Adaptive Digital Beamforming in Cellular CDMA ... - Semantic Scholar
Adaptive Digital Beamforming in Cellular CDMA Systems Using Noniterative Signal Subspace Tracking Wan Yi Shiu and Steven D. Blostein
Department of Electrical and Computer Engineering Queen's University, Kingston, Ontario Canada, K7L 3N6 [email protected], [email protected] Tel: (613)545-6561, Fax: (613)545-6615
Abstract We introduce a novel adaptive digital beamforming algorithm for application in cellular CDMA communications systems. A Noniterative Rank-one Signal Subspace eigenstructure Update (NR1SSU) algorithm enables low-complexity recursive processing of beamforming weights. A PN chip-level IS-95 CDMA reference model [3] has been used to test the beamforming algorithm via simulation. Our results show that accurate beamforming can be achieved to moving mobiles at one update per IS-95 frame of 192 data bits.
I. Introduction
Adaptive antenna array beamforming in future multimedia cellular CDMA communication systems holds great potential for improving signal-to-interference-noise ratios and achieving higher cell capacity. An adaptive beamforming network spatially lters the in-band interfering mobiles by adaptively updating the weight vector of each beamformer so as to steer the main beam lobe to the direction-of-arrival (DOA) of the desired mobile. Traditional eigen-based beamforming techniques require iterative and computationally expensive batch processing [5] [7] [6]. Our goal is to achieve higher capacity and coverage in future multimedia cellular CDMA systems using an adaptive antenna array beamforming network at the base station. To further reduce basestation costs, we propose to simplify the required computation by employing a noniterative rank-one signal subspace eigenstructure update algorithm [2]. Updating beamforming weights for each mobile in a multi-beamforming network can be very expensive depending on how the estimates are being calculated [5] [7] [6]. The Signal to Interference-Noise Ratio (SINR) optimum beamforming weight wopt requires (i) the array response of the lth multipath of the desired ith mobile, ai;l 2 CMx1 and (ii) the inverse of the batch processing interference-noise covariance matrix, R?IN1;i;l 2 CMxM , i.e., i;l
w
opt i;l
= R?IN1;i;l ai;l
where ai;l = [a1(i;l (t)); ::::; aM(i;l (t))]T 2 CMx1 is the ith mobile's lth multipath unit array response vector, i;l (t) is the time-varying DOA of the lth multipath of the ith mobile, M is the number of antenna elements, is a scalar, and C is the complex domain. Currently, estimation of (i) & (ii) requires the formation of a pre-correlation autocovariance This research was supported by a grant from the Canadian Institute for Telecommunications Research under the NCE program of the Government of Canada.
matrix Rxx as well as the ith mobile's lth multipath postcorrelation autocovariance matrix Rzz [7] [6]. This proposed adaptive beamforming method makes beamforming in CDMA unattractive and computationally very expensive, since there are 1228800 pre-correlation vectors x(t) per user per second (PN chips per second) and 307200 post-correlation vectors zi (n) per user per second (Walsh chips per second) in a CDMA IS-95 system. It is shown in [7] that R^ IN can be estimated as R^ IN = L ?L 1 (Rxx ? L1 Rzz ) where L is the code ltering processing gain, and ai;l can be estimated as ai;l = R?xx1=2ei;l The iterative power method is used to calculate the principal eigenvector ei;l corresponding to the largest eigenvalue of ?=2R R1=2, where it is shown in [7] that Rxx xx zz i
i;l
i;l
i;l
i
i;l
i
i
i
i;l
R?= R
1=2 1=2 ;i;l R? xx i Rxx i ai;l
= R1xx=2ai;l One drawback of an iterative eigen-decomposition method, such as the power method, is that its computational complexity may be unbounded, depending on the eigenvalue spread. In this paper, we develop a novel application of a computationally ecient and numerically stable Noniterative Rank1 Signal Subspace Update (NR1SSU) for array response estimation in a CDMA system. This application of NR1SSU is developed in section II. In addition, the NR1SSU uses a recursive estimation formulation, which dramatically reduces the computation compared to [7]. The NR1SSU algorithm and application to IS-95 is described in section III, where a detailed computational analysis is provided. Preliminary simulation results are shown in section IV. xx i
2
zz
i
II. System Model A PN chip-level IS-95 CDMA uplink model [4] has been used to test the beamforming algorithm via simulation. We have considered amplitude Rayleigh fading, multipath loss, shadowing, perfect power control in both a circular and a three-sector linear calibrated array IS-95 CDMA system. A. Case of zero multipath angle spread For the case of relatively small multipath angle spread, we will rst assume for simplicity that only one resolvable
DOA from each mobile is received. At time t, the baseband signal vector received at the antenna array network is xi (t) = [xi;1(t); xi;2(t); :::; xi;M(t)]T (1) =
N X
j =1
q
cj (t ? i;j )bj (t ? i;j ) Pj (t)aj (t) + n(t)
