Reduction of Intracell and Intercell Interference for Array MC-DS-CDMA David J. Sadler and A. Manikas Communications and Signal Processing Research Group Department of Electrical and Electronic Engineering Imperial College London London, SW7 2BT, U.K. E-mail:
[email protected]
Abstract— This paper is concerned with asynchronous, multicarrier DS-CDMA communications using a receiver with an antenna array. An estimator is proposed to find the composite channel vector for each intracell user; the only prerequisite being that each user’s spreading sequence is available at the receiver. In the uplink the base station typically does not have knowledge of the spreading sequences active in the neighbouring cells so that intercell interference must be blindly suppressed. Consequently, an implementation of the whitening zero-forcing receiver that reduces both intracell and intercell interference is proposed which only requires knowledge of the intracell users’ spreading sequences. The effectiveness of this receiver is demonstrated by computer simulations.
NOTATION a a A L {·} dim {·} T (·)
scalar vector matrix range space dimension transpose
H
(·) ⊗ exp (a) IN 0N ·
Hermitian transpose Kronecker product elementwise exponential N × N identity matrix N element zero vector round down to integer
I. INTRODUCTION The fading radio channel is a fundamental problem associated with mobile communication systems which can be ameliorated by diversity reception. Several varieties of multicarrier (MC) code-division multiple-access (CDMA) have recently been proposed to exploit the channel’s frequency diversity. In contrast to single carrier (SC) direct-sequence (DS) CDMA systems where the RAKE receiver accesses frequency diversity, multicarrier signalling can provide diversity without a RAKE structure—assuming the fading processes associated with different subcarriers are not fully correlated [1]. Furthermore, MC-DS-CDMA can outperform SC-DSCDMA even if the diversity order of the channel is the same in both cases [2]. In this paper MC-DS-CDMA is applied within a system where the base station employs an antenna array, so that processing is performed jointly in the spatial, temporal and multicarrier domains. Papers [3]–[5] are representative examples of work which combines MC-DS-CDMA with antenna arrays. Specifically in [3] and [4] receiver structures are proposed with a spatial weight for each subcarrier, the weights being adapted to reduce the multiple access interference (MAI). In
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[5] the authors consider a more comprehensive system model which includes frequency selective fading for each subcarrier, then a suitable blind zero-forcing receiver is proposed for this environment. If intercell transmissions reach the array receiver in addition to the wanted intracell signals then greater interference will occur. The intercell interference will be particularly significant when intercell users are near the cell boundary and when the frequency reuse factor is equal to one. The multiuser detection problem in this case is to detect the intracell signals whilst blindly suppressing the intercell interference and without requiring knowledge of the exact intercell pseudonoise (PN) sequences (typically not available at the base station). This paper will provide a solution to this problem using blind estimation techniques that are applied within a whitening zeroforcing receiver. Following this section the model of the MC-DS-CDMA system is described. In Section III blind composite channel vector estimation is detailed. Implementation of the whitening zero-forcing receiver is detailed in Section IV and supporting computer simulations are in Section V. The paper is concluded in Section VI. II. SYSTEM MODEL The baseband DS-CDMA signal for the ith transmitter is mi (t) =
+∞
bi [n] cP N,i (t − nTcs ) ,
(1)
n=−∞
where {bi [n] ∈ C} is the sequence of unity magnitude channel symbols, Tcs is the channel symbol period and nTcs ≤ t < (n + 1) Tcs . A single period of the PN-signal is modelled by cP N,i (t) =
N c −1
αi [m] pc (t − mTc ) ,
(2)
m=0
where {αi [m] ∈ ±1} is the ith user’s PN-sequence of length Nc = Tcs /Tc chips, pc (t) is a rectangular chip pulse of duration Tc and mTc ≤ t < (m + 1) Tc . The DS-CDMA signal modulates Nsc subcarriers, with the baseband frequency of the k th subcarrier relative to the lowest subcarrier frequency being Fk = F0 + k/Tc where k = 0, 1, . . . , Nsc − 1. The modulated subcarriers are summed
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Downconversion
Sampler
BÐ>Ñ
Tapped delay lines of length #P
X
=
X
=
X
=
Ð4#1J >Ñ
exp
Weight application
-
X
=
Ð4#1J >Ñ
exp
X
=
X
–L
=
Weight calculation
-
Base station array receiver block diagram.
complex received signal vector is x (t) =
PT x,i exp (j (2πFc t + ζi )) exp (j2πFk t) mi (t)
(3) in which PT x,i is the transmitted power, Fc is the carrier frequency and ζi is a random phase offset relative to the base station receiver. Each subcarrier experiences a multipath dispersive radio channel which is assumed to be quasi-stationary. Consequently, the array complex baseband channel impulse response for the k th subcarrier, j th path of the ith user can be represented by cijk (t) = βijk S ijk δ (t − τij ) .
