is L R?1. Without loss of generality, we assume that the rst user is the desired user. 2
Claim: The bit error probability for the rst user is obtained as
P1 = 12 (1 ? 1 + 1 ) 1 s
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where 1 is the average signal-to-noise ratio (SNR) for user 1 and given by
1 = and
1
(42)
2 E [R?1 ] 11 L
E [R?1]11 = Var[1 ]
22 2K ... . . . ... K2 KK
(43)
11 1K ... . . . ... K1 KK where E [:] denotes the expectation over Rayleigh random variables contained in R?1 , Var[.] represents variance, j:j is the determinant and ij = AiAj aHi aj ij . Proof: The (1,1) entry of R?1 is 1
R22 R2K [R?1]11 = jR1 j ... . . . ... RK2 RKK
Since the ijth entry of R is given by Rij = i j ij , using Hermitian symmetry, it can be easily obtained that
and
R22 R2K 22 2K ... . . . ... = j 2j2 j K j2 ... . . . ... RK2 RKK K2 KK
(44)
11 1K jRj = j 1j2 j K j2 ... . . . ... (45) K1 KK where j k j2 = k k , from which (43) follows. For Rayleigh fading channel and BPSK modulation, (41) is obtained using the results in [25]. DRAFT
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B. Cramer-Rao Lower Bound (CRLB)
To measure channel array response vector estimation performance, we derive the CramerRao lower bound. We collect the unknown parameters into the column vector a
= [aT1
aTK ]T
(46)
where ak = k a(k ) is the channel array response vector for the kth user. For an M-element antenna array, we de ne the M M matrix for the ith bit of the kth user, k = 1; ; K , as Bk (i) = diag[bk(i); ; bk (i)]. Proposition: The CRLB matrix for parameter vector a is given by 1 N G(i)]?1 CRLB(a) = [ (47) L i=1 where p 1 212B12(i) p 1 K 1K B1K (i)
111IM ... p 1 212B12(i) ...
222IM (48) G(i) = ... ... ... ... p 1 K 1K B1K (i)
K KK IM where Bkj (i) = BkH (i)Bj (i) = IM if bk (i) = bj (i) or ?IM if bk (i) = ?bj (i) (for i = 1; ; N and k; j = 1; ; K ), k = A2k =2 is the signal-to-noise ratio (SNR) corresponding to the kth user and ij is the spreading code cross-correlation de ned in (22). Outline of Proof: First, we de ne the parameter vector as = [ 2 aT a T ]T , where 2 is background noise covariance, a and a are the real and imaginary parts of a, respectively. Discarding the terms independent of , the log-likelihood function is given by N (49) ln = ?MLN ln 2 ? L2 [x(i) ? H b(i)]H [x(i) ? H b(i)]) i=1 We compute the derivative of (49) with respect to . Using the de nition of the Fisher ln )( @ ln )T ] and following the technique in [26], we obtain information matrix I( ) = E [( @ @ @ X
2
3
6 6 6 6 6 6 6 6 6 6 4
7 7 7 7 7 7 7 7 7 7 5
X
I( a) =
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E [( @ @lna )( @ @lna )T ] = 2L2
N
X
i=1
Re[B H (i)DH DB (i)] DRAFT
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@ ln
2 L H (i)DH DB (i)] T I( a) = E [( Re [ B )( ) ] = 2 @ a @ a i=1 X
) I( aa
= E [( @ @lna )( @ @lna )T ] = ? 2L2
N
X
i=1
Im[B H (i)DH DB (i)]
where B (i) = diag[B1(i); ; BK (i)] and D = [D1 ... ... DK ] with ML M matrix Dk = diag[Akck ; ; Akck ] (for k = 1; ; K ). De ne G(i) = 1 B H (i)DH DB (i), (48) follows. Since G(i) (for i = 1; ; N ) is real, we have I(aa) = 0 and 2
I( a)
= I(a) = 2L
N
X
i=1
G(i)
(50)
Thus, the Fisher information matrix is given by 2
) =
I(
6 6 6 6 6 6 4
I( 2 ) 0TMK 0TMK 0MK
I( a)
0
0MK
0
I( a)
3 7 7 7 7 7 7 5
(51)
Finally, using inverse of a partitioned matrix, we can obtain the Cramer-Rao lower bound (CRLB) matrix for channel array response vectors CRLB(a) = I(a)?1 + I(a)?1. Clearly, the Fisher information matrix, Ni=1 G(i), is a function of the SNR at the ith bit as well as the cross-correlation between the spreading codes. P
VI. Simulations
In this section, we present performance results for the proposed receivers. A randomly generated Gold sequence of length 31 is assigned to each user in our CDMA system. In the simulations, we transmit a 100-bit information sequence from each user and assume the channels remain unchanged during the data-block transmission. We insert one training bit in the rst bit position in each sequence. A uniform linear array with half-wavelength spacing is used at the base-station and the DOAs are uniformly distributed in [?60; 60]. The initial guesses of the channel and array response vectors are obtained using the training bits. Using (24), we obtain the initial detected information bit vectors. The simulation results DRAFT
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are computed from 10,000-150,000 trials, depending on the signal-to-noise ratio, so that the BER is calculated to within 5% with 90% con dence. The rst example is used to compare performance of the SAGE-based receiver, EM-based receiver [16] and spatial-temporal decorrelating receiver with perfect knowledge of channel and array response vectors. There are ve active users in the system and either a single antenna or a three-element base-station antenna array. For simplicity of analysis, all users have equal transmitted power. Both the SAGE-based and EM-based decorrelating receivers use eight iterations. From Figure 4, we can observe that both iterative receivers signi cantly outperform the conventional single-user receiver. When a single antenna is used, the iterative receivers do not converge to the optimal spatial-temporal decorrelating receiver. However, the performance of the SAGE-based receiver is greatly improved by using an antenna array. This is mainly due to the antenna array's improved channel estimates, as shown in Figure 5. When the SNR is larger than 20dB, the channel estimates of the SAGE-based receiver do not reach the CRLB. However, the loss in channel estimation performance at such a high SNR does not have strong e ect on symbol detection. At a BER of 10?2 , the three-element antenna array can achieve about 6 dB gain over a single antenna. To address the near-far resistance of the proposed receiver, we consider ve active users and a two-element base-station antenna array. We assume that the rst user is the desired user with 20 dB SNR and xed transmitted power. Again. all other interferers are assumed to have equal and varying transmitted power. At the power-ratio of 20 dB, the transmitted powers of the interferers are all 100 times stronger than that of the desired user. This represents an extreme near-far environment. Figure 6 shows that the SAGE-base decorrelating receiver converges to the spatial-temporal decorrelating receiver with perfect knowledge of channel and array response vectors and is therefore near-far resistant. Although the EMbased decorrelating receiver is an improvement over the conventional single-user receiver, neither receiver is near-far resistant as shown in Figure 6. From Figure 7, it is observed October 6, 1998