where N is the total number of in-band mobiles, cj (t ? i;j ) 2 f1; ?1g is the (square pulse) Pseudo Noise (PN) chips of the j th mobile with a chip period of Tc = 1=1228800 sec, bj (t ? i;j ) 2 f1; ?1g is the j th mobile's information bit sequence in Walsh chip period Tb = 1=307200 sec, j i;j j 2 U (0; Tc ) is the uniformly distributed di erential time delay of the j th mobile relative to the time delay of the ith mobile's time delay received at the base station, column vector n(t) = [n1(t); ::::; nM(t)]T N (0; 2 I) represents independent and identically distributed Gaussian thermal noise, M is the number of antenna elements in the base station array or sector, Pj (t) is the total power received at the base station of the j th mobile's DOA, aj (t) = [a1(j (t)); ::::; aM(j (t))]T 2 CMx1 is the time-varying column vector of the j th mobile's array response, j (t) is the time-varying DOA of the j th mobile, and []T denotes transpose. Without loss of generality, for all i = 1; 2; ::::; N and j = 1; 2; ::::; N, we assume that synchronization to the ith desired mobile possible at each ith branch of the network i.e. i;i = 0, and the signal bj (t ? i;j ), chip cj (t ? i;j ) and noise n(t) are mutually uncorrelated. Bits bj (t ? i;j ) and bk (t ? i;k ) are assumed uncorrelated as well as cj (t ? i;j ) and ck (t ? i;k ) for all k = 1; 2; ::::;N and k 6= j, and array response vectors aj (t) are assumed to be unchanged over one information bit period Tb . Under the above assumptions, the pre-correlation covariance matrix of eqn.(1) becomes
Rxx (t) = Piai(t) ai (t)y + i
N X
j =1;j 6=i
Pj aj (t) aj (t)y + 2 I
(2) = Piai (t) ai (t)y + RIN where RIN denotes the interference-plus-noise covariance matrix, I is the MxM identity matrix, ()y denotes transpose conjugate, and at information bit n the ith mobile's post-correlation vector, zi (n) of xi (t) over time interval Tb is Z T p n(t)ci(t)dt (3) zi(n) = Tb Pi bi(n)ai (n) + p1T b 0 Z T N X p p1 + Pj bj (t ? i;j )cj (t ? i;j )ci (t)aj (n)dt j =1;j 6=i Tb 0 i
i
b
b
The i mobile's post-correlation autocovariance matrix can be de ned as (4) Rzz (n) Efzi (n) zi(n)y g T1 th
i
c
where L = Tb =Tc is the "code ltering" processing gain. Using results in [10], " #2 1 E Z T c ( )c ( ? )d = 2 [ Tb2 ? Tb2 ] ' 2Tb i j i;j Tb Tb 3L 12L2 3L 0 b
After integration, eqn.(4) becomes, Rzz (n) N 2 X = LPiai (n) ai (n)y + 32 Pj aj (n) aj (n)y + T I c j =1;j 6=i (5) = LPiai (n) ai (n)y + 32 RIN + 3 ?3T2Tc 2I c If L is large enough, such that N X LPi ai (n) ai (n)y 32 Pj ai (n) ai (n)y (6) j =1;j 6=i i
i
we can consider eqn.(5) as a signal-plus-noise spherical subspace problem, where N 2 X Pj ai (n) ai (n)y + T I Rzz (n) = Rss (n) + 32 c j =1;j 6=i i
i
X = s (n)^ai (n) ^ai (n)y + m (n)um (n) um (n)y (7) M
m=2
where m are the eigenvalues of Rzz (n) and um (n) are the corresponding eigenvectors for m = 1; 2; :::; M and s = 1 is the signal subspace. In eqn.(7), the signal covariance matrix is rank one with Rss (n) = LPiai (n) ai (n)y = s(n)^ai (n) ^ai(n)y (8) and the interference-noise covariance matrix is N 2 X y + 2 I (9) P a (n) a (n) j j j 3 j =1;j 6=i Tc i
i
which is now approximately a spherical covariance matrix for large processing gain L. The principal eigenvector ^ai (n) corresponding to the largest eigenvalue of Rzz (n) is an accurate signal subspace array response vector estimate when eqn.(6) holds. B. Case of non-zero multipath angle spread In the case of non-zero multipath angle spread, with the assumption that all paths are assumed to be resolvable [9], the pre- and post-correlation covariance matrices with a maximum of P resolvable multipaths can be generalized to i
R
xx i;l
(t) =
N P X X
j =1 p=1
Pj;paj;p (t) ai;p (t)y + 2 I
= Pi;l ai;l (n) ai;l (n)y + RIN and the post-correlation covariance matrix is Rzz (n) = LPi;l ai;l (n) ai;l(n)y P X + 32 Pi;p ai;p (n) ai;p (n)y p=1;p6=l
i;l
(10)
i;l
+
P N X 2 X y + 2 I (11) P a (n) a (n) j;p j;p j;p 3 j =1;j 6=i p=1 Tc
where some of the Pj;p or Pi;p may be negligible. If L is large enough that P X LPi;l ai;l (n) ai;l (n)y 32 Pi;p ai;p (n) ai;p (n)y p=1;p6=l
N P X X Pj;paj;p (n) aj;p(n)y (12) + 32 j =1;j 6=i p=1
then eqn.(11) can be again considered as a signal plus noise subspace problem, i.e. Rzz (n) = Rss + 32 RIN + 3 ?3T2Tc 2I (13) c i;l
i;l
i;l
X = i;l (n)^ai;l (n) ^ai;l (n)y + m (n)um (n) um (n)y M
m=2
where the rank of Rss (n) is one and the eigenvector ^ai;l corresponding to the largest eigenvalues of Rzz (n) in eqn.(13) is again an accurate array response estimate of one of the resolvable multipath DOA's of the ith mobile as long as L is large enough such that eqn.(12) holds. i
i
where Q(2) = [q1; q2] is the 2x2 eigenvector matrix and 0 v 1 is a memory factor used to deemphasize old data, g = Q^ (k ? 1)T vi (r), f = Gy g, H is the block Householder transformation matrix which de ates the MxM eigen-problem into a noniterative 2x2 quadratic eigenproblem as well as stablizes the subspace tracking [2], and G = diag( jgg j ; jgg j ; ; jgg j ) is the unitary transformation matrix. From eqns.(2) and (5) or eqns.(10) and (13), RIN (n)?1 or RIN (n)?1 can be estimated from their pre- and postcorrelation matrice for both zero and non-zero multipath angle spread cases, i.e. (14) R^ IN (n)?1 = 3L R (n) ? Rzz (n) + 3 ? 2Tc 2 I?1 3L ? 2 xx L 3Tb ? 2Tc which is computationally very expensive. The subtraction in eqn.(14) also poses numerical problems. From eqns.(5) and (13), we nd an alternative method to estimate the inverse of RIN (n) can be expressed by 1
i
i
i
i
i
i
j =1;j 6=i
and L ( Nw NF + Nw NF )=( Nw NF ) = (Nw NF + 1) = 192:75, in which Nw = 4 is the number of PN chips in one Walsh chip, NF = 64 is number of Walsh chips in one Walsh function [4]. The feedback correlation algorithm with NR1SSU is summarized in the block diagram Figure 1. Instead of employing batch covariance matrix processing [7], we employed a rank-one eigendecomposition model which adaptively updates the time-varying data covariance matrix [2] [7] that allows for modelling of the motion of mobiles. Rvv (k) = v Rvv (k ? 1) + (1 ? v )vi(r) vi (r)y ^ (k ? 1)T) = Q^ (k ? 1)GHHy ( v D^ (k ? 1) + f f y)H(Hy Gy Q = Q^ (k ? 1)GH 3 2 v d(s) Ip?1 0 0 7 6 6 0 Q(2) 0s n0 Q(2)y 0 75 4 0 0 v d(n) IM?p?1 Hy Gy Q^ (k ? 1)T 1 4
i
2
1 4
2
1 3
m=1
3 2
um (n) um(n)y (m (n) ? ? TT ) (3 2 3
c)
(15)
2
c
