System and method for transmitting signals in cooperative base station multi-user mimo networks

[0001] This invention relates generally to multi-input, multi-output (MIMO) communication networks, and more particularly to transmitting signals to reduce inter-cell interference in cooperative base station, multi-user MIMO networks.

[0002] Spectral efficiency gains in multi-input, multi-output (MIMO) communication networks are significant for point-to-point links. However, the gains are limited in multi-user (transceiver) cellular networks. This is especially so in a downlink from a base station to a transceiver (mobile telephone) of a cellular network, which is also called a vector broadcast channel.

[0003] In conventional cellular networks, inter-cell, co-channel interference (CCI) is primarily handled by careful radio resource management techniques such as power control, frequency reuse, and assignments of spreading codes.

[0004] In recent years, a number of more advanced techniques that outperform conventional approaches have been described to reduce the effect of CCI in cellular MIMO networks. For example, multi-user detection (MUD) in mobile stations (MSs) offers considerable performance improvements, H. Dai, A. F. Molisch and H. V. Poor, “Downlink capacity of interference-limited MIMO networks with joint detection,” IEEE Trans. Wireless Communications, vol. 3, no. 2, pp. 442-453, March 2004. However, MUD is prohibitively complex for most practical networks.

[0005] In conventional single-cell transmission networks, where the signal intended for only one MS is transmitted from only one associated base station (BS), joint pre-coding optimizations among coordinated BSs has been described, A. Pascual-Iserte, A. I. Perez-Neira and M. A. Lagunas, “An approach to optimum joint beamforming design in a MIMO-OFDM multi-user system,” European Journal on Wireless Communications and Networking, 2004, no. 2, pp. 210-221, 4^th Quarter, 2004; and C. Windpassinger, R. F. H. Fischer, T. Vencel and J. B. Huber, “Pre-coding in multiantenna and multi-user communications,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1305-1316, July 2004.

[0006] However, very stringent dimension constraints severely limit the number of transceivers that can be handled by such networks. While joint pre-coding significantly increases the network complexity, its performance gains are limited. Dirty paper coding (DPC), when used for the joint transmission among all the cooperative base stations, can effectively eliminate the effect of CCI, A. F. Molisch, Wireless Communications, Wiley, 2005; A. Goldsmith, S. A. Jafar, N. Jindal and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE J. Select. Areas Commun., vol. 21, no. 5, pp. 684-702, June 2003; and S. Shamai and B. M. Zaidel, “Enhancing the cellular downlink capacity via co-processing at the transmission end,” Proc. 2001 Spring IEEE Vehicular Technology Conf., pp. 1745-1749, May 2001.

[0007] Tomlinson-Harashima pre-coding (THP), which yields results that are only slightly suboptimal, is another practical solution, A. Pascual-Iserte, A. I. Perez-Neira and M. A. Lagunas, “An approach to optimum joint beamforming design in a MIMO-OFDM multi-user system,” European Journal on Wireless Communications and Networking, 2004, no. 2, pp. 210-221, 4^th Quarter, 2004; and C. Windpassinger, R. F. H. Fischer, T. Vencel and J. B. Huber, “Pre-coding in multiantenna and multi-user communications,” IEEE Trans. Wireless Communication., vol. 3, no. 4, pp. 1305-1316, July 2004.

[0008] However, both DPC and THP are nonlinear pre-coding techniques, and are prohibitively complicated for cooperative BS MIMO networks. Therefore, linear pre-coding at the transmitter among cooperative BSs is an attractive solution given its relatively lower complexity requirements at both the BSs and the MSs, S. Shamai and B. M. Zaidel, “Enhancing the cellular downlink capacity via co-processing at the transmission end,” Proc. 2001 Spring IEEE Vehicular Technology Conf., pp. 1745-1749, May 2001; G. J. Foschini, H. Huang, K. Karakayali, R. A. Valenzuela and S. Venkatesan, “The value of coherent base station coordination,” Proc. 2005 Conference on Information Sciences and Systems (CISS 05), The Johns Hopkins University, Mar. 16-18, 2005; P. W. Baier, M. Meurer, T. Weber and H. Troeger, “Joint transmission (JT), an alternative rationale for the downlink of time division CDMA using multi-element transmit antennas,” Proc. 2000 IEEE 6th Int. Symp. Spread Spectrum Techniques, vol. 1, pp. 1-5, September. 2000; and B. L. Ng, J. S. Evans, S. V. Hanly and D. Aktas, “Transmit beamforming with cooperative base stations,” Proc. IEEE International Symposium on Information Theory, ISIT 05, pp. 1431-1435, September 2005.

[0009] Joint transmission among cooperative base stations not only reduces CCI effectively, but also exploits macro-diversity, and can avoid capacity bottlenecks in channels with severe spatial correlations. However, conventional joint transmission schemes invariably assume that both the desired signals and the interfering signals from different BSs arrive at each of the MSs synchronously. While this assumption enables the well-studied single cell downlink transmission model to be applied in a straightforward manner, it is fundamentally unrealizable in practical networks.

[0010] The BSs can align their transmissions so that the signals intended for any MS arrive at that MS synchronously. However, the BSs cannot also simultaneously control when these signals are also received as interference by other MSs. Thus, the interference signals do not arrive simultaneously at the MSs, even under the assumption of accurate synchronous BS cooperation. The impact of this inherent asynchronism is readily apparent in high data rate networks. It can significantly degrade the performance of the networks. Even the multi-BS pre-coding optimization, in which a linearly pre-coded signal for an MS is transmitted by only one BS, ignores the asynchronous arrival of interference signals.

[0011] To the best of our knowledge, this problem of asynchronous interference in multi-user MIMO networks has not been addressed in the prior art.