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that the SAGE-based receiver has improved near-far resistance than that of the EM-based receiver. The nal example is used to illustrate the convergence of the proposed iterative receiver. Five users with equal transmitted powers are considered in the system and there is a threeelement base-station antenna-array. Figure 8 shows convergence of the SAGE-based decorrelating receiver. Because the SAGE-based receiver converges very quickly in the low SNR region, for a required BER, three or four iterations can achieve acceptable performance. VII. Conclusions
We have derived a spatial-temporal decorrelator based on the discrete-time signal model using maximum likelihood criteria for CDMA uplink Rayleigh fading channel with a basestation antenna array. The spatial-temporal decorelator is near-far resistant. Numerical results show that the incorporation of the base-station antenna array results in signi cant performance improvement. An iterative receiver structure is obtained by applying the SAGE algorithm to implement the spatial-temporal decorrelating receiver. It is shown that using base-station antenna array can accelerate convergence rate of the proposed receiver. The bit error probability and CRLB are derived for the proposed receiver. Simulation results show that the SAGE-based decorrelating receiver can achieve signi cant performance gain over the conventional single-user receiver. It is observed that the SAGE-based receiver outperforms an EM-based receiver and converges to the BER of the spatial-temporal decorrelating receiver. The channel estimates of the SAGE-based receiver are also close to the CRLB. The simulated BER results for the spatial-temporal decorrelating receiver agree closely with analytical results. Extension of the present results to asynchronous and multipath fading systems is under investigation. References [1] R. Lupas, S. Verdu, \Near-far resistance of multiuser detectors in asynchronous channels", IEEE Trans. Commun., vol. 38, no. 4, pp. 496-508, Apr. 1990. DRAFT
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[2] S. Verdu, \Minimum probability of error for asynchronous Gaussian multiple-access channels", IEEE Trans. Inform. Theory, Vol. 32, N0. 1, pp.85-96, Jan. 1986. [3] Z. Xie, R. T. Short and C. K. Rushforth, \A family of suboptimum detectors for coherent multiuser communications", IEEE J. Select. Areas Commun., vol. 8, no. 4, pp. 683{690, May 1990. [4] M. K. Varanasi and B. Aazhang, \Multistage detection in asynchronous code-division multiple-access communications", IEEE Trans. Commun., vol. 38, no. 4, pp. 509{519, Apr. 1990. [5] A. Duel-Hallen, \Decorrelating decision-feedback multiuser detector for synchronous code-division multiple-access channel", IEEE Trans. Commun., vol. 41, no. 2, pp. 285{290, Feb. 1993. [6] Z. Zvonar and D. Brady, \Multiuser detection in single-path fading channels", IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1729{1739, Feb./Mar./Apr. 1994. [7] A. J. Paulraj and C. B. Papadias, \Space-time processing for wireless communications", IEEE Signal Processing Magazine, pp. 49{83, Nov. 1997. [8] A. F. Naguib and A. Paulraj, \Performance of wireless CDMA with M-ary orthogonal modulation and cell site antenna arrays", IEEE J. Select. Areas Commun., vol. 14, no. 9, pp. 1770{1783 , May 1996. [9] H. Liu and M. D. Zoltowski, \Blind equalization in antenna array CDMA systems", IEEE Trans. Signal Processing, vol. 45, no. 1, pp. 161{172, Jan. 1997. [10] R. Kohno, H. Imai, M. Hatori and S. Pasupathy, \Combination of an adaptive array antenna and a canceller of interference for direct-sequence spread-spectrum multiple-access system", IEEE J. Select. Areas Commun., vol. 8, no. 4, pp. 675{681, May 1990. [11] S. Y. Miller and S. C. Schwartz, \Integrated spatial-temporal detectors for asynchronous Gaussian multipleaccess channels", IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 396{411, Feb./Mar./Apr. 1995. [12] M. Feder and E. Weinstein, \Parameter estimation of superimposed signals using the EM algorithm", IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, no. 4, pp. 477{489, Apr. 1988. [13] C. N. Georghiades and J. C. Han, \Sequence Estimation in the Presence of Random Parameters Via the EM Algorithm", IEEE Trans. on Commun., vol. 45, no. 3, pp. 300-308, Mar. 1997. [14] L. B. Nelson and H. V. Poor, \Iterative multiuser receivers for CDMA channels: an EM-Based approach", IEEE Trans. Commun., vol. 44, no. 12, pp. 1700{1710, Dec. 1996. [15] U. Fawer and B. Aazhang, \A multiuser receiver for code division multiple access communications over multipath channels", IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. 1556-1565, Feb./Mar./Apr. 1995. [16] R. Wang and S. D. Blostein, \Maximum likelihood multi-user CDMA receiver for base-station antenna arrays using EM algorithm", Proc. 19th Biennial Symp. Commun., pp 110{114, Kingston, ON, June 1998. [17] J. A. Fessler and A. O. Hero, \Space-Alternating Generalized EM algorithm", IEEE Trans. Signal Processing, vol. 42, no. 10, pp. 2664{2677, Oct. 1994. [18] D. Dahlhaus, A. Jarosch, B. H. Fleury and R. Heddergott, \Joint demodulation in DS/CDMA systems exploiting the space and time diversity of the mobile radio channel", Proc. IEEE Int. Symp. on Personal, Indoor and Mobile Radio Communications, pp. 47{52, Helsinki, Finland, 1997. October 6, 1998
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[19] I. Sharfer and A. O. Hero, \Optimum multiuser CDMA detector using grouped coordinate ascent and the DWT", Proc. IEEE Workshop in Signal Processing Advances in Wireless Communications, Paris, Apr. 1997. [20] A. Duel-Hallen, J. Holtzman and Z. Zvonar, \Multiuser detection for CDMA systems", IEEE Personal Communications, pp 46{58, Apr. 1995. [21] R. Lupas, S. Verdu, \Linear multiuser detectors for synchronous code-division multiple-access channels", IEEE Trans. Inform. Theory, vol. 35, no. 1, pp. 123{136, Jan. 1989. [22] C. F. J. Wu, \On the convergence properties of the EM algorithm", Annals of Statistics, vol. 11, no. 1, pp. 95{103, Jan. 1983. [23] A. O. Hero and J. A. Fessler, \Convergence in norm for alternating expectation-maximization (EM) type algorithms", Statistica Sinica, vol. 5, no. 1, pp. 41{54, Jan. 1995. [24] A. M. Monk, M. Davis and L. B. Milstein, \A noise-whitening approach to multiple access noise rejection - part I: theory and background", IEEE J. Select. Areas Commun., vol. 12, no. 5, pp. 817, June 1994. [25] J. G. Proakis, Digital Communications, 3rd Ed., McGraw-Hill, Inc., 1995. [26] P. Stoica and A. Nehorai, \Music, maximum likelihood, and Cramer-Rao bound", IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, no. 5, pp. 720{741, May 1989.
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x(t) -
Matched Filter
t=gT
-
#1 Despreader ? ? ?
A AU x -
#K Despreader
y1 -
Coherent #1 Beamformer
yK -
? ? ?
