Joint Transitter-Receiver Beaforing in Downlink Cyclic Prefix-free Spatio-Teporal MC-CDMA Ho Huat Peh, Athanassios Manikas, Tjeng Thiang Tjhung and Wai-Choong Wong Institute for Infoco Research, A*STAR, Singapore Eail: {hhpeh, tjhungtt}@i2r.a-star.edu.sg Departent of Electrical and Electronic Engineering, Iperial College London, UK Eail: a.anikas@iperial.ac.uk Departent of Electrical and Coputer Engineering, National University of Singapore, Singapore Eail: elewwcl@nus.edu.sg Abstract The proble of jointly optiizing the transitter and receiver beaforing weights in the downlink of a cyclic prefix-free MC-CDMA syste over ultipath fading channels is addressed in this paper. Both the base station and user terinals are equipped with antenna arrays to leverage on the spatial inforation provided by the antenna arrays. The precoding and equalization atrices are designed with the ai of iniizing the overall ean-squared-error (MSE of the syste subject to a power constraint for each transitter antenna, which has the additional benefit of reducing the probability of clipping of the transitted signals fro the transitter. This is highly favorable for a MC-CDMA syste due to the ulticarrier odulation involved. The perforance of the optiization process is supported by siulation results. NOTATION a Scalar. a,a Colun vector. A Matrix. I N N N identity atrix. O M N M N zero atrix. N N eleent colun vector of zeros. ι N,k N 1 vector with 1 at the k th eleent and s elsewhere. ( H Heritian transpose. ( T Transpose. Kronecker product. diag(a Diagonalization of vector a. exp(a Eleentwise exponential of vector a. tr (A Trace of atrix A. I. INTRODUCTION In future counication systes, high data rates are required to support the anticipated ulti-edia intensive applications. Multiple-Input Multiple-Output (MIMO systes are an effective strategy to achieve the high data rates required in such applications. This is because MIMO systes take advantage of the spatial diversity that is presented through the use of ultiple antennas, and exploit the ultipath scattering present in the wireless channel which can be achieved by techniques such as spatial ultiplexing, e.g. V-BLAST [1] and space-tie coding [2]. In addition, ulti-carrier (MC odulation techniques such as orthogonal frequency division ultiplexing (OFDM and MC code division ultiple access (MC-CDMA [3], are expected to be ipleented in future MIMO systes, such as MIMO MC-CDMA systes, due to their ability to overcoe the detriental effects of the wireless channel such as ultipath effects and frequencyselective fading. In downlink MIMO systes, there has been considerable interest in the optiization of the transitter and receivers in ultiple access systes. In [4], a variety of design criteria were unified in a generalized fraework for the joint optiization process of a MC-MIMO syste. In MC-based systes, the peak-to-average power ratio (PAPR is an iportant area of concern as the peak transit power, often liited by regulatory or application restrictions [5], would have to be clipped when the threshold is exceeded. It was shown in [4] that the probability of clipping of an OFDM signal can be reduced through the iposition of a constraint on the transitted power. This paper focuses on the downlink of a cyclic prefix-free MC-CDMA syste, eploying antenna arrays at both the transitter and receivers, over a frequency-selective fading channel. The use of antenna arrays enables the geoetrical inforation of the antennas and channel to be leveraged upon as array signal processing techniques can be used. This is in contrast to the systes in [6], [7] which ignore the antenna array geoetry and so, do not have access to the geoetrical inforation of the channel for iproved signal processing capabilities. Furtherore, due to the MC nature of the syste, this paper seeks to reduce the probability of clipping of the transitted signals fro each antenna by aking use of the relation between the probability of clipping and the average transit signal power of each antenna. The joint optiization of the precoding and equalization atrices is thus perfored with the goal of iniizing the overall MSE of the syste, subject to a constraint on the transitted power of each antenna eleent. Moreover, instead of the subcarrier- 978-1-4244-3435-/9/$25. 29 IEEE