where 1 = 0 and m = m for m = 2; 3; :::;M. This will eliminate the batch processing computation and dramatically reduces the adaptive beamforming weight signal processing. However, accurate eigenvalues estimates will be needed since a small error in m will generate a large error in R^ ?i 1 . There is a trade-o between computation eciency and optimum SINR spatial ltering for the beamforming adaptation algorithm. That is, we choose wi (k) = ai (k). Calibrated Sensor Array
Beamformer
nt
w 1 (k)* i,1
LPF
θ
PN chips correlator
cos( ω t)
fro e av
Pj aj (k) aj (k)y + 21 Nw NF Tc 2 I
1
INi
W
+ 31 Nw NF Tc2
M X
R^ (n)? =
i
N X
M M
i;l
III.Noniterative Rank-One Signal Subspace Update
We remark that the conventional IS-95 CDMA system only gives a \code ltering" processing gain of L =4 [8], which is insucient for equations (6) and (12) to be satis ed. However, this problem may be overcome by feeding back the Walsh function correlation vector vi (r) [4]. In [4] it is shown that with an estimation delay of one frame (20 ms) the feedback Walsh function covariance matrix Rvv will increase the spatial processing gain L to nearly 200. Indeed, [4] shows that Rvv (k) = ( 41 Nw2 NF2 + 14 Nw NF )Tc2 Pi ai (k) ai (k)y
2 2
1
LPF
10 0011 1010
w 1 (k) * i,2
01
11 00
0110
cos( ω t)
00 11 11 00 11 00
00 11 11 00 01
1 Tb
c (t- τ i) i
Viterbi Decoder
Recovered Data Bit Sequences
Walsh function
LPF
w 1 (k)* i,M
3 4
cos( ω t)
Control Algorithm
1001 10 10
if (Weight Update == TRUE) do [ Dv(k) , Qv(k) ] NR1SSU ( Dv (k-1) , Q v (k-1) , vi (r) ) i i i i
pre-
PN chips correlator
x1(t)
1 Tb
LPF
cos( ω t) LPF
cos( ω t)
11 00 00 11 11 00
x 2(t)
c (t- τ i) i 1 Tb
c i (t- τ i)
x (t) M
1 Tb
LPF
cos( ω t)
00 11 01 11 00
c i (t- τ i)
z
0110 00 00 11 11 11 00 00 11
posti,1(n) 6144 bits time
vi,1(r)
delay block of one frame
v
z i,2 (n) 6144 bits time delay block of one frame
i,2 (r)
01 0101 00 11 z
i,M
(n) 6144 bits time delay block of one frame
v
i,M (r) Regenerate the
Walsh last 96 Walsh function function of 64 Walsh chips each
Adaptive Signal Processor
i
Figure 1: IS-95 CDMA NR1SSU beamformingblock diagram for tracking the ith mobile's in phase branch (ith in phase branch) of the network Since in practice, the number of samples are limited, the inverse of the interference-noise covariance matrix is not accurately estimated, the SINR weight vector may not always be steering to the desired mobile's DOA. We instead propose
to update the weight vector with an optimum SNR instead of an optimum SINR criteria (a suboptimal solution for the For the ith mobile of the CDMA network
Initialization phase f1st frame of a new callg, 1 k 0 fweight update indexg 2 [wi1(k); wi2 (k); :::; wip(k)] all ones 3 Rvvi (0) 0 4 for r = 1 to r = 96 fWalsh function indexg 5 do v r?r 1 6 Rvv (r) v Rvv (r ? 1) + (1 ? v )vi(r)vi(r)y ^ v (k); Q^ v (k)] ITER-EIGEN(Rvv (r)) 7 [D ^ v (k); Q^ v (k)) ^ 8 [Dv (k); Q^ v (k)] ORDER(D 9 wi1 (k) ^ai;1(k) Tracking phase ffor the rest of the callg 10 11 12 13 14 15 16
i
i
i
d(s) d(n)
d1
i
i
i
i
?
d2 +d3 + +dM M 1 (s)
d 0 D^ vi (k) 0 d I ? while call 6= end do if ((r modulus (96(update rate)))= 0) then k k + 1 ^ v (k); Q^ v (k)] [D i i NR1SSU(D^ vi (k); Q^ vi (k); vi(r); ) 17 wi (k) ^ai; (k) Principle Eigenvector
(n)
M 1
Flops per weight update per frame Step # # of ops 2 1-3 (4M + 2M + 1) + 5M + 4M 4 0:5M2 + 2:5M + 7 5-14 36 + (4M3 ? 6M2 + 20M - 4) + 2
v
1
1
1 1
2 2
M M
1
j
1
j
12 13 14 15
1 d(s) new (n) dnew " +(MM??2)1 d d(s) 0 new D^ (k) 0 d(n) new IM?1 return D^ (k), Q^ (k) 2
0
0 I
(n)
#
Table 2: The NR1SSU procedure
?
M 2
Table 3: Computation complexity of NR1SSU
IV. Simulation Results We now present simulation results to test the algorithm performance and verify the assumptions in eqn.(6) of our analysis in section II and the proposed algorithm stated in section III. All the simulation results assumed zero-angle spread thus far. Rather than present results for the array response vectors, we have instead calculated the DOA corresponding to a given array response, which lends to easier visual interpretation of the results. Spatial beam pattern: Circular array, IN_P, doa = 1.65,Rvv
Spatial beam pattern: Linear array, QUAD, doa = 42.8,Rvv
0
Normalized to # of antenna elements beam pattern in dB
Table 1: The tracking algorithm for the weight vector update for the ith mobile NR1SSU(D^ p (k-1), Q^ (k-1), y, ) ^ (k ? 1)y y 1 g (1 ?g )Q g g 2 G diag( jg j ; jg j ; ; jg j ) f (s) = f Gyg 3 f (n) 4 H HOUSEHOLDER-DEFLAT (f (s) ; f (n)) [2] 2 2 5 b (d(s) + d(n) ) + f ( s) + f (n) 6 c 2d(s) d(n) + (d(s) f (n) 2 + d(n)p f (s) 2 ) p 7-8 1 21(b + b2 ? 4c);2 12?(b ? b 2 ? 4c) (s) f (s) 0 ? I d (n) (n) 0 d f ; j = 1; 2 9-10 qj (s) d 0 ? I ? f (s) f (n) 0 d(n) ( s ) ( n ) ^ (k); Q^ (k)] 11 [Q ^ (s) (k ? 1)G(s)H(s); Q^ (n)(k ? 1)G(n)H(n) ] [Q [q1 q2 ] 0
0
Normalized to # of antenna elements beam pattern in dB
i
i
weighting vector). Examples of the spatial beam pattern di erence between the SINR and SNR criteria can be found in Figures 2a and 2b where eqn.(14) is used for optimum SINR beamforming. The application of the NR1SSU algorithm to the IS-95 CDMA system is summarized in Table 1, Table 2 and the block diagram representation is shown in Figure 1. In Table 1, only the initialization phase at update time k = 0 needs covariance matrix batch processing to generate the eigenstructure. From time k = 1 onwards, no batch processing covariance matrix calculation is necessary. In Table 3, the number of real oating-point multiplication ( ops) per weight update in function NR1SSU is presented. Comparing table 2 to table 4.4 in [6] , we note that our algorithm requires fewer ops than one power method iteration in [7] if M 6 ((4M ? 1:5)M2 versus 24M2 + O(M2 ) iterations). Note that in step 11, we may further reduce computation to 8:5M2 by updating only one of the noise eigenvectors [1] [2] , however simulations show that this would lead to an inaccurate array response estimate due to the fact that the noise subspace is not perfectly spherical.