Coherent #K Beamformer
z1 -
zK -
21
SpatialTemporal Decorrelator
........ ... ...
b -
Fig. 1. Spatial-Temporal Decorrelator Structure
Asymptotic Efficiency Comparison Between Spatial−Temporal Decorrelator and Coventional Detector 1
0.9
0.8
Asymptotic Efficiency
0.7
0.6
0.5
0.4
0.3 * decorrelator (single antenna) 0.2
+ decorrelator (antenna array) o conventional (single antenna)
0.1
0 −20
x conventional (antenna array)
−15
−10 −5 0 5 10 Power−Ratio (dB) (All Interferers Have Equal Transmitted Power)
15
20
Fig. 2. Asymptotic Eciencies for Single Antenna and a Three-Element Antenna Array
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. Y, MONTH 1999 Convergence Rate of SAGE−Based Receiver 0.7
0.6
0.5
Convergence Rate
Single Antenna 0.4
Three Antennas
0.3
0.2
Five Antennas
0.1
0
2
3
4
5 6 Number of Users
7
8
9
Fig. 3. Convergence Rate of SAGE-Based Receiver (A larger rate implies slower convergence. The algorithm converges faster as the number of antennas increases) Receiver Perpormance Comparison: Bit Error Probability
−1
10
Single Antenna
Antenna Array −2
Bit Error Rate of the Desired User
10
Single Antenna
Antenna Array
−3
10
−− conventional receiver * EM−based receiver o SAGE−based receiver x spatial−temporal decorrelating receiver (simulated results)
−4
10
solid line: spatial−temporal decorrelating receiver (analytical results) Five Users, Equal Transmitted Power Synchronous System, Rayleigh Fading Channel
−5
10
5
10
15 20 Signal−to−Noise Ratio (dB)
25
30
Fig. 4. Bit Error Rate for Single Antenna and a Three-Element Antenna Array DRAFT
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Receiver Performance Comparison: MSE of Channel Estimates
0
Mean Squared Error of Channel Estimates for the Desired User
10
Initial Estimates −1
10
−2
10
EM Estimates (single antenna)
EM Estimates (antenna array) −3
10
SAGE Estimates (single antenna)
SAGE Estimates (antenna array) −4
10
Five Users, Equal Transmitted Power Three−Element Antenna Array Synchronous System, Rayleigh Fading Channel
CRLB
−5
10
5
10
15 20 Signal−to−Noise Ratio (dB)
25
30
Fig. 5. Mean Squared Error of Channel Estimates for Single Antenna and a Three-Element Antenna Array Receiver Performance in Near−Far Environment: Bit Error Rate
0
10
Five Users, Two−Element Antenna Array SNR of the Desired User: 20 dB Synchronous System, Rayleigh Fading Channel
Bit Error Rate of the Desired User
solid line: spatial−temporal decorrelating receiver (analytical results) −1
10
x spatial−temporal decorrelating receiver (simulated results) −− conventional receiver * EM−based receiver o SAGE−based receiver
−2
10
−3
10 −20
−15
−10 −5 0 5 10 Power Ratio (dB) (All Interferers Have Equal Transmitted Powers)
15
20
Fig. 6. Bit Error Rate for a Two-Element Antenna Array in Near-Far Environment October 6, 1998
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. Y, MONTH 1999 Receiver Performance in Near−Far Environment: MSE of Channel Estimates
2
Mean Squared Error of Channel Estimates for the Desired User
10
Five Users, Two−Element Antenna Array
1
10
SNR of the Desired User: 20 dB Synchronous System, Rayleigh Fading Channel
0
10
Initial Estimates −1
10
EM Estimates −2
10
−3
10
SAGE Estimates CRLB
−4
10 −20
−15
−10 −5 0 5 10 Power Ratio (dB) ( All Interferers Have Equal Transmitted Power)
15
20
Fig. 7. Mean Squared Error of Channel Estimates a Two-Element Antenna Array in Near-Far Environment Convergence of SAGE−Based Receiver: Bit Error Rate
−1
Bit Error Rate of the Desired User
10
−2
10
Iteration 1
Iteration 2 Iteration 3 −3
10
Iteration 4 Iteration 5 Five Users, Equal Transmitted Power Three−element Antenna Array Synchronous System, Rayleigh Fading Channel
Iteration 6: x− Iteration 7: o− Iteration 8: solid line
−4
10
5
10
15 20 Signal−to−Noise Ratio (dB)
25
30
Fig. 8. Convergence of SAGE-Based Receiver for a Three-Element Antenna Array DRAFT
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1
A Spatial-Temporal Decorrelating Receiver for CDMA systems with Base-Station Antenna Arrays Ruifeng Wang, Student Member, IEEE and Steven D. Blosteiny, Senior Member, IEEE
Abstract We investigate multi-user signal detection with a base-station antenna array for a synchronous CDMA uplink using non-orthogonal codes in Rayleigh fading channels. We have developed a new formulation for a spatial-temporal decorrelating detector using the maximum-likelihood criteria. The detector is shown to be near-far resistant. We propose to implement the spatial-temporal decorrelating receiver iteratively by applying the space alternating generalized expectation-maximization (SAGE) algorithm. Simulation results show that the SAGE-based decorrelating receiver signi cantly outperforms the conventional single-user receiver and with performance close to that of a spatial-temporal decorrelating receiver with known channel parameters. We have observed that adding base-station antennas can actually improve convergence of the proposed iterative receiver.
I. Introduction
Code division multiple access (CDMA) systems use spreading codes to distinguish di erent mobile users. Because of the relative time delays among the active mobile users, CDMA systems su er from co-channel interference which results in the near-far problem [1]. However, the near-far problem is not inherent to CDMA systems, but due to the conventional singleuser receiver which models the interference from other users as noise. By jointly detecting all the users' signals, optimum multi-user signal detection for CDMA systems is near-far resistant and can achieve signi cant performance improvement over that of conventional single-user detection [2]. Because of the computational complexity of optimum multi-user detection, several suboptimum multi-user signal detectors have been proposed [1] [3] [4] [5] The authors are with the Department of Electrical and Computer Engineering, Queen's University, Kingston, Ontario, K7L 3N6 Canada. y denotes corresponding author. E-mail: [email protected]. This research has been supported by the Canadian Institute for Telecommunications Research under the NCE program of the Government of Canada. October 6, 1998
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for additive white Gaussian noise (AWGN) channels. In [6], multi-user signal detection is extended to fading channels. Since multi-user signal detection is near-far resistant, precise power control is not needed. Multi-access interference can also be reduced using array signal processing. By digital beamforming, a base-station antenna array can be used to improve CDMA communication system capacity and coverage (see [7] and references herein). Combined beamformer-RAKE single-user receivers have been proposed for multipath channels in [8] and [9], respectively. In [10], adaptive antenna array processing and interference cancellation approaches using the least mean squared (LMS) algorithm are analyzed and the convergence is found to be very slow, requiring several hundred training bits. The problem of integrating antenna array beamforming and multi-user signal detection is proposed for AWGN channels in [11], but channel estimation has not been addressed. In order to detect information symbols reliably, we have to estimate channel and antenna array response vectors. Feder and Weinstein apply the expectation-maximization (EM) algorithm to parameter estimation of superimposed signals [12]. Bit sequence detection with joint random parameter estimation using the EM algorithm is studied for single-user systems in [13]. Recently, applications of the EM algorithm to CDMA systems have been proposed for signal detection [14], channel estimation [15] and joint channel estimation and signal detection [16]. The space-alternating generalized EM (SAGE) algorithm has been developed to accelerate the convergence of the EM algorithm [17]. Applications of the SAGE algorithm in multi-user CDMA channels can be found in [14] for single antenna and known channels, in [18] for antenna array channel parameter estimation and in [19] for joint parameter estimation and signal detection based on the discrete wavelet transform for a single antenna system. In this paper, we investigate the integration of spatial signal processing with multi-user signal detection for the synchronous CDMA systems with non-orthogonal spreading codes over a single dominant-path Rayleigh fading channel. The motivation to focus on synchronous DRAFT