Fig. 1. Syste architecture of the downlink of a ulti-user cyclic prefix-free MC-CDMA arrayed MIMO counication syste. noncooperative approach, the subcarrier-cooperative approach is used here which iplies that the subcarriers are not required to be orthogonal. Thus, there can be flexibility in the allocation of channel sybols aong the subcarriers and transitter antennas to overcoe channel fading effects [4]. In Section II, the syste and channel odels are developed and explained for a downlink MIMO MC-CDMA. The joint optiization process is then developed in Section III and in Section IV, siulation results based on the proposed algorith are presented. Finally, the paper is concluded in Section V. II. SYSTEM MODELLING The antenna array-based downlink MIMO MC-CDMA syste is shown in Fig. 1. The base station transitter is equipped with an antenna array of N eleents, while each receiver terinal has an N eleent antenna array. Note that sybols with a bar at the top is associated with the transitter. The channel sybol (e.g. QPSK strea of the i th user is deultipexed into Ni sub streas, where Ni sub is the nuber of substreas, to for the n th channel sybol vector a i [n] = [ a i1 [n], a i2 [n],...,a in sub[n] ] T i. The vector-sybol sequence {a i [n], n} at point-a is then converted into the vector-signal at point-b, defined as: n=+ n= a i [n]c (t nt cs, nt cs t<(n +1T cs (1 where c(t is the rectangular pulse of duration T cs. The vectorsignal is first copied into parallel streas and the k th copy of the signal given by (1 then undergoes transit beaforing through the application of the transit beaforing atrix of the i th user at the k th sub N N subcarrier, W i,k C i, to obtain the signal at point-c of Fig. 1. Cyclic prefix-free MC-CDMA odulation is then perfored by ultiplying the k th branch with a corresponding k th chip α ik = ±1 of α i = [ ] T α i1,...,α ik,...,α insc, which is the PN sequence of length corresponding to the i th user. The resultant vector is then applied to a subcarrier with frequency given by F k = k F, where F =1/T cs is the subcarrier separation. The odulated subcarriers are then sued up to produce the baseband MC-CDMA vector-signal s i (t C N 1 : s i (t = n=+ n= with a i [n] k=1 α ik W i,k exp (j2πf k t (2 nt cs t<(n +1T cs For M users, the transitted signal is given by
Fig. 2. Vector-signal-input vector-signal-output (VIVO channel for downlink MIMO syste. (t = s i (t (3 The baseband signal s i (t is upconverted, using a carrier frequency F c, to a bandpass signal. However, in this paper, without any loss of generality, the carrier will be ignored and baseband transission will be assued. Based on (3, the total transitted power of M users is given by P = M (W tr i W H i, where [ ] W i = W T i,1,...,w T i,k,...,w T T i, (4 Also, the total transitted power fro p th transit antenna can be expressed as P p = ( M (W tr i W H i I Nsc ι N,p ι T. N,p The wireless channel of the array-based MIMO MC-CDMA syste can be odelled as a vector-signal-input vector-signaloutput (VIVO channel which is assued to be ultipath dispersive with a delay spread of T cs. The transitter and receiver antenna arrays are assued to have sall apertures which iplies that each subcarrier present in the signals leaving the transitter antenna array experiences a coon fading channel towards the receiver antenna array eleents. Thus, the channel is odelled as a function of the array anifold vectors of both the transitter and the receiver, as shown in Fig. 2. The array anifold vector S jk of a receive antenna array is dependent on the array geoetry, direction-of-arrival (DOA with aziuth θ j and elevation φ j and frequency (F c + F k of the received signal, as given by ( j2π (Fc + F k S jk =exp [r c 1,r 2,...,r N ] T u j where u j = [ cos (θ j cos(φ j, sin (θ j cos(φ j, sin (φ j ] T is a unit vector in the direction (θ j,φ j, c is the speed of light and the vector r i, i 1,...,N, denotes the Cartesian coordinates of the i th eleent of the receive antenna array. (5 A siilar expression can be also written for the array anifold of the transit antenna array, S jk, as a function of the direction-of-departure (DOD ( θ j,φ j. However, when ax k ( k F F c, the transit and receive array anifold vectors can be assued to be independent of the subcarrier frequency, i.e. F c +F k F c. Thus, as in Fig. 2, the dependence of the array anifold vectors on the subcarrier frequency has been reoved. In Fig. 2, τ j and β j represents the path delay and fading coefficient of the j th path with j 1, 2,...,K and K is the nuber of