[0012] The embodiments of the invention provide methods and systems for transmitting signals in cooperative base station, multi-user, multi-input, multi-output (MIMO) communications networks using linear pre-coding processes to reduce inter-cell interference.

[0013] The embodiments can also reduce timing advance inaccuracies or ‘jitter,’ which are inevitable in practical networks. By using jitter statistic-aware pre-coders, the impact of these timing advance inaccuracies can be significantly reduced.

[0014] In general, the invention enables the implementation of joint transmission in cooperative base station, multi-user MIMO networks.

[0023] FIG. 1 shows a cooperative base station, multi-user, multi-input, multi-output network according to an embodiment of our invention. The network includes B base stations (BSs) 100. Each BS has N_T antennas 102, where T is greater than one. The base stations transmit signals 101 in a cooperative manner. There are also K mobile stations (MSs) 103. Each MS has N_R antennas 104, where R is one or more. For example, the base station is located at a cellular site, and the mobile stations (users) are cellular transceivers (‘cell phones’). Each station can transmit or receive radio signals. The signals from the BSs partially overlap as shown in FIG. 1. That is, both MS₁ and MS₂ can receive signals from both BS₁ and BS₂.

[0024] The basic idea behind the invention is that signals transmitted by multiple base stations to a particular mobile station are synchronous with respect to each other, and asynchronous with respect to signals transmitted by the base stations to any other mobile station.

[0025] As shown in FIG. 4, each BS 100 transmits cooperatively L_k data streams 401 to MS_k, for k=1, . . . , K. In baseband, the data streams 401 are an output of a transmit RF chain 500 and include symbols, which can be drawn from a signal constellation such as well known modulation formats, e.g., QPSK, M-QAM, etc. Such chains use what is known as finite modulation alphabets.

[0026] As shown in FIG. 5 for a 3GPP downlink channel, the transmit RF chain 500 can include the following stages: channel coding 501, interleaving 502, scrambling 503, spreading 504, and channel mapping 600. The input is some signal 510, e.g., a signal derived from a microphone in a cellular telephone, and the outputs are the data streams 401. FIG. 6 shows the channel mapping stage 600, which includes a serial to parallel module 601, and a modulation mapper 602.

[0027] Referring back to FIG. 4, the data streams 401 are transcoded 410, as described herein, so that the transcoded signals 411 arrive synchronously at an intended mobile station, and asynchronously at other, non-intended mobile stations.

[0028] Cooperative transmitting means that the signals are synchronous. The links (channels) 101 between the different transmit and receive antenna pairs are assumed to be independent of each other and undergo frequency-flat Rayleigh fading. Therefore, H_k^(b), the base-band representation of the channel from BS_b to MS_k, has Gaussian independent identically distributed (i.i.d.) entries. Let b_k denote the index of the BS closest to the MS_k.

[0029] For any MS, the BSs cooperate and synchronously transmit the signals intended for the MS. As stated above, we use joint linear pre-coding at the BSs. The transmit vector (signal) for MS_k from BS_b is linearly pre-coded using the matrix T_k^(b) of size N_T×L_k. The transmit vector takes a form x_k^(b)(m)=T_k^(b)s_k (m), where s_k(m) denotes the m^th zero-mean data vector, of size L_k×1 for MS_k.

[0030] In a slowly fading environment, we assume that each BS has the required channel state information for all the subchannels to all the MSs. This can be achieved, for example, by an initial joint training phase that involves all the coordinated base stations, or by means of adaptive signal tracking and feedback processes implemented at the MSs.

[0031] We assume a block-fading channel model with a large enough coherence time so that the channel fade remains substantially constant over a duration when the pre-coding matrix T_k^(b) is used. The coherence time is typically much longer than the propagation delay between any BS-MS pair. In order to maximize the information transmission rate for each MS, a Gaussian code book is used for the transmit data vectors.

[0032] Power is normalized according to E[s_k(m)s_k^H(m)]=I_Lk, where the operator H represents a matrix conjugate transpose, and I_L denotes an L×L identity matrix. The power constraints on the transmitted signals can be defined by the norms of the pre-coding matrices T. Furthermore, the code books for different MSs are independent of each other, i.e., E[s_k(m)s_l^H(m)]=0, for k≠l.

[0033] We assume accurate inter-BS synchronization of the transmitted signals, which can be realized by, e.g., the Global Positioning System (GPS) or through on-air signaling methods, M. Capaccioli, and D. Rispo, “A technique to realize base stations on-air frame synchronization in TD-SCDMA system,” Proc. IEEE Vehicular Technology Conference, 2003, VTC-Fall 03, vol. 2, pp. 982-986, October 2003; and M. Rudlf and B. Jechoux, “Design of concatenated extended complementary sequences for inter-base station synchronization in WCDMA TDD mode,” Proc. IEEE Global Telecommunications Conference, 2001, GLOBECOM 01, vol. 1, pp. 674-679, November 2001, both incorporated herein by reference, or by a wire-line backbone, a technique already in place in CDMA2000 and IS-95 networks to facilitate soft handoffs, H. Dai, A. F. Molisch and H. V. Poor, “Downlink capacity of interference-limited MIMO systems with joint detection,” IEEE Trans. Wireless Communications, vol. 3, no. 2, pp. 442-453, March 2004, incorporated herein by reference.

[0034] The channel state information (CSI) available at each BS also includes the knowledge of the propagation delay from each BS to each of the MSs. Given synchronous clocks among cooperative BSs, achieved by inter-BS synchronization, this delay knowledge enables the use of the timing-advance mechanism in the downlink.

[0035] Specifically, the BSs advance their transmission times in order to ensure that the signals arrive at the desired MS_k synchronously. However, as stated above, due to the broadcast nature of the wireless channel, MS_k also inevitably receives signals intended for other MSs. As elaborated upon above, these signals arrive at MS_k with different delay offsets, and are asynchronous with the data streams intended for MS_k.