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systems is as follows: a K -user asynchronous system can be modelled as a synchronous system with K N users [1], where N is the number of bits in each transmitted block. Most importantly, the synchronous problem formulation simpli es the derivation and analysis of the new algorithm and the results obtained can be generalized to the case of higher complexity asynchronous multipath systems [20]. We rst formulate a spatial-temporal decorrelator which we show to be near-far resistant. Then, we apply the SAGE algorithm to jointly estimate channel array response vectors and detect information symbol sequences. The bit error probability and Cramer-Rao Lower Bound (CRLB) for the estimated channel are then derived. This paper is organized as follows. The discrete-time system model is developed in Section II. In Section III, we derive a spatial-temporal decorrelator using the maximum likelihood criteria and analyze its asymptotic eciency. An iterative spatial-temporal decorrelating receiver is proposed in Section IV by applying the SAGE algorithm. The bit error probability and CRLB are derived in Section V. Section VI presents simulation results. II. System Model
We consider a synchronous frequency non-selective Rayleigh fading channel for an uplink CDMA communication system with a base-station antenna array. There are K active users in the system. An N-bit transmitted signal from the kth user is
sk (t) = Ak
N
X
i=1
bk (i)ck (t ? iTb)
(1)
where Ak is the amplitude of the kth user, bk (i) 2 f?1; 1g is the ith transmitted bit of the kth user with equal probability and ck (t) represents the spreading waveform of the kth user, which is given by L?1 ck (t) = ck (l)p(t ? lTc) (2) X
l=0
where ck (l) 2 f?1; +1g (l = 1 L ? 1) is the spreading code, p(t) is the chip pulse, Tc is the chip interval, Tb is the bit interval and processing gain L is de ned as L = Tb=Tc. October 6, 1998
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We assume that the information bits from K users are independent, the spreading code sequences for K users are independent and the spreading waveform has normalized energy, i.e., Tb jck (t)j2 = 1. The transmitted signal passes through a fading channel and arrives at a base-station antenna array with M elements. The impulse response of the channel from transmitter to antenna array output can be modelled as R
0
t) = k (t)a(k (t))(t)
gk (
(3)
where k (t) represents channel attenuation for user k and a(k (t)) is the M-dimensional array response vector with direction-of-arrival (DOA) k (t) from the kth user. The channel fading attenuations for K users are assumed to be mutually independent and also independent of information bit symbols. The received composite signal at the base-station antenna array from K users is given by
t) =
x(
K
X
k=1
t) =
xk (
K
X
k=1
sk (t) gk (t) + n(t)
(4)
where n(t) is the AWGN vector with zero mean and covariance matrix 2IM , where IM is an M M identity matrix. From here on, we assume that k (t) and k (t) remain unchanged over the N-bit duration. We denote these quantities as k and k , respectively. From (1),
sk (t) gk (t) = Ak k a(k )
N
X
i=1
bk (i)ck (t ? iTb)
(5)
We denote the channel impulse response vector for user k as fk
= Ak k a(k )
(6)
where the mth component of fk is fkm = Ak k am(k ) and am(k ) is the mth component of array response vector a(k ). The received signal at the mth array element (for m = 1; ; M ) is given by K N (7) xm(t) = fkm bk (i)ck (t ? iTb) + nm(t) DRAFT
X
X
k=1
i=1
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where nm (t) is AWGN at the mth array element. The received signal at each element rst passes through a lter matched to the chip waveform, and is then sampled at the chip rate. The received discrete-time signal at the ith bit interval from the mth element can be obtained for sample g 2 f0; 1; ; L ? 1g as
xm(i; g) =
Z (g +1)T c
t=gTc
xm(t)p(t)dt
(8)
Finally, the gth sample of the ith bit for the mth antenna element is obtained in terms of the sampled chips ck (g) by substituting (7) into (8) yielding
xm(i; g) =
K
X
k=1
fkm ck (g)bk (i)=L + nm(i; g)
(9)
where nm (i; g) = t ggTc Tc nm(t)p(t)dt is Gaussian distributed with zero mean and variance 2=L. We denote the code vector for the kth user as R ( +1) =
ck
= [ck (0)=L ck (1)=L ck (L ? 1)=L]T
(10)
The matched lter output at the mth element for m 2 f1; ; M g can be written in vector form as K (11) fkmbk (i)ck + nm (i) xm (i) = X
k=1 nm(i; L ? 1)]T .
where We de ne the overall system impulse response vector for user k, including the fading channel, array response vector and spreading code vector de ned above as nm (i) = [nm(i; 0)
nm(i; 1)
i) = fkm ck
hm k(
(12)
Equation (11) can be written in terms of (12) as xm (
i) =
K
X
hm k k(
k=1 (xM (i))T ]T , hk
b i) + nm(i)
(13)
Denote x(i) = [(x1(i))T = [(h1k )T (hMk )T ]T and n(i) = [(n1(i))T (nM (i))T ]T . The received discrete-time signal from the antenna array is given by
i) =
x(
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K
X
k=1
i) =
xk (
K
X
k=1
b i) + n(i)
hk k (
(14) DRAFT
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where xk (i) is the received signal from the kth user and n(i) is AWGN vector with zero mean and covariance matrix L IML, where IML is an ML ML identity matrix. Vector hk models the spatial and temporal channel characteristics of our system. The spatial-temporal channel vector hk can also be decomposed as 2
2
hk
=
6 6 6 6 6 6 6 6 6 6 4
ck
0
0
ck
32
0
... 0 ... . . . . . . ...
76 76 76 76 76 76 76 76 76 76 54
fk1 fk2 ...
3 7 7 7 7 7 7 7 7 7 7 5
= C k fk
(15)
fkM where Ck is an ML M spreading code sequence matrix of the kth user. Denote matrix H = [h1 ... ... hK ] and vector b(i) = [b1(i) bK (i)]T , (14) can be expressed as x(i) = H b(i) + n(i) (16) 0
0
ck
A necessary condition that K users are identi able is that H be of full column rank, i.e., ML > K . III. Spatial-Temporal Decorrelator
The log-likelihood of the received signal x(i) conditioned on the bit vector b(i) and the spatial-temporal channel matrix H is given by (for the sake of simplicity, we omit the index i in this section)
(b; H ) = ? 21 (x ? H b)H (x ? H b) (17) =L
where the superscript H denotes conjugate transpose. The unknown parameters are the information bit vector b and channel and array response matrix H. If H is available, the bit vector decision variable b^ can be obtained by maximizing (17) ^d b
= arg max
(b; H ) b
(18)
Equating the derivative of (17) with respect to b to zero, we obtain ^ b DRAFT
= signf[H H H ]?1H H xg
(19) October 6, 1998
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where signfag = 1 if a 0 or -1 if a < 0. We de ne spatial-temporal cross-correlation matrix as H
R = H H = C1f1 C2f2 CK fK H
C1f1 C2f2 CK fK
(20)
Denoting
fij = fiH fj and
ij = cHi cj
8 > >
> :
2
3
6 6 6 6 6 6 6 6 6 6 4
7 7 7 7 7 7 7 7 7 7 5
2
z
=H
H
x
=
6 6 6 6 6 6 6 6 6 6 4
fH
1
0T
0T
f2H
0T
0T
. . . 0T ... . . . . . . ...