resolvable ultipaths fro the base station to the th user. The baseband received signal x (t at the th user can be written as K x (t = β j S j S H j (t τ j+n (t (6 j=1 where n (t denotes the Additive White Gaussian Noise (AWGN of power σ 2 present in the channel. With reference to Fig. 1, the signal at point-f fro each receive antenna is sapled with a rate of 1 T s, where T s = Tcs. Hence, saples are obtained per channel sybol period and a tapped delay line of length 2 (equivalent to 2T cs applied at each antenna enables the asynchronous receiver to capture the contribution of a full sybol (current sybol. A vector of length 2N, x[n], is then fored at point-g of Fig. 1 by concatenating the contents of the tapped delay lines, 1 read at a rate of T cs. A subcarrier vector, f k [l], given by f k [l] = exp(j2πf k ( lt s exp(j2πf k (1 lt s. exp(j2πf k ( 1 lt s odels the tie variation of the k th subcarrier after undergoing a path delay of l saple periods where l, 1,...,( 1. Thus, (6 can be rewritten as shown in (8. K x [n] = β j S j j=1 k=1 ( α [ ] ik J T J l j fk [l j ] S H j W i,ka i [n 1] Nsc [ ] fk [l +α ik J lj j ] S H j W i,ka i [n] Nsc [ ] +α ik (J Nsc fk [l J lj j ] S H j W i,ka i [n +1] Nsc +n [n] (8 The atrix J (or J T isa2 2 tie down-shift (or up-shift operator defined by [ ] T J = 2Nsc 1 (9 I 2Nsc 1 2Nsc 1 and it is used to odel the delay l when J l or ( J T l is applied [ ] T on the colun vector f T k [l], T. A ulti-carrier space-tie array (MC-STAR anifold a- (7
trix for the j th path of i th user, as observed at the th user, is defined as ( [ ] H,ij = S j J lj Flj diag (α i ( I O Nsc SH j Nsc (1 which is a function of the receive and transit array anifold vectors, S j and S j, respectively, and the path delay l j for the j th path as observed at the th user as well as the PN sequence of the i th user and F lj = [ f 1 [l j ],...,f k [l j ],...,f Nsc [l j ] ] (11 Thus, H,ij encopasses the spatial-teporal channel characteristics (DOA and TOA for the received signal at the th user and blind channel estiation algoriths such as that in [8] can be used to obtain estiates of the spatial-teporal channel paraeters. Thus, (8 can be expressed as ( a i [n 1] x [n] = V,i I3 W i a i [n] +n [n] (12 a i [n +1] where V,i = [ H prev ],i H,i H next,i and H,i, H prev,i and H next,i are defined as K H,i = β j H,ij j=1 H prev,i = (I N ( J T H,i ; H next,i = ( I N J Nsc H,i In the th receiver, the receive beaforing weight, W H,is used to obtain the soft decision variable vector, â [n], which is given by â [n] =W H x [n] (13 III. JOINT TRANSMIT-RECEIVE BEAMFORMER OPTIMIZATION The objective of the optiization process is to deterine the precoder and equalization atrices, W and W, respectively, for each user with a predeterined nuber of substreas for each user. Thus, based on the soft decision variable vector, â [n] in (13, the MSE atrix of the th receiver can be expressed as where E = E { (â [n] a [n] (â [n] a [n] H} (14 = W H R xx, W + I N sub W H H, W W H H H,W R xx, = E { x [n] x H [n]} (15 = V,i (I 3 W i W H i V H,i + σi 2 2NNsc denotes the received signal covariance atrix of the th receiver. The optiization objective of iniizing the MSE of all users can thus be expressed as the following constraint optiization proble where s.t. =1 in (W 1,...,W M ;W 1,...,W M =1 ( ( tr W W H I Nsc ι N,p ι T N,p with p =1,...,N P p = ( ln A 2 clip 1 1 (1 P 1 L tr (E (16 P p, (17 and Pr { p (t >A clip } = P denotes the probability that the aplitude of a saple of the transitted signal of the p th antenna, given by p (t, is greater than the clipping aplitude, A clip. In obtaining (17, it is assued that there are L saples in the transitted MC-CDMA signal and due to the assuption of a Rayleigh distribution for p (t [9], the above expression is obtained for the constraint P p instead of the expression in [4] which is based on a Gaussian assuption. The optiization proble stated in (16 shows that the constraint involves liiting the total transitted power fro each transitter antenna and this is in contrast to [1] which iposes a constraint on the total transit power for a DS- CDMA syste. In the following, it is assued that both the transitter and receivers have perfect channel knowledge and so, by using the Lagrange Multiplier ethod, the cost function for the optiization proble can be expressed as: J = + tr (E (18 =1 ( N M μ p p=1 =1 ( ( tr W W H I Nsc ι N,p ι T P N,p p In order to obtain the solution for the precoding atrix W of the th receiver, it is assued that the receive ( decoding atrix W i, i are fixed and thus, by evaluating J W W =W =, the solution W is then given by { ( W M = L H ( ( V H,i Wi W H i V,i L + N μ p (I Nsc ι N,p ι T N,p p=1 where L =1 3 I NNsc. 1 H H,W (19 Also, the solution for the Lagrange Multiplier for the p th antenna, μ p, is given by