[0036] As shown in FIG. 1 for two BSs and two MSs, the propagation delay from BS_b to MS_k is denoted as τ_k^(b). To guarantee synchronous receptions of the transmitted signal {x_k^(b)}_{b=1 . . . B} at MS_k, the BS_b advances the time when the signal x_k^(b)(m) is transmitted by an interval Δτ_k^(b)=τ_k^(b)−τ_k^(bk). This ensures that the transmitted signals {x_k^(b)}_{b=1 . . . B} arrive at the MS_k with the same delay, τ_k^(bk). The equivalent received base-band signal at MS_k, when a linear modulation with a base-band signature waveform g(t) defined in a range [0, (α+1)T_s] where α is a roll-off factor of the pulse shape, is given by

[0037] $\begin{array}{c}{r}_k\left(t\right)=\sum _m\left\{\sum _{b}g\left(t-{\mathrm{mT}}_S-{\tau }_k^{\left(b\right)}+{\mathrm{\Delta \tau }}_k^{\left(b\right)}\right)H_k^{\left(b\right)}{x}_k^{\left(b\right)}\left(m\right)\right\}+\\ \sum _m\left\{\sum _{\underset{\left(j\ne k\right)}{j}}\sum _{b}g\left(t-{\mathrm{mT}}_S-{\tau }_k^{\left(b\right)}+{\mathrm{\Delta \tau }}_j^{\left(b\right)}\right)H_k^{\left(b\right)}{x}_j^{\left(b\right)}\left(m\right)\right\}+n_k\left(t\right)\\ =\sum _{m}g\left(t-{\mathrm{mT}}_S-{\tau }_k^{\left(b_k\right)}\right)H_k{x}_k\left(m\right)+\\ \sum _m\left\{\sum _{\underset{\left(j\ne k\right)}{j}}\sum _{b}g\left(t-{\mathrm{mT}}_S-{\tau }_k^{\left(b\right)}+{\mathrm{\Delta \tau }}_j^{\left(b\right)}\right)H_k^{\left(b\right)}{x}_j^{\left(b\right)}\left(m\right)\right\}+n_k\left(t\right),\end{array}$
where H_k=[H_k⁽¹⁾, H_k⁽²⁾, . . . , H_k^(B)] are all channel matrices from the B BSs to the K MSs, x_k(m)=[x_k⁽¹⁾(m)^H, . . . , x_k^(B)(m)^H]^H is the jointly transmitted signal, and n_k(t) represents an additive white Gaussian noise vector.

[0038] At MS_k, the received base-band signal r_k(t) is passed through a filter matched to g(t−mT_s−τ_k^(bk)) to generate channel statistics. Note that the matched filter is also delayed by τ_k^(bk). The delay is obtained by a symbol synchronization mechanism at the MS_k. The autocorrelation of g(t) is denoted as

[0039] $\rho \left(\tau \right)={\int }_0^{\left(\alpha +1\right)T_S}g\left(t\right)g\left(t-\tau \right)dt,\mathrm{with}\rho \left(0\right)=1,$
and the corresponding received signal component vector at MS_k is H_kT_ks_k(m), where T_k=[T_k^(1)H, T_k^(2)H, . . . T_k^(B)H]^H are the transmit pre-coding matrices used by the B base stations for MS_k.

[0040] Then, the total discrete received signal after matched filtering is

[0041] $\begin{array}{cc}{y}_k\left(m\right)=H_k{T}_k{s}_k\left(m\right)+\sum _{\underset{\left(j\ne k\right)}{j}}\sum _b{H}_k^{\left(b\right)}{T}_j^{\left(b\right)}{i}_{\mathrm{jk}}^{\left(b\right)}+n_k,& \left(1\right)\end{array}$
where n_k is the discrete noise vector satisfying E[n_kn_k^H]=N₀I, and i_jk^(b) is the asynchronous interference at the MS_k from the signal transmitted by BS_b for MS_j. The strength of the interference depends on the delay difference, τ_jk^(b), between the timing-advances used by BS_b for MS_j and for MS_k:
τ_jk^(b)=(τ_k^(b)−Δτ_j^(b))−τ_k^(bk)=Δτ_k^(b)−Δτ_j^(b).  (2)

[0042] FIG. 2 shows the desired signal m 200 transmitted by BS_b, and the asynchronous interference at MS_k due to the signal transmitted by BS_b to MS_j, due to two adjacent symbol transmissions 201-202 with indices m_jk^(b) and m_jk^(b)=1, where m_jk^(b)=m−┌τ_jk^(b)/T_S┐ and ┌x┐ denotes the smallest integer larger than or equal to x. Let 0≦τ′_jk^(b)≦T_S denote the delay offset τ_jk^(b) modulo the symbol duration T_S. Then, we have
i_jk^(b)=ρ(τ′_jk^(b)−T_S)s_j(m_jk^(b))+ρ(τ′_jk^(b))s_j(m_jk^(b)+1).  (3)

[0043] The first and second moments for the interference i_jk^(b) are given as follows. From Equation (3), it can be seen that E[i_jk^(b)]=0. As the information signals intended for two different MSs j₁ and j₂ are independent of each other, we have E[i_j1k^(b1)i_j2k^(b2)H)]=0, for j₁≠j₂≠k.