3
32
0T
H fK
76 76 76 76 76 76 76 76 76 76 54
y1 y2
...
yK
7 7 7 7 7 7 7 7 7 7 5
= F Hy
(24)
using (15), yk = CkH x for k = 1; 2; ; K . Thus, yk actually represents the despread output for the kth user's decision variable. In (24), the despread outputs for all K users are beamformed by a maximum SNR criterion. This detector structure is a conventional October 6, 1998
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single-user detector with maximum SNR beamforming, which we refer to as the conventional single-user detector. Using (19), (20) and (24), the detected bit vector can be written in the form ^ = signfR?1 F H yg b (25) Therefore, the nal data decision vector is obtained by decorrelating the conventional singleuser detector output. Eq. (25) can also be viewed as a maximum SINR beamforming. Matrix R includes both temporal cross-correlation due to non-orthogonal spreading codes and instantaneous spatial correlation because of the spatial distribution of the active users in the system. The detector structure is illustrated in Figure 1. This structure di ers from that of [11] since the matched- lter (despreader) is at the front end. Substituting (16) into (19) and using de nition of R (c.f. (20)), the spatial-temporal decorrelator output can be simpli ed to ^ b
= signfb + R?1 ndg = signfb + wg
(26)
where w is the spatial-temporal decorrelator output AWGN vector with zero mean and variance L R?1 . If there is no base-station antenna array, the spatial-temporal detector reduces to the classic decorrelating detector proposed in [1]. 2
A. Near-far Resistance
The asymptotic eciency is a performance measure for multi-user signal detection in the limit as the background noise goes to zero. The kth user's asymptotic eciency is de ned as [2] prw e k ( ) (27) Pk ()=Q( k ) < +1g k = lim ! ! wk = supf0 r 1 : lim 0
0
where ek() is the kth user's e ective energy required to achieve the same error probability in the absence of interferers, wk is the received energy for user k, Pk () represents bit-error-rate (BER) of the kth user when the variance of AWGN is 2 and Q(x) = x1 exp(p?2y =2) dy. R
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Because the asymptotic eciency of the temporal decorrelator in a single-path Rayleigh fading channel is the same as that of an AWGN channel [6], we only consider AWGN channel here. In this case, fk = Akak (k ). The kth user error probability for the spatial-temporal decorrelator can be obtained as
Pkd = Q(
p L
1 ) (R?1 )kk
(28)
q
where (R?1 )kk represents the kth diagonal element of matrix R?1 [3]. Thus, the asymptotic eciency of the spatial-temporal decorrelator for the kth user is given by
kd = max2f0;
1 1 g = Rkk (R?1 )kk Rkk (R?1 )kk
(29)
q
By formulating the bit error probability for the conventional signal-user detector [21], the kth user's asymptotic eciency of the conventional single-user detector can be obtained
c k
= max2f0;
pR ? jR j=pR kk kk pi6 Rk ik g = max2f0; 1 ? P
=
kk
X
6
i=k
jRik j g Rkk
(30)
The kth user's near-far resistance is de ned as its worst case asymptotic eciency over all possible energies of the interferers and given by k = inf wi 0;i6=k k [1]. Example 1: To numerically compare the asymptotic eciency for the spatial-temporal decorrelator and the conventional single-user detector, we consider a four-user system based on a set of spreading codes from Gold sequences of length seven. The corresponding crosscorrelation matrix of the spreading codes is taken from [5]. We assume a uniform linear array (ULA) with half-wavelength spacing at the base-station. The direction-of-arrivals (DOA) for four active users with known array responses are [25; 5; ?15; ?35] with respect the array boresight. We consider two cases: a single antenna and a three-element antenna array. We refer the rst user as the desired user and de ne the power ratio as A2k =A21, for k = 2; ; K . In this example, we assume that all interferers have equal transmitted power. Figure 2 shows that performance of the spatial-temporal decorrelator is independent of the received energies from interferers and is near-far resistant. Also note that the asymptotic eciencies October 6, 1998
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for both conventional single-user detectors tends to zero as the interferers become stronger: the conventional single-user detector is not near-far resistant even when an antenna array is used. From Figure 2, it can be observed that using an antenna array can potentially improve the asymptotic eciency for either the spatial-temporal decorrelator and the conventional single-user detector. The above analysis is based on the assumption that matrix R is available, implying that channel array response vectors, fk (for k = 1; 2; ; K ), are known at the receiver. This is not the case in a practical application. We now investigate the problem of joint channel estimation and signal detection. IV. Decorrelating Receiver via SAGE Algorithm
In this section, we address the key implementation issue of the proposed spatial-temporal decorrelating receiver, namely the joint parameter estimation and spatial-temporal signal decorrelation. We derive a sequential decision-feedback receiver structure based on the space alternating generalized EM (SAGE) algorithm. The available observed data at the receiver is a data vector set fx(i); 1; ; N g. The SAGE algorithm can achieve improved performance, such as faster convergence, over the standard EM algorithm [17]. The key idea for the SAGE algorithm is to choose a less informative hidden-data space to improve convergence rate. Starting with an initial parameter vector estimate at iteration j = 0, iterate the following: 1. Choose new index set S = S j , a subset of the parameters, which de nes an admissible hidden-data space xSj ; 2. Compute the following conditional expectation of xSj given observed data x and the previous estimate of parameter vector ^j
(Sj j^j ) = E [ln f (xSj jxSj ; ^Sj )jx; ^j ] +1
(31)
3. Obtain the next estimate by maximizing over the chosen subset while keeping the other DRAFT
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parameters xed
^Sj+1 = arg maxSj (Sj j^j ) j ^Sj+1 = ^Sjj j where the index set Sj is the complement of S j . 4. Increment j and go to step 1. +1
+1
(32)
+1
+1
+1
A. Application to CDMA
For the CDMA system model considered, we would like to estimate each user's channel parameter and detect the information symbol individually. In step 1, we choose user index k as the index set. Thus, the admissible hidden-data space for index k, for k = 1; ; K and i = 1; ; N , is given by (14) 2 i) N(hk bk (i); L IML)
xSk (
(33)
The log-likelihood function after removing the terms independent of hk and bk (i) is straightforward N (34)
(xSk (1); ; xSk (N )) = ? 21=L (xSk ? hk bk )H (xSk ? hk bk ) X
i=1
Given the estimation results at the jth iteration, the conditional expectation of xSk (i) (for k = 1; ; K and i = 1; ; N ) in step 2 is given by
i) = h^ jk^bjk (i) + (x(i) ?
^ Sk ( x
K X
i=1
^ i ^i ( h j bj
i)) = x(i) ?
X
i6=k
^ i ^i ( h j bj
i)
(35)
Finally, substituting (35) into (34), we obtain the conditional expectation of the log-likelihood function of xSk (i). The maximization results at the next iteration, (32) in step 3, is given by N
f? (^xSk (i) ? hk bk (i))H (^xSk (i) ? hk bk (i))g [h^ jk+1; ^bjk+1(i)] = arg hmax ;b (i) X
k k
i=1
(36)
Thus, SAGE-based receiver for joint array response vector estimation and information symbol detection is obtained as for j = 1; 2 October 6, 1998
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k = (j modulo K ) E-step: compute the conditional expectation of hidden-data for k = 1; ; K and i = 1; ; N ^ Sk x
j
(i) = x(i) ?