(4)
βijk is the complex path coefficient which encompasses fading and a random phase shift. The path delay is τij and S ijk S (θij , Fk ) is the array manifold vector at a frequency of Fc + Fk for the path arriving at an azimuth angle of θij . Without a loss of generality, the users are restricted to the horizontal (x-y) plane. The form of S ijk for an N element (not necessarily linear) array of omnidirectional antennas is T ∈ C N ×1 S ijk = exp −j 2π(Fc c +Fk ) [r1 , r2 , . . . , rN ] uij (5) where the vectors r1 , r2 , . . . , rN contain the antenna positions in metres using Cartesian co-ordinates, T uij = [cos (θij ) , sin (θij ) , 0] is a unit vector pointing in the direction θij , and c is the speed of light. At the base station antenna array, see Fig. 1, the superimposed radio signals for all users, paths and subcarriers will be received. Let Min represent the number of intracell users and Mout the number of significant intercell interferers, with M = Min + Mout. After removal of the carrier the N × 1
Ki N M sc −1
βijk S ijk
(6)
i=1 j=1 k=0
k=0
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3
38
=
and then upconverted to the carrier frequency to produce the transmitted radio frequency signal yi (t) =
Ö,s Ò8Ó× 3œ"áQ
.Ò8Ó
-=
X
X
Fig. 1.
N sc −1
BÒ8Ó
Detection
· exp (j2πFk (t − τij )) mi (t − τij ) + n (t) where βijk has been redefined to include the complex factor PT x,i exp (j (ζi − 2πFc τij )), and n (t) represents complex additive white Gaussian noise (AWGN) of power σn2 . The received signal is discretized by a bank of N samplers operating at a rate of 1/Ts where Ts = Tc /qNsc and q is the oversampling factor. This implies that the path delay is modelled as τij = (lij + ρij ) Ts with lij ∈ {0, 1, . . . , L − 1} and ρij ∈ [0, 1), where L = qNc Nsc is the number of temporal samples per symbol. This means that the model can accommodate path delays that lie within the range [0, Tcs ). The samples pass into a bank of N tapped delay lines of length 2L, necessary due to the asynchronous transmissions. A discrete vector x [n] is formed by concatenating the contents of the tapped delay lines for all antennas and reading the entries every symbol period. This multicarrier space-time received signal vector contains the signals associated with the nth (current) symbol, furthermore, contributions from the previous and next data symbols are present due to the lack of synchronization. A complete representation is given by previous
x [n] =
Ki N M sc −1
L βijk S ijk ⊗ JT Jlij aik [lij ] bi [n − 1]
i=1 j=1 k=0
+ Jlij aik [lij ] bi [n] + JL Jlij aik [lij ] bi [n + 1] + n [n] current
next
(7) Note that the complex factor exp (−j2πFk ρij Ts ) due to the fractional path delay has also been absorbed into the path
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coefficient, βijk . Furthermore, T (8) aik [ ] = aik [0, ] , aik [1, ] , . . . , aik [L − 1, ] , 0TL is the temporal vector for the k th subcarrier of the ith user, for paths arriving with an integer delay of sample periods, and contains elements defined by
m aik [m, ] = αi exp (j2πFk (m − ) Ts ) . (9) qNsc In Equation 7 J is a 2L × 2L shift matrix which is used to represent the different multipath delays in a compact way,
T 02L−1 0 J= . (10) I2L−1 02L−1 To simplify the representation of x [n] the multicarrier spatio-temporal array (STAR) manifold vector for the k th subcarrier, j th path of the ith user is defined:
hijk S ijk ⊗ Jlij aik [lij ] ∈ C 2N L×1 , (11) so that x [n] is written more compactly as M bi [n − 1] (prev) (next) Hi x [n] = β i , Hi β i , Hi β i bi [n] +n [n] i=1 bi [n + 1] (12) where the vector β i ∈ C Ki Nsc ×1 contains the path coefficients βijk ∀j, k and the matrix Hi ∈ C 2N L×Ki Nsc has columns containing the multicarrier STAR manifold vectors hijk [n] ∀j, k. (prev)
(next)
= J(prev) H and H = J(next) Hi Furthermore, Hi
T L i (next)i (prev) and J with J = IN ⊗ J = IN ⊗ JL . Finally, the composite channel vector for the ith user is a linear combination of the multicarrier STAR manifold vectors defined by (13) hi Hi β i ∈ C 2N L×1 ,
so that the composite channel matrix is H = [h1 , h2 , . . . , hM ]. Hence, (14) x [n] = Hb [n] + n [n] , (prev) where H = J H, H, J(next) H and b [n] contains the corresponding previous, current and next symbols from all the users. Based on the above modelling, the objective is to design a weight matrix W ∈ C 2N L×3Min which coherently combines wanted signal contributions for each of the Min intracell users whilst simultaneously reducing interference. III. COMPOSITE CHANNEL VECTOR ESTIMATION In this section an estimator for the composite channel vector, hi of Equation 13, is described which does not require the transmission of a training sequence and is resistant to interference from other received signals. The technique proposed uses two non-orthogonal constraint subspaces to uniquely identify a normalized vector hi that approximates hi up to a complex scaling factor, i.e. hi ≈ γi hi with γi ∈ C. The first constraint subspace is identified from the second order statistics of x [n]. Performing eigen-decomposition on
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the received signal covariance matrix and partitioning into the signal and noise subspaces produces H Rxx = Es Ds EH s + En Dn En .
(15)
All paths for a particular user will occupy a single dimension of the signal subspace. Hence, the overall dimensionality of the signal subspace is 3M , one dimension per previous, current and next data symbol for every user. A consequence of this fact is that Ds is a diagonal matrix with the 3M principle eigenvalues of Rxx on its main diagonal. Similarly, Dn is a diagonal matrix with the other 2N L − 3M eigenvalues on its main diagonal. The Es and En matrices contain the corresponding normalized eigenvectors. The composite channel vector for each user is in the range space of Es , so the signal subspace constraint is defined by hi ∈ L {Es }. The matrix projector into the signal subspace is Ps = Es EH s
(16)
since the columns of Es are orthonormal. In order to identify the second subspace constraint based on the code for the ith user and the structure of the multicarrier signal, the code submatrix of the k th subcarrier for paths arriving with an integer delay of Ts is defined as
Cik [ ] = IN ⊗ J aik [ ] ∈ C 2N L×N . (17) The overall code matrix for the ith user is denoted Ci and is created by concatenating all of the submatrices Cik [ ] for k = 0, 1, . . . , Nsc − 1 and = 0, 1, . . . , L − 1. Additionally, the composite array response subvector of the k th subcarrier is βijk S ijk . (18) sik [ ] = all paths with lij =
It should be noted that for any particular integer delay Ts there may be no paths present, a single path, or multiple paths. The total composite array response vector for the ith user, si , is formed by stacking up all of the subvectors sik [ ] for k = 0, 1, . . . , Nsc − 1 and = 0, 1, . . . , L − 1. Using the above definitions it is possible to represent the composite channel vector as hi = Hi β i = Ci si .