[0044] Moreover, the correlation between i_jk^(b1) and i_jk^(b2), for j≠k, is given by

[0045] $\begin{array}{cc}E\left[i_{\mathrm{jk}}^{\left(b1\right)}{i}_{\mathrm{jk}}^{\left(b2\right)H}\right)]={\beta }_{\mathrm{jk}}^{\left(b1,b2\right)}{I}_{L_k},& \left(4\right)\\ \mathrm{where}& \\ {\beta }_{\mathrm{jk}}^{\left(b1,b2\right)}=\{\begin{array}{cc}0,& m_{\mathrm{jk}}^{\left(b2\right)}>m_{\mathrm{jk}}^{\left(b1\right)}+1\\ \rho \left({{\tau }^{\prime }}_{\mathrm{jk}}^{\left(b1\right)}\right)\rho \left({{\tau }^{\prime }}_{\mathrm{jk}}^{\left(b2\right)}-T_S\right),& m_{\mathrm{jk}}^{\left(b2\right)}=m_{\mathrm{jk}}^{\left(b1\right)}+1\\ \rho \left({{\tau }^{\prime }}_{\mathrm{jk}}^{\left(b1\right)}\right)\rho \left({{\tau }^{\prime }}_{\mathrm{jk}}^{\left(b2\right)}\right)+\rho \left({{\tau }^{\prime }}_{\mathrm{jk}}^{\left(b1\right)}-T_S\right)\rho \left({{\tau }^{\prime }}_{\mathrm{jk}}^{\left(b2\right)}-T_S\right),& m_{\mathrm{jk}}^{\left(b2\right)}=m_{\mathrm{jk}}^{\left(b1\right)}\end{array},\mathrm{for}j\ne k.& \left(5\right)\end{array}$

[0046] When b1=b2=b, we have β_jk^(b,b)=ρ(τ′_jk^(b))²+ρ(τ′_jk^(b)−T_S)². Furthermore, β_kk^(b1,b2)=1 for all b1 and b2.

[0047] We maximize the information rates that can be achieved by linear pre-coding. From Equation (1), the bandwidth-normalized information rate R_k at MS_k is given by
R_k=log|I+Φ_k⁻¹H_kT_kT_k^HH_k^H|,  (6)
where Φ_k is the covariance matrix of noise plus interference in Equation (1), and takes the form

[0048] $\begin{array}{cc}{\Phi }_k=N_{0}I+\sum _{\underset{\left(j\ne k\right)}{j}}\sum _{\left(b1,b2\right)}{\beta }_{\mathrm{jk}}^{\left(b1,b2\right)}{H}_k^{\left(b1\right)}{T}_j^{\left(b1\right)}{T}_j^{\left(b2\right)H}{H}_k^{\left(b2\right)H}.& \left(7\right)\end{array}$

[0049] Because all the K MSs use the same waveform g(t),

[0050] $\left\{{\beta }_{\mathrm{jk}}^{\left(b1,b2\right)}\right\},$
corresponding different timing parameters can be predetermined and stored in a look-up table. Note that in Equations (6) and (7), the linear pre-coding matrices {T_k}_{k=1, . . . , K} are functions of the channel matrices H₁˜H_K, of all the K MSs.

[0051] Our goal is to jointly optimize the transmitter pre-coding matrices {T_k}_{k=1 . . . K} so as to maximize the sum of information rates over all the K MSs, given the channel state H₁˜H_K.

[0052] To ensure fairness between users, a per-MS power constraint of the following form can be used: Trace(T^H_kT_k)≦P_T, where the power constraint threshold P_T is predetermined. In addition to ensuring fairness among users, this also has the advantage that it leads to analytically tractable solutions. Another constraint can be a per-BS power constraint, which simplifies the design of the power amplifiers in the BS. The per-MS power constraint is compatible with the per-BS constraint so long as the transmit power allocation for all the involved co-channel MSs does not make each of the cooperative BSs surpass their transmit power upper limit. However, it should be noted that the embodiments of the invention can also be applied to cases where there is no power constraint on the mobile stations or with other power constraints.

[0053] An additional constraint follows from the constraints on channel spatial dimensions: BN_T≧Σ_kL_k. Therefore, the optimization problem can be defined by a target function

[0054] $\begin{array}{cc}{\left\{T_{k\_\mathrm{opt}}\right\}}_{k=1\dots K}=\underset{{\left\{T_k\right\}}_{k=1\dots K}}{\arg \max }\left\{\sum _k{R}_k\right\},& \left(8\right)\end{array}$
subject to the power constraint Trace(T^H_kT_k)≦P_T, for k=1, . . . , K.

[0055] This optimization problem is non-linear and, even, non-convex. Resorting to conventional brute-force numerical optimizations techniques involves searching over an extremely large space of dimension BN_TΣ_kL_k. Such techniques include the Nelder-Mead method, J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, “Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM Journal of Optimization, vol. 9, no. 1, pp. 112-147, 1998; and a simulated annealing method, L. Shao and S. Roy, “Downlink Multicell MIMO-OFDM: An Architecture for Next Generation Wireless Networks,” Proc. IEEE Wireless Communications and Networking Conference (WCNC), March 2005. Those methods are prohibitively expensive computationally and do not provide a viable option.

[0056] Therefore, we provide alternative sub-optimal techniques to determine the pre-coding matrices.

[0057] One possible solution could be based on a nullification method, G. J. Foschini, H. Huang, K. Karakayali, R. A. Valenzuela and S. Venkatesan, “The value of coherent base station coordination,” Proc. 2005 Conference on Information Sciences and Systems (CISS 05), Mar. 16-18, 2005; P. W. Baler, M. Meurer, T. Weber and H. Troeger, “Joint transmission (JT), an alternative rationale for the downlink of time division CDMA using multi-element transmit antennas,” Proc. 2000 IEEE 6th Int. Symp. Spread Spectrum Techniques, vol. 1, pp. 1-5, September 2000, and B. L. Ng, J. S. Evans, S. V. Hanly and D. Aktas, “Transmit beamforming with cooperative base stations,” Proc. IEEE International Symposium on Information Theory, ISIT 05, pp. 1431-1435, September 2005, all incorporated herein by reference.