−5
−10
−15
−20 optimum SINR using eqn.(14) estimate optimum SINR using eqn.(15) estimate
−5
−10
−15
−20 optimum SINR using eqn.(14) estimate optimum SINR using eqn.(15) estimate
Optimum SNR using array response −25
−150
−100
−50
0 Angle in degree
50
100
Optimum SNR using array response 150
−25
−150
−100
−50
0 Angle in degree
50
100
150
Figure 2: Weight vector spatial beam pattern for the SNR and SINR criteria in a) Circular array, b) Three-sector linear array, both with 14 interfering mobiles and perfect power control receiving at SNR = 7dB From the beam patterns shown in Figures 2a and 2b, the optimum SNR beamforming pattern is close to that of the optimum SINR algorithm. Therefore, we propose to use the optimum SNR beamformer to further reduce computation. In addition, we do not have enough samples to obtain
an accurate estimate of the asymptotic covariance matrix due to the time-varying array response of moving mobiles. Figures 3 and 4 show a Monte-Carlo evaluation of 50 trials over 200 frames of a moving mobile with 9 in-band interfering moving mobiles. In comparison, Figures 5 and 6 show another Monte-Carlo evaluation of a faster moving mobile, having two interferers, with 50 repeated trials over 500 frames. For both simulations, one update is performed per IS-95 frame (192 data bits) for each trial (there are 50 frames per second in IS-95). Note that the DOA is tracked to within 0.25 degrees or three degrees depending on the motion of the mobile.
that eqn.(6) holds. With the proposed NR1SSU algorithm, the computationally expensive formation of a batch covariance matrix is required only on initialization. We propose to update the weight vector wi (k) once per IS-95 frame. However, depending on the application, the weight vector update rate can be adjusted for di erent time-varying scenarios, as shown in Figures 3,4,5, and 6. More frequently weight vector updating reduces time-delay in the array response vector estimate. Root Mean Square Error of Rvv est. with slower mobile movement 3
2.5 In phase DOA est. Quadrature DOA est. 2
RMSE in degree
Mean of Rvv est. with50 trials with slower mobile movement 6.65 6.6 6.55
1.5
1
Mean in degree
6.5 6.45
0.5 6.4 6.35
0 0
6.3
50
100
150 200 250 300 350 Weight update index k in frame
400
450
500
True DOA In phase DOA est.
6.25
Quadrature DOA est. 6.2 6.15 0
20
40
60
80 100 120 140 Weight update index k in frame
160
180
200
Figure 3: Mean of DOA estimates, perfect power control received at SNR = 7dB, 3-element 3-sector linear array with 9 moving interferers updated once per frame with 50 trials Root Mean Square Error of Rvv est. with slower mobile movement 0.25 In phase DOA est. Quadrature DOA est.
RMSE in degree
0.2
0.15
0.1
0.05
0 0
20
40
60
80 100 120 140 Weight update index k in frame
160
180
200
Figure 4: Root Mean Square Error of Figure 3 Mean of Rvv est. with50 trials with slower mobile movement 20
18
Mean in degree
16
14
12 True DOA 10
In phase DOA est. Quadrature DOA est.
8
6 0
50
100
150 200 250 300 350 Weight update index k in frame
400
450
500
Figure 5: Mean of DOA estimates, perfect power control received at SNR = 7dB, 3-element 3-sector linear array with 2 moving interferers updated once per frame with 50 trials
V. Conclusions We have shown that a NR1SSU is a promising technique that may be used to track the desired mobile's array response vectors in an IS-95 CDMA system using Walsh function feedback to ensure a large enough spatial processing gain L such
Figure 6: Root Mean Square Error of DOA estimates in Figure 5
References
[1] J. Bunch, C. Nielsen, and D. Sorensen. Rank-one modi cation of the symmetric eigenproblem. Numer. Math., 3:111{129, 1978. [2] R. D. DeGroat. Noniterative subspace tracking. IEEE Trans. Sig. Proc., 40(3):571{577, 1992. [3] A. M. Earnshaw and S. D. Blostein. \A chip-level IS95compliant cellular CDMA simulator: Design, implementation, and analysis". Technical report, Dept. of Elec. & Comp. Engineering, Queen's University, 1996. [4] A. M. Earnshaw and S. D. Blostein. Investigating the e ects of imperfect digital beamforming on cell capacity in a cellular CDMA communication system. In ICUPC, pages 458{462, 1996. [5] R. A. Monzingo and T. W. Miller. Introduction to Adaptive Arrays. A Wiley-Interscience Publication, 1980. [6] A. F. Naguib. Adaptive Antennas for CDMA Wireless Networks. PhD thesis, Dept. of Electrical Engineering, Stanford University, 1995. [7] A. F. Naguib and A. Paulraj. Recursive adaptive beamforming for wireless cdma. In IEEE ICC, volume 3 of 3, pages 1515{1519, 1995. [8] Qualcomm Inc. Mobile station-base compatibility standard for dual-mode wideband spread system, 1993. [9] G. L. Turin. Introduction to spread-spectrum antimultipath techniques and their application to urban digital radio. Proc. IEEE, 68(3):328{353, 1980. [10] G. L. Turin. The e ects of multipath and fading on the performance of direct-sequence cdma systems. IEEE JSAC, SAC-2(4):597{603, July 1984.