μ p = 1 ( tr{ I Nsc ι P N,p ι T (H H N,p,W W H p =1 L H ( V H (,i Wi W H i V,i LW W H } (2 Conversely, the precoding atrices are fixed in order to obtain the optial equalization atrices, which can be expressed as W = R 1 xx,h, W (21 corresponding to the MMSE receiver. Based on the derivations for the precoder and equalization atrices as well as the Lagrangian Multipliers, an iterative algorith is an attractive option since the precoder and equalization atrices as well as the Lagrangian Multipliers are dependent on each other. Thus, the joint optiization process can be perfored iteratively. 1. Initialization: (iterative index i = the precoder atrix for each user, W, is first initialized as W (i= = P MN sub U N sub where the coluns of U N sub are the general eigenvectors corresponding to the N sub largest general eigenvectors of the atrix H H P,H,. The coefficient is MN sub required to ensure an equal distribution of power to each user and substrea. 2. The equalization atrix W (i is then calculated for each user =1,...M based on (21. 3. Using the i th precoder and equalization atrices, the Lagrangian ultipliers μ (i p, p = 1,...N are then calculated using (2. 4. The equalization atrices and Lagrangian ultipliers obtained in the previous 2 steps are used to update the precoder atrices W (i+1 by using (19. 5. i is replaced by i +1 and Steps 2 4 are repeated until a convergence criterion is satisfied. IV. SIMULATION RESULTS The perforance of the proposed iterative joint optiization algorith is investigated in this section. The transitter is assued to be equipped with an unifor linear array (ULA of N =4eleents while the receivers are each equipped with aulaofn =2eleents, all with half-wavelength spacing. Each user is assigned a Gold sequence and Ni sub = 2 i. It is also assued that the transitted signal arrives at each receiver via K i =4ultipaths. The DOD and DOA associated with each path are assued to be uniforly distributed over [, 18 while the path delay is assued to be uniforly distributed over [, ( 1 T s ]. A block of 25 channel sybols is collected for processing at each tie, during which the channel is assued to be stationary and both the transitter and receivers have access to the channel state inforation. Average MSE per substrea.35.3.25.2.15.1.5 Convergence of the Joint Optiization Iterative Process Transit SNR = db Transit SNR = 8dB Transit SNR = 16dB 5 1 15 2 25 3 35 4 Iteration Fig. 3. Study of convergence of the proposed algorith with transit SNR =, 8, 16dB in a (6, 4, 2 cyclic prefix-free MC-CDMA arrayed MIMO syste ( = 15. The convergence behavior of the proposed iterative algorith is first studied for a cyclic prefix-free MC-CDMA arrayed MIMO syste with M =6users, hereafter known as (6, 4, 2, i.e. ( M,N,N cyclic prefix-free MC-CDMA arrayed MIMO syste. The convergence behavior is investigated for different transit SNR values, defined as P σ.fig.3shows 2 the average MSE per substrea per user as a function of the iterative process. It can be seen that the convergence property of the proposed iterative algorith iproves with the transit SNR. In Fig. 4, the BER perforance of the iterative joint optiization algorith is copared with the transit precoding schee described in Step 1 of the proposed algorith coupled with equalization atrices based on RAKE and MMSE receivers for a (6, 4, 2 MIMO MC-CDMA syste while Fig. 5 shows the BER perforance for a (1, 4, 2 MIMO MC-CDMA syste. The precoder and decoder atrices are obtained fro 3 iterations. In Figs. 4, and 5, the effect of frequency diversity is also investigated by considering the use of Gold sequences of lengths 7 and 15 in the siulations. As shown in the figures, the iterative process results in a uch better BER perforance copared to the case where the precoder and decoder atrices are not obtained jointly. This is because of the adaptation of the precoder and decoder atrices, according to the syste conditions, during the optiization process. In addition, it has been shown that with frequency diversity, the BER perforance of the proposed iterative approach can be iproved. The probability of clipping of the transitted MC-CDMA signal for various transit SNR for a (6,4,2 cyclic prefix-free MC-CDMA arrayed MIMO syste with =15is next shown in Fig. 6. It is assued that the transitted signal is oversapled by a factor of 4 so that the sapled transitted signal can approxiate the continuous tie signal. It can be seen that the iterative process results in a lower probability of