K
X
i=1;6=k
^ i ^i ( h j bj
i)
(37)
M-step: obtain the maximum-likelihood estimates h^ k for k = 1; ; K and ^bk (i) for k = 1; ; K and i = 1; ; N N ^hjk+1 = 1 x^ Sk j (i)^bjk (i) N i=1 X
(38)
^bjk+1(i) = signf(h^ jk+1 )H x^ Sk j (i)g ^ jk+1 k0 6= k h 0 = h^ jk0 ; ^bjk+1 0 (i) = ^bjk0 (i); k 0 6= k
(39)
The above steps can be interpreted as follows: substituting (37) into (39), we obtain ^bjk+1(i) = signf((h^ jk+1 )H x^ k (i) ?
K
X
k0 =1;k0 6=k
f^kkj+10 kk0 ^bji (i))g
(40)
where f^kkj+10 = (^fkj+1 )H ^fkj0 is the estimated instantaneous spatial correlation de ned in (21) at the (j+1)st iteration and kk0 is the cross-correlation de ned in (22). Equation (40) implies that information symbol detection at each iteration involves explicit interference cancellation given the current channel estimation and previous bit detection results. Therefore, the SAGE-based receiver has a sequential interference cancellation structure. B. Convergence
From (17), it can be easily veri ed that the log-likelihood function is continuous and differentiable with respect to the unknown parameters. Since the likelihood function increases monotonically with each iteration, it is bounded above and the proposed receiver converges to a xed stationary point or local/global maxima depending on the initial guess of the unknown parameters [22]. The convergence rate of the SAGE-based receiver can be shown to DRAFT
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be the largest eigenvalue of [FxSk ? Fx](FxSk )?1 [23], where FxSk and Fx are Fisher information matrices of xSk and x, respectively. To investigate the e ect of the antenna array on receiver convergence, for simplicity, we assume that the channel array response vectors are given. From (17) and (35), Fx = 4L R and FxSk = 4L Rkk IK . 2
2
Example 2: We again consider a system with a uniform linear array at the base-station over a AWGN channel. The inter-element spacing of antenna array is half-wavelength and the DOAs of the active users are uniformly distributed in [?60; 60]. Gold sequences of length 31 from [24] are assigned to mobile users. All active users have equal transmitted power. The convergence rate curves are plotted in terms of number of users and number of antennas and averaged over 5,000 trials. From Figure 3, it can be observed that the convergence becomes slower as the number of active users increases in the system. An interesting remark is that using a base-station antenna array can accelerate the convergence of the proposed receiver. V. Receiver Performance
In this section, we assess the performance of the proposed spatial-temporal decorrelating receiver. Since we propose to jointly estimate the channel array response vectors and detect the transmitted information symbols, we determine the bit-error rate (BER) for symbol detection and the Cramer-Rao lower bound (CRLB) for channel estimation. A. Bit Error Rate (BER)
From (26), it is observed that the bit decision vector is an unbiased estimator containing a zero-mean noise vector w whose covariance matrix is L R?1. Without loss of generality, we assume that the rst user is the desired user. 2
Claim: The bit error probability for the rst user is obtained as
P1 = 12 (1 ? 1 + 1 ) 1 s
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(41) DRAFT
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where 1 is the average signal-to-noise ratio (SNR) for user 1 and given by
1 = and
1
(42)
2 E [R?1 ] 11 L
E [R?1]11 = Var[1 ]
22 2K ... . . . ... K2 KK
(43)
11 1K ... . . . ... K1 KK where E [:] denotes the expectation over Rayleigh random variables contained in R?1 , Var[.] represents variance, j:j is the determinant and ij = AiAj aHi aj ij . Proof: The (1,1) entry of R?1 is 1
R22 R2K [R?1]11 = jR1 j ... . . . ... RK2 RKK
Since the ijth entry of R is given by Rij = i j ij , using Hermitian symmetry, it can be easily obtained that
and
R22 R2K 22 2K ... . . . ... = j 2j2 j K j2 ... . . . ... RK2 RKK K2 KK
(44)
11 1K jRj = j 1j2 j K j2 ... . . . ... (45) K1 KK where j k j2 = k k , from which (43) follows. For Rayleigh fading channel and BPSK modulation, (41) is obtained using the results in [25]. DRAFT
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B. Cramer-Rao Lower Bound (CRLB)
To measure channel array response vector estimation performance, we derive the CramerRao lower bound. We collect the unknown parameters into the column vector a
= [aT1
aTK ]T
(46)
where ak = k a(k ) is the channel array response vector for the kth user. For an M-element antenna array, we de ne the M M matrix for the ith bit of the kth user, k = 1; ; K , as Bk (i) = diag[bk(i); ; bk (i)]. Proposition: The CRLB matrix for parameter vector a is given by 1 N G(i)]?1 CRLB(a) = [ (47) L i=1 where p 1 212B12(i) p 1 K 1K B1K (i)
111IM ... p 1 212B12(i) ...