Department of Electrical and Computer Engineering Queen's University, Kingston, Ontario Canada, K7L 3N6 [email protected], [email protected] Tel: (613)545-6561, Fax: (613)545-6615
Abstract We introduce a novel adaptive digital beamforming algorithm for application in cellular CDMA communications systems. A Noniterative Rank-one Signal Subspace eigenstructure Update (NR1SSU) algorithm enables low-complexity recursive processing of beamforming weights. A PN chip-level IS-95 CDMA reference model [3] has been used to test the beamforming algorithm via simulation. Our results show that accurate beamforming can be achieved to moving mobiles at one update per IS-95 frame of 192 data bits.
I. Introduction
Adaptive antenna array beamforming in future multimedia cellular CDMA communication systems holds great potential for improving signal-to-interference-noise ratios and achieving higher cell capacity. An adaptive beamforming network spatially lters the in-band interfering mobiles by adaptively updating the weight vector of each beamformer so as to steer the main beam lobe to the direction-of-arrival (DOA) of the desired mobile. Traditional eigen-based beamforming techniques require iterative and computationally expensive batch processing [5] [7] [6]. Our goal is to achieve higher capacity and coverage in future multimedia cellular CDMA systems using an adaptive antenna array beamforming network at the base station. To further reduce basestation costs, we propose to simplify the required computation by employing a noniterative rank-one signal subspace eigenstructure update algorithm [2]. Updating beamforming weights for each mobile in a multi-beamforming network can be very expensive depending on how the estimates are being calculated [5] [7] [6]. The Signal to Interference-Noise Ratio (SINR) optimum beamforming weight wopt requires (i) the array response of the lth multipath of the desired ith mobile, ai;l 2 CMx1 and (ii) the inverse of the batch processing interference-noise covariance matrix, R?IN1;i;l 2 CMxM , i.e., i;l
w
opt i;l
= R?IN1;i;l ai;l
where ai;l = [a1(i;l (t)); ::::; aM(i;l (t))]T 2 CMx1 is the ith mobile's lth multipath unit array response vector, i;l (t) is the time-varying DOA of the lth multipath of the ith mobile, M is the number of antenna elements, is a scalar, and C is the complex domain. Currently, estimation of (i) & (ii) requires the formation of a pre-correlation autocovariance This research was supported by a grant from the Canadian Institute for Telecommunications Research under the NCE program of the Government of Canada.
matrix Rxx as well as the ith mobile's lth multipath postcorrelation autocovariance matrix Rzz [7] [6]. This proposed adaptive beamforming method makes beamforming in CDMA unattractive and computationally very expensive, since there are 1228800 pre-correlation vectors x(t) per user per second (PN chips per second) and 307200 post-correlation vectors zi (n) per user per second (Walsh chips per second) in a CDMA IS-95 system. It is shown in [7] that R^ IN can be estimated as R^ IN = L ?L 1 (Rxx ? L1 Rzz ) where L is the code ltering processing gain, and ai;l can be estimated as ai;l = R?xx1=2ei;l The iterative power method is used to calculate the principal eigenvector ei;l corresponding to the largest eigenvalue of ?=2R R1=2, where it is shown in [7] that Rxx xx zz i
i;l
i;l
i;l
i
i;l
i
i
i
i;l
R?= R
1=2 1=2 ;i;l R? xx i Rxx i ai;l
= R1xx=2ai;l One drawback of an iterative eigen-decomposition method, such as the power method, is that its computational complexity may be unbounded, depending on the eigenvalue spread. In this paper, we develop a novel application of a computationally ecient and numerically stable Noniterative Rank1 Signal Subspace Update (NR1SSU) for array response estimation in a CDMA system. This application of NR1SSU is developed in section II. In addition, the NR1SSU uses a recursive estimation formulation, which dramatically reduces the computation compared to [7]. The NR1SSU algorithm and application to IS-95 is described in section III, where a detailed computational analysis is provided. Preliminary simulation results are shown in section IV. xx i
2
zz
i
II. System Model A PN chip-level IS-95 CDMA uplink model [4] has been used to test the beamforming algorithm via simulation. We have considered amplitude Rayleigh fading, multipath loss, shadowing, perfect power control in both a circular and a three-sector linear calibrated array IS-95 CDMA system. A. Case of zero multipath angle spread For the case of relatively small multipath angle spread, we will rst assume for simplicity that only one resolvable
DOA from each mobile is received. At time t, the baseband signal vector received at the antenna array network is xi (t) = [xi;1(t); xi;2(t); :::; xi;M(t)]T (1) =
N X
j =1
q
cj (t ? i;j )bj (t ? i;j ) Pj (t)aj (t) + n(t)
where N is the total number of in-band mobiles, cj (t ? i;j ) 2 f1; ?1g is the (square pulse) Pseudo Noise (PN) chips of the j th mobile with a chip period of Tc = 1=1228800 sec, bj (t ? i;j ) 2 f1; ?1g is the j th mobile's information bit sequence in Walsh chip period Tb = 1=307200 sec, j i;j j 2 U (0; Tc ) is the uniformly distributed di erential time delay of the j th mobile relative to the time delay of the ith mobile's time delay received at the base station, column vector n(t) = [n1(t); ::::; nM(t)]T N (0; 2 I) represents independent and identically distributed Gaussian thermal noise, M is the number of antenna elements in the base station array or sector, Pj (t) is the total power received at the base station of the j th mobile's DOA, aj (t) = [a1(j (t)); ::::; aM(j (t))]T 2 CMx1 is the time-varying column vector of the j th mobile's array response, j (t) is the time-varying DOA of the j th mobile, and []T denotes transpose. Without loss of generality, for all i = 1; 2; ::::; N and j = 1; 2; ::::; N, we assume that synchronization to the ith desired mobile possible at each ith branch of the network i.e. i;i = 0, and the signal bj (t ? i;j ), chip cj (t ? i;j ) and noise n(t) are mutually uncorrelated. Bits bj (t ? i;j ) and bk (t ? i;k ) are assumed uncorrelated as well as cj (t ? i;j ) and ck (t ? i;k ) for all k = 1; 2; ::::;N and k 6= j, and array response vectors aj (t) are assumed to be unchanged over one information bit period Tb . Under the above assumptions, the pre-correlation covariance matrix of eqn.(1) becomes