[0058] That method is widely applied in intra-cell multi-user scenarios due to its simplicity and relatively good performance. The nullification method makes the transmitters satisfy the constraint: H_kT_j=0, ∀k≠j. However, that constraint cannot annul the interference term in Equation (1) due to the inter-cell asynchronous interference described earlier. Thus, significant interference persists.

[0059] Another possible option would enforce a stronger per-base station constraint H_k^(b)T_j^(b)=0 for k≠j, H. Kaaranen, A. Ahtiainen, L. Laitinen, S. Naghian and V. Niemi, UMTS Networks: Architecture, Mobility and Services, Wiley, 2005. While that constraint completely cancels interference, it severely restricts the number of users. Only K≦N_T/N_R users can be supported, which is clearly undesirable.

[0060] Therefore, according to embodiments of our invention, we provide joint transmitter linear pre-coding methods, which lead to closed-form or simplified solutions. The methods reduce the amount of interference at the input of all the mobile stations. The methods are based on sub-optimal optimization criteria.

[0061] We first assume that the timing-advance mechanisms work accurately, so that the desired signals, but not the interference signals, arrive synchronously at each MS. Inaccurate timing-advances, which can even affect the desired signals, are described below.

[0062] As described above, it is difficult to apply conventional direct optimization and nullification approaches to the signal modeled according to Equation (1). Therefore, our transmitter pre-coding matrices {T_k}_{k=1 . . . K} use a Wiener smoothing criterion, such that an overall mean square error (MSE) for all the K MSs is minimized at the input of their receivers.

[0063] FIG. 7 shows the signals transmitted at each base station, and the SINR at each base station. In FIG. 7, the signal T₁⁽¹⁾s₁+T₂⁽¹⁾s₂ is sent by BS₁, where with T₁⁽¹⁾s₁ is the signal component intended for MS₁, and T₂⁽¹⁾s₂ is the signal component intended for MS₂. Similarly, the signal T₁⁽²⁾s₁+T₂⁽²⁾s₂ is transmitted by BS₂, where T₁⁽²⁾s₁ is the signal component intended for MS₁ and T₂⁽²⁾s₂ is the signal component intended for MS₂. Then, MS₁ receives the first signal H₁T₁s₁+H₁⁽¹⁾T₂⁽¹⁾t₂₁⁽¹⁾+H₁⁽²⁾T₂⁽²⁾t₂₁⁽²⁾ and MS₂ receives the second signal H₂T₂s₂+H₂⁽¹⁾T₁⁽¹⁾t₁₂⁽¹⁾+H₂⁽²⁾T₁⁽²⁾t₁₂⁽²⁾. The signal to interference plus noise ratios at MS1 and MS2 are respectively given by:

[0064] $\begin{array}{c}{\mathrm{SINR}}_1=\frac{\mathrm{Trace}\left[T_1^H{H}_1^H{H}_1{T}_1\right]}{N_0{N}_R+\mathrm{Trace}\left[\sum _{b1=1}^2\sum _{b2=1}^2{\beta }_{21}^{\left(b1,b2\right)}{T}_2^{\left(b1\right)H}{H}_1^{\left(b1\right)H}{H}_1^{\left(b2\right)}{T}_2^{\left(b2\right)}\right]}\\ {\mathrm{SINR}}_2=\frac{\mathrm{Trace}\left[T_2^H{H}_2^H{H}_2{T}_2\right]}{N_0{N}_R+\mathrm{Trace}\left[\sum _{b1=1}^2\sum _{b2=1}^2{\beta }_{12}^{\left(b1,b2\right)}{T}_1^{\left(b1\right)H}{H}_2^{\left(b1\right)H}{H}_2^{\left(b2\right)}{T}_1^{\left(b2\right)}\right]}\end{array}$

[0065] JWF has been used in context of intra-cell interference reduction, G. J. Foschini, H. Huang, K. Karakayali, R. A. Valenzuela, and S. Venkatesan, “The value of coherent base station coordination,” Proc. 2005 Conference on Information Sciences and Systems (CISS 05), Mar. 16-18, 2005; P. W. Baier, M. Meurer, T. Weber and H. Troeger, “Joint transmission (JT), an alternative rationale for the downlink of time division CDMA using multi-element transmit antennas,” Proc. 2000 IEEE 6th Int. Symp. Spread Spectrum Techniques, vol. 1, pp. 1-5, September 2000; and B. L. Ng, J. S. Evans, S. V. Hanly and D. Aktas, “Transmit beamforming with cooperative base stations,” Proc. IEEE International Symposium on Information Theory, ISIT 05, pp. 1431-1435, September 2005, all incorporated herein by reference.

[0066] However, extending the JWF technique to our cooperative BS network with asynchronous interference is not at all obvious.

[0067] Although our technique is sub-optimal for maximizing the information rate, as an advantage, it avoids a complex iterative process and provides closed-form solutions that exploit the gains of BS cooperation.