222IM (48) G(i) = ... ... ... ... p 1 K 1K B1K (i)
K KK IM where Bkj (i) = BkH (i)Bj (i) = IM if bk (i) = bj (i) or ?IM if bk (i) = ?bj (i) (for i = 1; ; N and k; j = 1; ; K ), k = A2k =2 is the signal-to-noise ratio (SNR) corresponding to the kth user and ij is the spreading code cross-correlation de ned in (22). Outline of Proof: First, we de ne the parameter vector as = [ 2 aT a T ]T , where 2 is background noise covariance, a and a are the real and imaginary parts of a, respectively. Discarding the terms independent of , the log-likelihood function is given by N (49) ln = ?MLN ln 2 ? L2 [x(i) ? H b(i)]H [x(i) ? H b(i)]) i=1 We compute the derivative of (49) with respect to . Using the de nition of the Fisher ln )( @ ln )T ] and following the technique in [26], we obtain information matrix I( ) = E [( @ @ @ X
2
3
6 6 6 6 6 6 6 6 6 6 4
7 7 7 7 7 7 7 7 7 7 5
X
I( a) =
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E [( @ @lna )( @ @lna )T ] = 2L2
N
X
i=1
Re[B H (i)DH DB (i)] DRAFT
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@ ln
2 L H (i)DH DB (i)] T I( a) = E [( Re [ B )( ) ] = 2 @ a @ a i=1 X
) I( aa
= E [( @ @lna )( @ @lna )T ] = ? 2L2
N
X
i=1
Im[B H (i)DH DB (i)]
where B (i) = diag[B1(i); ; BK (i)] and D = [D1 ... ... DK ] with ML M matrix Dk = diag[Akck ; ; Akck ] (for k = 1; ; K ). De ne G(i) = 1 B H (i)DH DB (i), (48) follows. Since G(i) (for i = 1; ; N ) is real, we have I(aa) = 0 and 2
I( a)
= I(a) = 2L
N
X
i=1
G(i)
(50)
Thus, the Fisher information matrix is given by 2
) =
I(
6 6 6 6 6 6 4
I( 2 ) 0TMK 0TMK 0MK
I( a)
0
0MK
0
I( a)
3 7 7 7 7 7 7 5
(51)
Finally, using inverse of a partitioned matrix, we can obtain the Cramer-Rao lower bound (CRLB) matrix for channel array response vectors CRLB(a) = I(a)?1 + I(a)?1. Clearly, the Fisher information matrix, Ni=1 G(i), is a function of the SNR at the ith bit as well as the cross-correlation between the spreading codes. P
VI. Simulations
In this section, we present performance results for the proposed receivers. A randomly generated Gold sequence of length 31 is assigned to each user in our CDMA system. In the simulations, we transmit a 100-bit information sequence from each user and assume the channels remain unchanged during the data-block transmission. We insert one training bit in the rst bit position in each sequence. A uniform linear array with half-wavelength spacing is used at the base-station and the DOAs are uniformly distributed in [?60; 60]. The initial guesses of the channel and array response vectors are obtained using the training bits. Using (24), we obtain the initial detected information bit vectors. The simulation results DRAFT
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are computed from 10,000-150,000 trials, depending on the signal-to-noise ratio, so that the BER is calculated to within 5% with 90% con dence. The rst example is used to compare performance of the SAGE-based receiver, EM-based receiver [16] and spatial-temporal decorrelating receiver with perfect knowledge of channel and array response vectors. There are ve active users in the system and either a single antenna or a three-element base-station antenna array. For simplicity of analysis, all users have equal transmitted power. Both the SAGE-based and EM-based decorrelating receivers use eight iterations. From Figure 4, we can observe that both iterative receivers signi cantly outperform the conventional single-user receiver. When a single antenna is used, the iterative receivers do not converge to the optimal spatial-temporal decorrelating receiver. However, the performance of the SAGE-based receiver is greatly improved by using an antenna array. This is mainly due to the antenna array's improved channel estimates, as shown in Figure 5. When the SNR is larger than 20dB, the channel estimates of the SAGE-based receiver do not reach the CRLB. However, the loss in channel estimation performance at such a high SNR does not have strong e ect on symbol detection. At a BER of 10?2 , the three-element antenna array can achieve about 6 dB gain over a single antenna. To address the near-far resistance of the proposed receiver, we consider ve active users and a two-element base-station antenna array. We assume that the rst user is the desired user with 20 dB SNR and xed transmitted power. Again. all other interferers are assumed to have equal and varying transmitted power. At the power-ratio of 20 dB, the transmitted powers of the interferers are all 100 times stronger than that of the desired user. This represents an extreme near-far environment. Figure 6 shows that the SAGE-base decorrelating receiver converges to the spatial-temporal decorrelating receiver with perfect knowledge of channel and array response vectors and is therefore near-far resistant. Although the EMbased decorrelating receiver is an improvement over the conventional single-user receiver, neither receiver is near-far resistant as shown in Figure 6. From Figure 7, it is observed October 6, 1998
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that the SAGE-based receiver has improved near-far resistance than that of the EM-based receiver. The nal example is used to illustrate the convergence of the proposed iterative receiver. Five users with equal transmitted powers are considered in the system and there is a threeelement base-station antenna-array. Figure 8 shows convergence of the SAGE-based decorrelating receiver. Because the SAGE-based receiver converges very quickly in the low SNR region, for a required BER, three or four iterations can achieve acceptable performance. VII. Conclusions
We have derived a spatial-temporal decorrelator based on the discrete-time signal model using maximum likelihood criteria for CDMA uplink Rayleigh fading channel with a basestation antenna array. The spatial-temporal decorelator is near-far resistant. Numerical results show that the incorporation of the base-station antenna array results in signi cant performance improvement. An iterative receiver structure is obtained by applying the SAGE algorithm to implement the spatial-temporal decorrelating receiver. It is shown that using base-station antenna array can accelerate convergence rate of the proposed receiver. The bit error probability and CRLB are derived for the proposed receiver. Simulation results show that the SAGE-based decorrelating receiver can achieve signi cant performance gain over the conventional single-user receiver. It is observed that the SAGE-based receiver outperforms an EM-based receiver and converges to the BER of the spatial-temporal decorrelating receiver. The channel estimates of the SAGE-based receiver are also close to the CRLB. The simulated BER results for the spatial-temporal decorrelating receiver agree closely with analytical results. Extension of the present results to asynchronous and multipath fading systems is under investigation. References [1] R. Lupas, S. Verdu, \Near-far resistance of multiuser detectors in asynchronous channels", IEEE Trans. Commun., vol. 38, no. 4, pp. 496-508, Apr. 1990. DRAFT