Rxx (t) = Piai(t) ai (t)y + i
N X
j =1;j 6=i
Pj aj (t) aj (t)y + 2 I
(2) = Piai (t) ai (t)y + RIN where RIN denotes the interference-plus-noise covariance matrix, I is the MxM identity matrix, ()y denotes transpose conjugate, and at information bit n the ith mobile's post-correlation vector, zi (n) of xi (t) over time interval Tb is Z T p n(t)ci(t)dt (3) zi(n) = Tb Pi bi(n)ai (n) + p1T b 0 Z T N X p p1 + Pj bj (t ? i;j )cj (t ? i;j )ci (t)aj (n)dt j =1;j 6=i Tb 0 i
i
b
b
The i mobile's post-correlation autocovariance matrix can be de ned as (4) Rzz (n) Efzi (n) zi(n)y g T1 th
i
c
where L = Tb =Tc is the "code ltering" processing gain. Using results in [10], " #2 1 E Z T c ( )c ( ? )d = 2 [ Tb2 ? Tb2 ] ' 2Tb i j i;j Tb Tb 3L 12L2 3L 0 b
After integration, eqn.(4) becomes, Rzz (n) N 2 X = LPiai (n) ai (n)y + 32 Pj aj (n) aj (n)y + T I c j =1;j 6=i (5) = LPiai (n) ai (n)y + 32 RIN + 3 ?3T2Tc 2I c If L is large enough, such that N X LPi ai (n) ai (n)y 32 Pj ai (n) ai (n)y (6) j =1;j 6=i i
i
we can consider eqn.(5) as a signal-plus-noise spherical subspace problem, where N 2 X Pj ai (n) ai (n)y + T I Rzz (n) = Rss (n) + 32 c j =1;j 6=i i
i
X = s (n)^ai (n) ^ai (n)y + m (n)um (n) um (n)y (7) M
m=2
where m are the eigenvalues of Rzz (n) and um (n) are the corresponding eigenvectors for m = 1; 2; :::; M and s = 1 is the signal subspace. In eqn.(7), the signal covariance matrix is rank one with Rss (n) = LPiai (n) ai (n)y = s(n)^ai (n) ^ai(n)y (8) and the interference-noise covariance matrix is N 2 X y + 2 I (9) P a (n) a (n) j j j 3 j =1;j 6=i Tc i
i
which is now approximately a spherical covariance matrix for large processing gain L. The principal eigenvector ^ai (n) corresponding to the largest eigenvalue of Rzz (n) is an accurate signal subspace array response vector estimate when eqn.(6) holds. B. Case of non-zero multipath angle spread In the case of non-zero multipath angle spread, with the assumption that all paths are assumed to be resolvable [9], the pre- and post-correlation covariance matrices with a maximum of P resolvable multipaths can be generalized to i
R
xx i;l
(t) =
N P X X
j =1 p=1
Pj;paj;p (t) ai;p (t)y + 2 I
= Pi;l ai;l (n) ai;l (n)y + RIN and the post-correlation covariance matrix is Rzz (n) = LPi;l ai;l (n) ai;l(n)y P X + 32 Pi;p ai;p (n) ai;p (n)y p=1;p6=l
i;l
(10)
i;l
+
P N X 2 X y + 2 I (11) P a (n) a (n) j;p j;p j;p 3 j =1;j 6=i p=1 Tc
where some of the Pj;p or Pi;p may be negligible. If L is large enough that P X LPi;l ai;l (n) ai;l (n)y 32 Pi;p ai;p (n) ai;p (n)y p=1;p6=l
N P X X Pj;paj;p (n) aj;p(n)y (12) + 32 j =1;j 6=i p=1
then eqn.(11) can be again considered as a signal plus noise subspace problem, i.e. Rzz (n) = Rss + 32 RIN + 3 ?3T2Tc 2I (13) c i;l
i;l
i;l
X = i;l (n)^ai;l (n) ^ai;l (n)y + m (n)um (n) um (n)y M
m=2
where the rank of Rss (n) is one and the eigenvector ^ai;l corresponding to the largest eigenvalues of Rzz (n) in eqn.(13) is again an accurate array response estimate of one of the resolvable multipath DOA's of the ith mobile as long as L is large enough such that eqn.(12) holds. i
i
where Q(2) = [q1; q2] is the 2x2 eigenvector matrix and 0 v 1 is a memory factor used to deemphasize old data, g = Q^ (k ? 1)T vi (r), f = Gy g, H is the block Householder transformation matrix which de ates the MxM eigen-problem into a noniterative 2x2 quadratic eigenproblem as well as stablizes the subspace tracking [2], and G = diag( jgg j ; jgg j ; ; jgg j ) is the unitary transformation matrix. From eqns.(2) and (5) or eqns.(10) and (13), RIN (n)?1 or RIN (n)?1 can be estimated from their pre- and postcorrelation matrice for both zero and non-zero multipath angle spread cases, i.e. (14) R^ IN (n)?1 = 3L R (n) ? Rzz (n) + 3 ? 2Tc 2 I?1 3L ? 2 xx L 3Tb ? 2Tc which is computationally very expensive. The subtraction in eqn.(14) also poses numerical problems. From eqns.(5) and (13), we nd an alternative method to estimate the inverse of RIN (n) can be expressed by 1
i
i
i
i
i
i
j =1;j 6=i
and L ( Nw NF + Nw NF )=( Nw NF ) = (Nw NF + 1) = 192:75, in which Nw = 4 is the number of PN chips in one Walsh chip, NF = 64 is number of Walsh chips in one Walsh function [4]. The feedback correlation algorithm with NR1SSU is summarized in the block diagram Figure 1. Instead of employing batch covariance matrix processing [7], we employed a rank-one eigendecomposition model which adaptively updates the time-varying data covariance matrix [2] [7] that allows for modelling of the motion of mobiles. Rvv (k) = v Rvv (k ? 1) + (1 ? v )vi(r) vi (r)y ^ (k ? 1)T) = Q^ (k ? 1)GHHy ( v D^ (k ? 1) + f f y)H(Hy Gy Q = Q^ (k ? 1)GH 3 2 v d(s) Ip?1 0 0 7 6 6 0 Q(2) 0s n0 Q(2)y 0 75 4 0 0 v d(n) IM?p?1 Hy Gy Q^ (k ? 1)T 1 4
i
2
1 4
2
1 3
m=1
3 2
um (n) um(n)y (m (n) ? ? TT ) (3 2 3
c)
(15)
2
c
where 1 = 0 and m = m for m = 2; 3; :::;M. This will eliminate the batch processing computation and dramatically reduces the adaptive beamforming weight signal processing. However, accurate eigenvalues estimates will be needed since a small error in m will generate a large error in R^ ?i 1 . There is a trade-o between computation eciency and optimum SINR spatial ltering for the beamforming adaptation algorithm. That is, we choose wi (k) = ai (k). Calibrated Sensor Array