[0068] To simplify this description, we omit the symbol index m in Equation (1). The received signal vector for over all MSs is y=[y₁^H,y₂^H, . . . ,y_K^H]^H, and the data vector over all MSs is s=[s₁^H, s₂^H, . . . , s_K^H]^H. If y and s are of the same dimension, e.g., when L_k=N_R for all k, then the overall network-wide MSE can be expressed as:

[0069] $\begin{array}{cc}\mathrm{MSE}=E\left\{{y-s}^2\right\}=\sum _{k=1}^{K}E\left\{{y_k-s_k}^2\right\}=\sum _{k=1}^K{\mathrm{MSE}}_k,& \left(9\right)\end{array}$
where MSE_k stands for the MSE of MS_k, and the expectation E{ . . . } is over the random data vectors {s_k}_{k=1 . . . K} and the noise, {n_k}_{k=1 . . . K}. Then, the optimization criterion becomes:

[0070] $\begin{array}{cc}\underset{\left\{T_k^{\left(b\right)}\right\}k=1\dots K,b=1\dots B}{\min }\sum _{k=1}^K{\mathrm{MSE}}_k\mathrm{subject}\mathrm{to}& \\ \mathrm{Trace}\left(T_k^{*}{T}_k\right)=\mathrm{Trace}\left(\sum _{b=1}^B{T}_k^{{\left(b\right)}^{*}}{T}_k^{\left(b\right)}\right)\le P_T\mathrm{for}k=1\dots K.& \left(10\right)\end{array}$

[0071] If N_R ≠L_K for some k, then the MSE minimization in Equation (9) is not valid. In this case, certain transmitter-dependent and channel-dependent decoding structures need to be assumed at the receivers. This may require complicated iterative processes, see Foschini et al., Baier et al., and Ng et al. above.

[0072] By denoting the multi-user interference (MUI) term in Equation (1) as

[0073] $\begin{array}{cc}{J}_k=\sum _{\stackrel{j}{\left(j\ne k\right)}}\sum _b{H}_k^{\left(b\right)}{T}_j^{\left(b\right)}{i}_{jk}^{\left(b\right)},& \left(10.1\right)\end{array}$
the MSE_k is given by:

[0074] $\hspace{1em}\begin{array}{cc}\begin{array}{c}MS{E}_k=E\left\{{y_k-s_k}^2\right\}=E\left\{{\left[y_k-s_k\right]}^H\left[y_k-s_k\right]\right\}\\ =E\{{\left[\left(H_k{T}_k{s}_k-s_k\right)+J_k+n_k\right]}^H\\ \left[\left(H_k{T}_k{s}_k-s_k\right)+J_k+n_k\right]\}\\ =E\left\{{\left(H_k{T}_k{s}_k-s_k\right)}^H\left(H_k{T}_k{s}_k-s_k\right)\right\}+\\ E\left\{J_k^H{J}_k\right\}+E\left\{n_k^H{n}_k\right\}\\ =\mathrm{Trace}\{\sum _{b=1}^B{H}_k^{\left(b\right)}{T}_k^{\left(b\right)}{T}_k^{\left(b\right)H}{H}_k^{\left(b\right)H}-\sum _{b=1}^B{H}_k^{\left(b\right)}{T}_k^{\left(b\right)}-\\ \sum _{b=1}^B{T}_k^{\left(b\right)H}{H}_k^{\left(b\right)H},+\\ \left(N_0+1\right)I_{N_R}+\sum _{j\ne k}\sum _{\left(b1,b2\right)}{\beta }_{jk}^{\left(b1,b2\right)}{H}_k^{\left(b1\right)}{T}_j^{\left(b1\right)}{T}_j^{\left(b2\right)H}{H}_k^{\left(b2\right)H}\},\end{array}& \left(11\right)\end{array}$
where we used the identity E{J_k^HJ_k}=Trace{E[J_kJ_k^H]}. To solve Equation (10) in closed-form, we apply a Lagrange objective function:

[0075] $\begin{array}{cc}\hspace{1em}A\left({\left\{T_k^{\left(b\right)}\right\}}_{k=1\dots K,b=1\dots B}\right)=\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\sum _{k=1}^{K}MS{E}_k+\sum _{k=1}^K{\kappa }_k\left[\mathrm{Trace}\left(\sum _{b=1}^B{T}_k^{\left(b\right)H}{T}_k^{\left(b\right)}\right)-P_T\right],\hspace{1em}\hspace{1em}& \left(12\right)\end{array}$
where κ₁˜κ_K are the Lagrange multipliers associated with the power constraint for MSs 1˜K respectively. Therefore, Equation (9) can be solved by minimizing Equation (12).

[0076] Using the analysis in Appendix A, we obtain a closed-form solution for the pre-coding matrix T_k:
T_k=[C_k+κ_kI]⁻¹H_k^H,  (13)
where c_k, which corresponds to MS_k, is a block matrix of the form

[0077] $\begin{array}{cc}{C}_k=\left[\begin{array}{cccc}{C}_k^{\left(11\right)}& C_k^{\left(12\right)}& \cdots & C_k^{\left(1B\right)}\\ C_k^{\left(21\right)}& C_k^{\left(22\right)}& \cdots & C_k^{\left(2B\right)}\\ ⋮& ⋮& ⋰& ⋮\\ C_k^{\left(B1\right)}& C_k^{\left(B2\right)}& \cdots & C_k^{\left(BB\right)}\end{array}\right].& \left(14\right)\end{array}$

[0078] The submatrices of C_k are given by

[0079] $\begin{array}{cc}{C}_k^{\left(b1b2\right)}=\{\begin{array}{c}\sum _{\stackrel{j}{\left(j\ne k\right)}}^K{\beta }_{kj}^{\left(b1b2\right)}{H}_j^{\left(b1\right)H}{H}_j^{\left(b2\right)},\mathrm{for}b1\ne b2\\ \sum _{j=1}^K{\beta }_{kj}^{\left(bb\right)}{H}_j^{\left(b\right)H}{H}_j^{\left(b\right)},\mathrm{for}b1=b2=b,\end{array}& \left(15\right)\end{array}$

[0080] Appendix A also derives the expression for the Lagrangian multipliers κ_k. The Lagrange multipliers κ₁˜κ_K are jointly selected to minimize the network-wide MSE in Equation (9).

[0081] An alternative sub-optimal objective function, which has the advantage of allowing a station-by-station optimization can be obtained by considering interference leakage suppression. For MS_k, the pre-coding matrix T_k maximizes a ratio of the power of its desired signal received at MS_k over the sum of the noise and the total power due to ‘leakage’ of the signal x_k at other MSs. We call this the signal-to-leakage-plus-noise-ratio (SLNR).