Average BER 1 1 1 1 2 1 3 1 4 = 15 (6,4,2 MIMO MC CDMA = 7 Eigen + RAKE Eigen + MMSE Proposed P clipping of Transitted Signal 1 1 1 1 2 (6,4,2 MIMO MC CDMA P clipping =.1 P clipping =.1 Eigen Proposed 1 5 1 5 5 1 15 Transit SNR (db 1 3 1 8 6 4 2 2 4 6 8 1 Transit SNR (db Fig. 4. BER perforance of the the proposed iterative algorith and the eigendecoposition-based transit precoding with RAKE or MMSE receiver equalization for a (6,4,2 cyclic prefix-free MC-CDMA arrayed MIMO syste. Fig. 6. Average probability of clipping of transitted MC-CDMA signal fro each transitter antenna with target P clipping =.1 and.1 for various transit SNR ( =7. Average BER 1 1 1 1 2 1 3 1 4 = 15 (1,4,2 MIMO MC CDMA = 7 1 5 Eigen + RAKE Eigen + MMSE 1 6 Proposed 1 5 5 1 15 Transit SNR (db Fig. 5. BER perforance of the the proposed iterative algorith and the eigendecoposition-based transit precoding with RAKE or MMSE receiver equalization for a (1,4,2 cyclic prefix-free MC-CDMA arrayed MIMO syste. clipping of the transitted signal copared to the non-iterative case. V. CONCLUSION The proble of joint optiization of precoder and equalization atrices in a downlink MIMO MC-CDMA syste over ultipath fading channels has been investigated in this paper. A subcarrier-cooperative approach has been taken in the optiization of the MC precoding atrices which is a departure fro conventional ethods where the orthogonality of the subcarriers is required. The proposed approach is based on iniizing the overall MSE with a per-antenna power constraint which can also reduce the probability of clipping of the transitted MC-CDMA signals. An iterative solution has been provided and siulations have been carried out which show the validity of the solution. Perfect channel knowledge is required for the proposed algorith while the ipact of iperfect CSI on the perforance of the algorith will be addressed in future works. REFERENCES [1] P. Wolniansky, G. Foschini, G. Golden, and R. Valenzuela, V-blast: an architecture for realizing very high data rates over the rich-scattering wireless channel, 1998 URSI International Syposiu on Signals, Systes, and Electronics, 1998. ISSSE 98., pp. 295 3, 29 Sep-2 Oct 1998. [2] S. M. Alaouti, A siple transit diversity technique for wireless counications, IEEE Journal on Selected Areas in Counications, vol. 16, no. 8, pp. 1451 1458, October 1998. [3] N. Yee, J.-P. Linnartz, and G. Fettweis, Multicarrier CDMA in Indoor Wireless Radio Networks, in 2nd IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications, 1993. PIMRC 1993., vol. 1, Septeber 1993, pp. 19 113. [4] D. P. Paloar, J. M. Cioffi, and M. A. Lagunas, Joint tx-rx beaforing design for ulticarrier io channels: a unified fraework for convex optiization, IEEE Transactions on Signal Processing, vol. 51, no. 9, pp. 2381 241, Septeber 23. [5] H. H. Seung and H. L. Jae, An overview of peak-to-average power ratio reduction techniques for ulticarrier transission, IEEE Wireless Counications, vol. 12, no. 2, pp. 56 65, April 25. [6] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, Zero-forcing ethods for downlink spatial ultiplexing in ultiuser io channels, Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing, IEEE Transactions on], vol. 52, no. 2, pp. 461 471, 24. [7] S. Serbetli and A. Yener, Transceiver optiization for ultiuser io systes, Signal Processing, IEEE Transactions on, vol. 52, no. 1, pp. 214 226, 24. [8] F. Rashid, H. Peh, and A. Manikas, Diffused channel estiation and reception for cyclic prefix-free c-cda arrayed-io counication systes, International Journal of Wireless Inforation Networks, 28. [9] H. Ochiai and H. Iai, On the distribution of the peak-to-average power ratio in ofd signals, IEEE Transactions on Counications, vol. 49, no. 2, pp. 282 289, February 21. [1] X. Wang and A. Manikas, Joint linear precoder-decoder optiization over io-cda ultipath channels, in Personal, Indoor and Mobile Radio Counications, 27. PIMRC 27. IEEE 18th International Syposiu on, 27, pp. 1 5.