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[2] S. Verdu, \Minimum probability of error for asynchronous Gaussian multiple-access channels", IEEE Trans. Inform. Theory, Vol. 32, N0. 1, pp.85-96, Jan. 1986. [3] Z. Xie, R. T. Short and C. K. Rushforth, \A family of suboptimum detectors for coherent multiuser communications", IEEE J. Select. Areas Commun., vol. 8, no. 4, pp. 683{690, May 1990. [4] M. K. Varanasi and B. Aazhang, \Multistage detection in asynchronous code-division multiple-access communications", IEEE Trans. Commun., vol. 38, no. 4, pp. 509{519, Apr. 1990. [5] A. Duel-Hallen, \Decorrelating decision-feedback multiuser detector for synchronous code-division multiple-access channel", IEEE Trans. Commun., vol. 41, no. 2, pp. 285{290, Feb. 1993. [6] Z. Zvonar and D. Brady, \Multiuser detection in single-path fading channels", IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1729{1739, Feb./Mar./Apr. 1994. [7] A. J. Paulraj and C. B. Papadias, \Space-time processing for wireless communications", IEEE Signal Processing Magazine, pp. 49{83, Nov. 1997. [8] A. F. Naguib and A. Paulraj, \Performance of wireless CDMA with M-ary orthogonal modulation and cell site antenna arrays", IEEE J. Select. Areas Commun., vol. 14, no. 9, pp. 1770{1783 , May 1996. [9] H. Liu and M. D. Zoltowski, \Blind equalization in antenna array CDMA systems", IEEE Trans. Signal Processing, vol. 45, no. 1, pp. 161{172, Jan. 1997. [10] R. Kohno, H. Imai, M. Hatori and S. Pasupathy, \Combination of an adaptive array antenna and a canceller of interference for direct-sequence spread-spectrum multiple-access system", IEEE J. Select. Areas Commun., vol. 8, no. 4, pp. 675{681, May 1990. [11] S. Y. Miller and S. C. Schwartz, \Integrated spatial-temporal detectors for asynchronous Gaussian multipleaccess channels", IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 396{411, Feb./Mar./Apr. 1995. [12] M. Feder and E. Weinstein, \Parameter estimation of superimposed signals using the EM algorithm", IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, no. 4, pp. 477{489, Apr. 1988. [13] C. N. Georghiades and J. C. Han, \Sequence Estimation in the Presence of Random Parameters Via the EM Algorithm", IEEE Trans. on Commun., vol. 45, no. 3, pp. 300-308, Mar. 1997. [14] L. B. Nelson and H. V. Poor, \Iterative multiuser receivers for CDMA channels: an EM-Based approach", IEEE Trans. Commun., vol. 44, no. 12, pp. 1700{1710, Dec. 1996. [15] U. Fawer and B. Aazhang, \A multiuser receiver for code division multiple access communications over multipath channels", IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. 1556-1565, Feb./Mar./Apr. 1995. [16] R. Wang and S. D. Blostein, \Maximum likelihood multi-user CDMA receiver for base-station antenna arrays using EM algorithm", Proc. 19th Biennial Symp. Commun., pp 110{114, Kingston, ON, June 1998. [17] J. A. Fessler and A. O. Hero, \Space-Alternating Generalized EM algorithm", IEEE Trans. Signal Processing, vol. 42, no. 10, pp. 2664{2677, Oct. 1994. [18] D. Dahlhaus, A. Jarosch, B. H. Fleury and R. Heddergott, \Joint demodulation in DS/CDMA systems exploiting the space and time diversity of the mobile radio channel", Proc. IEEE Int. Symp. on Personal, Indoor and Mobile Radio Communications, pp. 47{52, Helsinki, Finland, 1997. October 6, 1998
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[19] I. Sharfer and A. O. Hero, \Optimum multiuser CDMA detector using grouped coordinate ascent and the DWT", Proc. IEEE Workshop in Signal Processing Advances in Wireless Communications, Paris, Apr. 1997. [20] A. Duel-Hallen, J. Holtzman and Z. Zvonar, \Multiuser detection for CDMA systems", IEEE Personal Communications, pp 46{58, Apr. 1995. [21] R. Lupas, S. Verdu, \Linear multiuser detectors for synchronous code-division multiple-access channels", IEEE Trans. Inform. Theory, vol. 35, no. 1, pp. 123{136, Jan. 1989. [22] C. F. J. Wu, \On the convergence properties of the EM algorithm", Annals of Statistics, vol. 11, no. 1, pp. 95{103, Jan. 1983. [23] A. O. Hero and J. A. Fessler, \Convergence in norm for alternating expectation-maximization (EM) type algorithms", Statistica Sinica, vol. 5, no. 1, pp. 41{54, Jan. 1995. [24] A. M. Monk, M. Davis and L. B. Milstein, \A noise-whitening approach to multiple access noise rejection - part I: theory and background", IEEE J. Select. Areas Commun., vol. 12, no. 5, pp. 817, June 1994. [25] J. G. Proakis, Digital Communications, 3rd Ed., McGraw-Hill, Inc., 1995. [26] P. Stoica and A. Nehorai, \Music, maximum likelihood, and Cramer-Rao bound", IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, no. 5, pp. 720{741, May 1989.
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x(t) -
Matched Filter
t=gT
-
#1 Despreader ? ? ?
A AU x -
#K Despreader
y1 -
Coherent #1 Beamformer
yK -
? ? ?
Coherent #K Beamformer
z1 -
zK -
21
SpatialTemporal Decorrelator
........ ... ...
b -
Fig. 1. Spatial-Temporal Decorrelator Structure
Asymptotic Efficiency Comparison Between Spatial−Temporal Decorrelator and Coventional Detector 1
0.9
0.8
Asymptotic Efficiency
0.7
0.6
0.5
0.4
0.3 * decorrelator (single antenna) 0.2
+ decorrelator (antenna array) o conventional (single antenna)
0.1
0 −20
x conventional (antenna array)
−15
−10 −5 0 5 10 Power−Ratio (dB) (All Interferers Have Equal Transmitted Power)
15
20
Fig. 2. Asymptotic Eciencies for Single Antenna and a Three-Element Antenna Array
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. Y, MONTH 1999 Convergence Rate of SAGE−Based Receiver 0.7
0.6
0.5
Convergence Rate
Single Antenna 0.4
Three Antennas
0.3
0.2
Five Antennas
0.1
0
2
3
4
5 6 Number of Users
7
8
9
Fig. 3. Convergence Rate of SAGE-Based Receiver (A larger rate implies slower convergence. The algorithm converges faster as the number of antennas increases) Receiver Perpormance Comparison: Bit Error Probability
−1
10
Single Antenna
Antenna Array −2
Bit Error Rate of the Desired User
10
Single Antenna
Antenna Array
−3
10
−− conventional receiver * EM−based receiver o SAGE−based receiver x spatial−temporal decorrelating receiver (simulated results)
−4
10
solid line: spatial−temporal decorrelating receiver (analytical results) Five Users, Equal Transmitted Power Synchronous System, Rayleigh Fading Channel
−5
10
5
10
15 20 Signal−to−Noise Ratio (dB)
25
30
Fig. 4. Bit Error Rate for Single Antenna and a Three-Element Antenna Array DRAFT
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Receiver Performance Comparison: MSE of Channel Estimates
0
Mean Squared Error of Channel Estimates for the Desired User
10
Initial Estimates −1
10
−2
10
EM Estimates (single antenna)
EM Estimates (antenna array) −3
10
SAGE Estimates (single antenna)
SAGE Estimates (antenna array) −4
10
Five Users, Equal Transmitted Power Three−Element Antenna Array Synchronous System, Rayleigh Fading Channel
CRLB
−5
10
5
10
15 20 Signal−to−Noise Ratio (dB)
25
30
Fig. 5. Mean Squared Error of Channel Estimates for Single Antenna and a Three-Element Antenna Array Receiver Performance in Near−Far Environment: Bit Error Rate
0
10
Five Users, Two−Element Antenna Array SNR of the Desired User: 20 dB Synchronous System, Rayleigh Fading Channel
Bit Error Rate of the Desired User
solid line: spatial−temporal decorrelating receiver (analytical results) −1
10
x spatial−temporal decorrelating receiver (simulated results) −− conventional receiver * EM−based receiver o SAGE−based receiver
−2
10
−3
10 −20
−15
−10 −5 0 5 10 Power Ratio (dB) (All Interferers Have Equal Transmitted Powers)
15
20
Fig. 6. Bit Error Rate for a Two-Element Antenna Array in Near-Far Environment October 6, 1998
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. Y, MONTH 1999 Receiver Performance in Near−Far Environment: MSE of Channel Estimates
2
Mean Squared Error of Channel Estimates for the Desired User
10
Five Users, Two−Element Antenna Array
1
10
SNR of the Desired User: 20 dB Synchronous System, Rayleigh Fading Channel
0
10
Initial Estimates −1
10
EM Estimates −2
10
−3
10
SAGE Estimates CRLB
−4
10 −20
−15
−10 −5 0 5 10 Power Ratio (dB) ( All Interferers Have Equal Transmitted Power)
15
20
Fig. 7. Mean Squared Error of Channel Estimates a Two-Element Antenna Array in Near-Far Environment Convergence of SAGE−Based Receiver: Bit Error Rate
−1
Bit Error Rate of the Desired User
10
−2
10
Iteration 1
Iteration 2 Iteration 3 −3
10
Iteration 4 Iteration 5 Five Users, Equal Transmitted Power Three−element Antenna Array Synchronous System, Rayleigh Fading Channel
Iteration 6: x− Iteration 7: o− Iteration 8: solid line
−4
10
5
10
15 20 Signal−to−Noise Ratio (dB)
25
30
Fig. 8. Convergence of SAGE-Based Receiver for a Three-Element Antenna Array DRAFT
October 6, 1998