(19)
Equation 19 indicates that hi lies in the range space of Ci , i.e. hi ∈ L {Ci }. The operator that projects into this subspace is generated by
† H Ci , (20) Pi = Ci CH i Ci †
where the pseudoinverse, (·) , is required because Ci does not have full column rank. The approach proposed to estimate the composite channel vector applies the theorem of alternating projection to find the intersection of the two constraint subspaces [6]. Formulating the iterative algorithm:
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[p+1] [p] = Pi Ps hi , hi
(21)
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10
1
V. COMPUTER SIMULATIONS For the results presented in this section, Gold codes of length 15 have been used for the PN-sequences and differential QPSK modulation was used to overcome the phase ambiguity due to blind estimation. Other system parameters include the use of 2 subcarriers and 2 receiving antennas spaced by half a wavelength. A 500 symbol block size was utilized in the processing and the oversampling factor was unity. All users’ transmissions were subjected to a 3 path, dispersive channel. A 120 ˚ sector was considered, with the paths for each user clustered around a nominal user direction generated from a uniform pdf within this sector. The directions of the three paths for a particular user were then generated from a Gaussian pdf with a 10 ˚ standard deviation and a mean equal to the nominal user direction. Path delays were set by a uniform pdf in the range of 0 to Tcs seconds. The magnitude of each path coefficient was drawn from a Rayleigh pdf and the phase was set with a uniform pdf. In Fig. 3 the normalized RMS error for the estimated received power of user-1 is plotted against the total number of users at an Eb /N0 of 9 dB. When the near-far ratio is 0 dB, i.e. the received power of all users is equal, the subspace method outperforms the least squares method. The difference in performance is even more obvious when the nearfar ratio is 10 dB (user-1 received with 10 dB less power than each of the other signals). This supports the claim that the subspace estimator is near-far resistant and should be used in the proposed receiver. The results in Fig. 4 demonstrate the performance of three different receivers when there are 12 intracell users and 4 significant intercell interferers. In all cases the composite channel vectors have been blindly estimated. The 3D RAKE receiver is generally ineffective due to uncancelled interference. The intracell zero-forcing receiver is found by using Equation 31 and setting Rn,out = I2N L so that intracell MAI is cancelled but intercell interference is ignored. This receiver is seen to outperform the 3D RAKE but it is inferior to the proposed whitening zero-forcing receiver when the Eb /N0 is above 6 dB. Consequently, the proposed receiver is demonstrated to be the most suitable when intercell interference is present. VI. CONCLUSION In this paper an array MC-DS-CDMA communications system has been described. The alternating projections algorithm is applied in a new subspace based composite channel vector estimator. Furthermore, a received power estimator has been detailed which—together with the composite channel estimator—are applied to blindly formulate a whitening zeroforcing receiver that cancels intracell ISI and MAI, and suppresses intercell interference. Simulation results demonstrate the effectiveness of the estimators and the proposed receiver, even in the presence of strong interference. ACKNOWLEDGEMENT The authors would like to thank Roke Manor Research Ltd. and the EPSRC for their support of this work.
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0 dB near-far ratio 10 dB near-far ratio
ro rre no tia 10 m tis er e ow p S M R de 10 izl a m ro N
Least squares method
0
-1
10
Subspace method
-2
1
2 Fig. 3.
4 Number of users
8
16
Accuracy of the power estimators.
10
0
10
-1
R EB 10
-2
10
3D RAKE Intracell zero-forcing (ignoring intercell interference) Proposed zero-forcing
-3
0
2
Fig. 4.
4
6
8
10 12 Eb/N0 [dB]
14
16
18
20
Comparative performance of different receivers.
R EFERENCES [1] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans. Commun., vol. 44, no. 2, pp. 238–246, Feb. 1996. [2] W. Xu and L. B. Milstein, “On the performance of multicarrier RAKE systems,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1812–1823, Oct. 2001. [3] T. M. Lok, T. F. Wong and J. S. Lehnert, “Blind adaptive signal reception for MC-CDMA systems in Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, no. 3, pp. 464–471, Mar. 1999. [4] Y. Sanada, M. Padilla and K. Araki, “Performance of adaptive array antennas with multicarrier DS/CDMA in a mobile fading environment,” IEICE Trans. Commun., vol. E81-B, no. 7, pp. 1392–1400, July 1998. [5] D. J. Sadler and A. Manikas, “Blind reception of multicarrier DS-CDMA using antenna arrays,” accepted for publication in IEEE Trans. Wireless Commun. [6] H. Stark and Y. Yang, Vector Space Projections, 1st ed., New York: John Wiley & Sons, 1998. [7] S. E. Bensley and B. Aazhang, “Subspace-based channel estimation for code division multiple access communication systems,” IEEE Trans. Commun., vol. 44, no. 8, pp. 1009–1020, Aug. 1996. [8] A. Klein, G. K. Kaleh and P. W. Baier, “Zero forcing and minimum mean-square-error equalization for multiuser detection in code-division multiple-access channels,” IEEE Trans. Veh. Technol., vol. 45, no. 2, pp. 276–297, May 1996.
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