Beamformer
nt
w 1 (k)* i,1
LPF
θ
PN chips correlator
cos( ω t)
fro e av
Pj aj (k) aj (k)y + 21 Nw NF Tc 2 I
1
INi
W
+ 31 Nw NF Tc2
M X
R^ (n)? =
i
N X
M M
i;l
III.Noniterative Rank-One Signal Subspace Update
We remark that the conventional IS-95 CDMA system only gives a \code ltering" processing gain of L =4 [8], which is insucient for equations (6) and (12) to be satis ed. However, this problem may be overcome by feeding back the Walsh function correlation vector vi (r) [4]. In [4] it is shown that with an estimation delay of one frame (20 ms) the feedback Walsh function covariance matrix Rvv will increase the spatial processing gain L to nearly 200. Indeed, [4] shows that Rvv (k) = ( 41 Nw2 NF2 + 14 Nw NF )Tc2 Pi ai (k) ai (k)y
2 2
1
LPF
10 0011 1010
w 1 (k) * i,2
01
11 00
0110
cos( ω t)
00 11 11 00 11 00
00 11 11 00 01
1 Tb
c (t- τ i) i
Viterbi Decoder
Recovered Data Bit Sequences
Walsh function
LPF
w 1 (k)* i,M
3 4
cos( ω t)
Control Algorithm
1001 10 10
if (Weight Update == TRUE) do [ Dv(k) , Qv(k) ] NR1SSU ( Dv (k-1) , Q v (k-1) , vi (r) ) i i i i
pre-
PN chips correlator
x1(t)
1 Tb
LPF
cos( ω t) LPF
cos( ω t)
11 00 00 11 11 00
x 2(t)
c (t- τ i) i 1 Tb
c i (t- τ i)
x (t) M
1 Tb
LPF
cos( ω t)
00 11 01 11 00
c i (t- τ i)
z
0110 00 00 11 11 11 00 00 11
posti,1(n) 6144 bits time
vi,1(r)
delay block of one frame
v
z i,2 (n) 6144 bits time delay block of one frame
i,2 (r)
01 0101 00 11 z
i,M
(n) 6144 bits time delay block of one frame
v
i,M (r) Regenerate the
Walsh last 96 Walsh function function of 64 Walsh chips each
Adaptive Signal Processor
i
Figure 1: IS-95 CDMA NR1SSU beamformingblock diagram for tracking the ith mobile's in phase branch (ith in phase branch) of the network Since in practice, the number of samples are limited, the inverse of the interference-noise covariance matrix is not accurately estimated, the SINR weight vector may not always be steering to the desired mobile's DOA. We instead propose
to update the weight vector with an optimum SNR instead of an optimum SINR criteria (a suboptimal solution for the For the ith mobile of the CDMA network
Initialization phase f1st frame of a new callg, 1 k 0 fweight update indexg 2 [wi1(k); wi2 (k); :::; wip(k)] all ones 3 Rvvi (0) 0 4 for r = 1 to r = 96 fWalsh function indexg 5 do v r?r 1 6 Rvv (r) v Rvv (r ? 1) + (1 ? v )vi(r)vi(r)y ^ v (k); Q^ v (k)] ITER-EIGEN(Rvv (r)) 7 [D ^ v (k); Q^ v (k)) ^ 8 [Dv (k); Q^ v (k)] ORDER(D 9 wi1 (k) ^ai;1(k) Tracking phase ffor the rest of the callg 10 11 12 13 14 15 16
i
i
i
d(s) d(n)
d1
i
i
i
i
?
d2 +d3 + +dM M 1 (s)
d 0 D^ vi (k) 0 d I ? while call 6= end do if ((r modulus (96(update rate)))= 0) then k k + 1 ^ v (k); Q^ v (k)] [D i i NR1SSU(D^ vi (k); Q^ vi (k); vi(r); ) 17 wi (k) ^ai; (k) Principle Eigenvector
(n)
M 1
Flops per weight update per frame Step # # of ops 2 1-3 (4M + 2M + 1) + 5M + 4M 4 0:5M2 + 2:5M + 7 5-14 36 + (4M3 ? 6M2 + 20M - 4) + 2
v
1
1
1 1
2 2
M M
1
j
1
j
12 13 14 15
1 d(s) new (n) dnew " +(MM??2)1 d d(s) 0 new D^ (k) 0 d(n) new IM?1 return D^ (k), Q^ (k) 2
0
0 I
(n)
#
Table 2: The NR1SSU procedure
?
M 2
Table 3: Computation complexity of NR1SSU
IV. Simulation Results We now present simulation results to test the algorithm performance and verify the assumptions in eqn.(6) of our analysis in section II and the proposed algorithm stated in section III. All the simulation results assumed zero-angle spread thus far. Rather than present results for the array response vectors, we have instead calculated the DOA corresponding to a given array response, which lends to easier visual interpretation of the results. Spatial beam pattern: Circular array, IN_P, doa = 1.65,Rvv
Spatial beam pattern: Linear array, QUAD, doa = 42.8,Rvv
0
Normalized to # of antenna elements beam pattern in dB
Table 1: The tracking algorithm for the weight vector update for the ith mobile NR1SSU(D^ p (k-1), Q^ (k-1), y, ) ^ (k ? 1)y y 1 g (1 ?g )Q g g 2 G diag( jg j ; jg j ; ; jg j ) f (s) = f Gyg 3 f (n) 4 H HOUSEHOLDER-DEFLAT (f (s) ; f (n)) [2] 2 2 5 b (d(s) + d(n) ) + f ( s) + f (n) 6 c 2d(s) d(n) + (d(s) f (n) 2 + d(n)p f (s) 2 ) p 7-8 1 21(b + b2 ? 4c);2 12?(b ? b 2 ? 4c) (s) f (s) 0 ? I d (n) (n) 0 d f ; j = 1; 2 9-10 qj (s) d 0 ? I ? f (s) f (n) 0 d(n) ( s ) ( n ) ^ (k); Q^ (k)] 11 [Q ^ (s) (k ? 1)G(s)H(s); Q^ (n)(k ? 1)G(n)H(n) ] [Q [q1 q2 ] 0
0
Normalized to # of antenna elements beam pattern in dB
i
i
weighting vector). Examples of the spatial beam pattern di erence between the SINR and SNR criteria can be found in Figures 2a and 2b where eqn.(14) is used for optimum SINR beamforming. The application of the NR1SSU algorithm to the IS-95 CDMA system is summarized in Table 1, Table 2 and the block diagram representation is shown in Figure 1. In Table 1, only the initialization phase at update time k = 0 needs covariance matrix batch processing to generate the eigenstructure. From time k = 1 onwards, no batch processing covariance matrix calculation is necessary. In Table 3, the number of real oating-point multiplication ( ops) per weight update in function NR1SSU is presented. Comparing table 2 to table 4.4 in [6] , we note that our algorithm requires fewer ops than one power method iteration in [7] if M 6 ((4M ? 1:5)M2 versus 24M2 + O(M2 ) iterations). Note that in step 11, we may further reduce computation to 8:5M2 by updating only one of the noise eigenvectors [1] [2] , however simulations show that this would lead to an inaccurate array response estimate due to the fact that the noise subspace is not perfectly spherical.
−5
−10
−15
−20 optimum SINR using eqn.(14) estimate optimum SINR using eqn.(15) estimate
−5
−10
−15
−20 optimum SINR using eqn.(14) estimate optimum SINR using eqn.(15) estimate
Optimum SNR using array response −25
−150
−100
−50
0 Angle in degree
50
100
Optimum SNR using array response 150
−25
−150
−100
−50
0 Angle in degree
50
100
150
Figure 2: Weight vector spatial beam pattern for the SNR and SINR criteria in a) Circular array, b) Three-sector linear array, both with 14 interfering mobiles and perfect power control receiving at SNR = 7dB From the beam patterns shown in Figures 2a and 2b, the optimum SNR beamforming pattern is close to that of the optimum SINR algorithm. Therefore, we propose to use the optimum SNR beamformer to further reduce computation. In addition, we do not have enough samples to obtain
an accurate estimate of the asymptotic covariance matrix due to the time-varying array response of moving mobiles. Figures 3 and 4 show a Monte-Carlo evaluation of 50 trials over 200 frames of a moving mobile with 9 in-band interfering moving mobiles. In comparison, Figures 5 and 6 show another Monte-Carlo evaluation of a faster moving mobile, having two interferers, with 50 repeated trials over 500 frames. For both simulations, one update is performed per IS-95 frame (192 data bits) for each trial (there are 50 frames per second in IS-95). Note that the DOA is tracked to within 0.25 degrees or three degrees depending on the motion of the mobile.
that eqn.(6) holds. With the proposed NR1SSU algorithm, the computationally expensive formation of a batch covariance matrix is required only on initialization. We propose to update the weight vector wi (k) once per IS-95 frame. However, depending on the application, the weight vector update rate can be adjusted for di erent time-varying scenarios, as shown in Figures 3,4,5, and 6. More frequently weight vector updating reduces time-delay in the array response vector estimate. Root Mean Square Error of Rvv est. with slower mobile movement 3
2.5 In phase DOA est. Quadrature DOA est. 2
RMSE in degree
Mean of Rvv est. with50 trials with slower mobile movement 6.65 6.6 6.55
1.5
1
Mean in degree
6.5 6.45
0.5 6.4 6.35
0 0
6.3
50
100
150 200 250 300 350 Weight update index k in frame
400
450
500
True DOA In phase DOA est.
6.25
Quadrature DOA est. 6.2 6.15 0
20
40
60
80 100 120 140 Weight update index k in frame
160
180
200
Figure 3: Mean of DOA estimates, perfect power control received at SNR = 7dB, 3-element 3-sector linear array with 9 moving interferers updated once per frame with 50 trials Root Mean Square Error of Rvv est. with slower mobile movement 0.25 In phase DOA est. Quadrature DOA est.
RMSE in degree
0.2
0.15
0.1
0.05
0 0
20
40
60
80 100 120 140 Weight update index k in frame
160
180
200
Figure 4: Root Mean Square Error of Figure 3 Mean of Rvv est. with50 trials with slower mobile movement 20
18
Mean in degree
16
14
12 True DOA 10
In phase DOA est. Quadrature DOA est.
8
6 0
50
100
150 200 250 300 350 Weight update index k in frame
400
450
500
Figure 5: Mean of DOA estimates, perfect power control received at SNR = 7dB, 3-element 3-sector linear array with 2 moving interferers updated once per frame with 50 trials
V. Conclusions We have shown that a NR1SSU is a promising technique that may be used to track the desired mobile's array response vectors in an IS-95 CDMA system using Walsh function feedback to ensure a large enough spatial processing gain L such
Figure 6: Root Mean Square Error of DOA estimates in Figure 5
References
[1] J. Bunch, C. Nielsen, and D. Sorensen. Rank-one modi cation of the symmetric eigenproblem. Numer. Math., 3:111{129, 1978. [2] R. D. DeGroat. Noniterative subspace tracking. IEEE Trans. Sig. Proc., 40(3):571{577, 1992. [3] A. M. Earnshaw and S. D. Blostein. \A chip-level IS95compliant cellular CDMA simulator: Design, implementation, and analysis". Technical report, Dept. of Elec. & Comp. Engineering, Queen's University, 1996. [4] A. M. Earnshaw and S. D. Blostein. Investigating the e ects of imperfect digital beamforming on cell capacity in a cellular CDMA communication system. In ICUPC, pages 458{462, 1996. [5] R. A. Monzingo and T. W. Miller. Introduction to Adaptive Arrays. A Wiley-Interscience Publication, 1980. [6] A. F. Naguib. Adaptive Antennas for CDMA Wireless Networks. PhD thesis, Dept. of Electrical Engineering, Stanford University, 1995. [7] A. F. Naguib and A. Paulraj. Recursive adaptive beamforming for wireless cdma. In IEEE ICC, volume 3 of 3, pages 1515{1519, 1995. [8] Qualcomm Inc. Mobile station-base compatibility standard for dual-mode wideband spread system, 1993. [9] G. L. Turin. Introduction to spread-spectrum antimultipath techniques and their application to urban digital radio. Proc. IEEE, 68(3):328{353, 1980. [10] G. L. Turin. The e ects of multipath and fading on the performance of direct-sequence cdma systems. IEEE JSAC, SAC-2(4):597{603, July 1984.