[0082] This approach minimizes the signal-to-interference-ratio due to data streams intended for another MS instead of the interference that arrives at that MS. This problem is much more general than the simple case of a single data stream per MS and one that does not model BS cooperation with asynchronous interference.

[0083] FIG. 8 shows the signals transmitted at each base station, and the SLNR at each base station. In FIG. 7, the signal T₁⁽¹⁾s₁+T₂⁽¹⁾s₂ is sent by BS₁, where T₁⁽¹⁾s₁ is the signal component intended for MS₁, and T₂⁽¹⁾s₂ is the component intended for MS₂. Similarly, the signal T₁⁽²⁾s₁+T₂⁽²⁾s₂ is transmitted by BS₂, where T₁⁽²⁾s₁ is the component intended for MS₁ and T₂⁽²⁾s₂ is the component intended for MS₂. Then, MS₁ receives the first signal H₁T₁s₁+H₁⁽¹⁾T₂⁽¹⁾t₂₁⁽¹⁾+H₁⁽²⁾T₂⁽²⁾t₂₁⁽²⁾ and MS₂ receives the second signal H₂T₂s₂+H₂⁽¹⁾T₁⁽¹⁾t₁₂⁽¹⁾+H₂⁽²⁾T₁⁽²⁾t₁₂⁽²⁾. The signal to leakage plus noise ratios due to the signals of MS₁ and MS₂ are respectively given by:

[0084] $SLN{R}_1=\frac{\mathrm{Trace}\left[T_1^H{H}_1^H{H}_1{T}_1\right]}{N_0{N}_R+\mathrm{Trace}\left[\sum _{b1=1}^2\sum _{b2=1}^2{\beta }_{12}^{\left(b1,b2\right)}{T}_1^{\left(b1\right)H}{H}_2^{\left(b1\right)H}{H}_2^{\left(b2\right)}{T}_1^{\left(b2\right)}\right]}$ $SLN{R}_2=\frac{\mathrm{Trace}\left[T_2^H{H}_2^H{H}_2{T}_2\right]}{N_0{N}_R+\mathrm{Trace}\left[\sum _{b1=1}^2\sum _{b2=1}^2{\beta }_{12}^{\left(b1,b2\right)}{T}_1^{\left(b1\right)H}{H}_2^{\left(b1\right)H}{H}_2^{\left(b2\right)}{T}_1^{\left(b2\right)}\right]}$

[0085] For analytical tractability, we restrict the set of the pre-coding matrices T_k to be scaled versions of semi-unitary matrices. Thus,

[0086] $T_k=\sqrt{\frac{P_T}{L_k}}{Q}_k,$
where the N_TB×L_k semi-unitary matrix Q_k contains orthonormal columns. Orthonormality ensures that there is no cross-talk among the L_k data streams for MS_k, and simplifies the detector at MS_k. Then, the received signal at MS_k takes the form

[0087] $y_k=\sqrt{\frac{P_T}{L_k}}{H}_k{Q}_k{s}_k+\sqrt{\frac{P_T}{L_k}}\sum _{\stackrel{j}{\left(j\ne k\right)}}\sum _b{H}_k^{\left(b\right)}{Q}_j^{\left(b\right)}{i}_{jk}^{\left(b\right)}+n_k,$
where the matrix Q_k^(b) includes the rows in the matrix Q_k that are associated with the b^th BS. Then, the signal component power is given by

[0088] $\begin{array}{cc}{P}_k=\frac{P_T}{L_k}E\left[s_k^H{Q}_k^H{H}_k^H{H}_k{Q}_k{s}_k\right]=\frac{P_T}{L_k}\mathrm{Trace}\left[Q_k^H{H}_k^H{H}_k{Q}_k\right].& \left(16\right)\end{array}$

[0089] The asynchronous interference leakage due to the signal x_k, i.e., the signal that is meant for MS_k but that is also received at MS_j is

[0090] $\sum _b{H}_j^{\left(b\right)}{T}_k^{\left(b\right)}{i}_{kj}^{\left(b\right)}.$
The power P_L—kj of the signal, is given by

[0091] $\begin{array}{cc}\begin{array}{c}{P}_{L\_\mathrm{kj}}=E\left[\frac{P_T}{L_k}\sum _{\left(b1,b2\right)}{i}_{\mathrm{kj}}^{\left(b1\right)H}{Q}_k^{\left(b1\right)H}{H}_j^{\left(b1\right)H}{H}_j^{\left(b2\right)}{Q}_k^{\left(b2\right)}{i}_{\mathrm{kj}}^{\left(b2\right)}\right]\\ =\frac{P_T}{L_k}\sum _{\left(b1,b2\right)}{\beta }_{\mathrm{kj}}^{\left(b1,b2\right)}\mathrm{Trace}\left(Q_k^{\left(b1\right)H}{H}_j^{\left(b1\right)H}{H}_j^{\left(b2\right)}{Q}_k^{\left(b2\right)}\right),\end{array}& \left(17\right)\end{array}$

[0092] The power of the noise at MS_k is P_Nk=N₀N_R. Combining Equations (16) and (17), the SLNR for MS_k is given by:

[0093] $\begin{array}{cc}\begin{array}{c}{\mathrm{SLNR}}_k=\frac{P_k}{P_{\mathrm{Nk}}+\sum _{\underset{\left(j\ne k\right)}{j}}{P}_{L\_\mathrm{kj}}},\\ =\frac{\mathrm{Trace}\left[Q_k^H{M}_k{Q}_k\right]}{\begin{array}{c}{N}_0{N}_R{L}_k+\\ \sum _{\underset{\left(j\ne k\right)}{j}}{P}_T\mathrm{Trace}\left[\sum _{\left(b1,b2\right)}{\beta }_{\mathrm{kj}}^{\left(b1,b2\right)}{Q}_k^{\left(b1\right)H}{H}_j^{\left(b1\right)H}{H}_j^{\left(b2\right)}{Q}_k^{\left(b2\right)}\right]\end{array}},\\ =\frac{\mathrm{Trace}\left[Q_k^H{M}_k{Q}_k\right]}{\mathrm{Trace}\left[Q_k^H{N}_k{Q}_k\right]}=\frac{\sum _{l=1}^{L_k}{q}_{\mathrm{kl}}^H{M}_k{q}_{\mathrm{kl}}}{\sum _{l=1}^{L_k}{q}_{\mathrm{kl}}^H{N}_k{q}_{\mathrm{kl}}},\end{array}& \left(18\right)\end{array}$
where q_kl is the l^th column of the matrix Q_k, M_k=P_TH_k^HH_k, and

[0094] $\begin{array}{cc}{N}_k=N_0{N}_{R}I+\sum _{\underset{\left(j\ne k\right)}{j}}{P}_T{A}_{\mathrm{kj}},\mathrm{where}\mathrm{the}\mathrm{matrix}{A}_{\mathrm{kj}}\mathrm{is}\mathrm{given}\mathrm{by}{A}_{\mathrm{kj}}=\left[\begin{array}{cccc}{\beta }_{\mathrm{kj}}^{\left(11\right)}{H}_j^{\left(1\right)H}{H}_j^{\left(1\right)}& {\beta }_{\mathrm{kj}}^{\left(12\right)}{H}_j^{\left(1\right)H}{H}_j^{\left(2\right)}& \cdots & {\beta }_{\mathrm{kj}}^{\left(1B\right)}{H}_j^{\left(1\right)H}{H}_j^{\left(B\right)}\\ {\beta }_{\mathrm{kj}}^{\left(21\right)}{H}_j^{\left(2\right)H}{H}_j^{\left(1\right)}& {\beta }_{\mathrm{kj}}^{\left(22\right)}{H}_j^{\left(2\right)H}{H}_j^{\left(2\right)}& \cdots & {\beta }_{\mathrm{kj}}^{\left(2B\right)}{H}_j^{\left(2\right)H}{H}_j^{\left(B\right)}\\ ⋮& ⋮& ⋰& ⋮\\ {\beta }_{\mathrm{kj}}^{\left(B1\right)}{H}_j^{\left(B\right)H}{H}_j^{\left(1\right)}& {\beta }_{\mathrm{kj}}^{\left(B2\right)}{H}_j^{\left(B\right)H}{H}_j^{\left(2\right)}& \cdots & {\beta }_{\mathrm{kj}}^{\left(\mathrm{BB}\right)}{H}_j^{\left(B\right)H}{H}_j^{\left(B\right)}\end{array}\right].& \left(19\right)\end{array}$

[0095] The optimization of the pre-coding matrices T₁, . . . , T_K is decoupled. Even so, directly maximizing Equation (18) with respect to q_k1˜q_kLk is still intractable. Therefore, we derive a lower bound that can be maximized analytically. Based on an inequality property, the SNLR_k in Equation (18) can be lower bounded as:

[0096] $\begin{array}{cc}{\mathrm{SLNR}}_k\ge \underset{l=1\dots L_k}{\min }\left(\frac{q_{\mathrm{kl}}^H{M}_k{q}_{\mathrm{kl}}}{q_{\mathrm{kl}}^H{N}_k{q}_{\mathrm{kl}}}\right).& \left(20\right)\end{array}$

[0097] Then we derive the following lemma:

[0098] Lemma:

[0099] The following values for the columns q_k1˜q_kLk maximizes the lower bound of the SLNR_k in Equation (20):
q_kl=v_l(N_k⁻¹M_k),  (21)
where v_l(A) represents the eigenvector of the matrix A corresponding to the l^th largest eigenvalue.

[0100] Proof:

[0101] Because q_kl˜q_kLk are orthonormal vectors, the vector space V=span {q_kl˜q_kLk} has a dimension dim V=L_k. Because the matrix M_k is Hermitian and N_k is positive-definite, the Courant-Fischer Max-Min theorem can be applied, J. R. Schott, Matrix analysis for statistics, 2^nd ed, Wiley, 2004; C. D. Meyer, Matrix analysis and applied linear algebra, SIAM, 2000; and G. Golob and C. V. Loan, Matrix computations, 3^rd edition, The John Hopkins University Press, 1996, incorporated herein by reference.

[0102] Then, the maximum value of the lower bound in Equation (20) is given by

[0103] $\begin{array}{cc}\begin{array}{c}\underset{q_{k1}~q_{{\mathrm{kL}}_k}}{\max }\left\{\underset{l=1\dots L}{\min }\left(\frac{q_{\mathrm{kl}}^H{M}_k{q}_{\mathrm{kl}}}{q_{\mathrm{kl}}^H{N}_k{q}_{\mathrm{kl}}}\right)\right\}=\underset{V}{\max }\left\{\underset{\underset{\dim V=L_k}{q\in V}}{\min }\left(\frac{q^H{M}_{k}q}{q^H{N}_{k}q}\right)\right\}\\ ={\lambda }_{L_k}\left(N_k^{-1}{M}_k\right),\end{array}& \left(22\right)\end{array}$
where λ_L(A) is the L^th largest eigenvalue of the matrix A. By inspection, Equation (21) satisfies Equation (22) in equality.

[0104] Consequently, the single closed-form solution in Equation (21) makes it less complex than JWF, c.f. Appendix A.

[0105] As a special case, when L_k=1, Equation (18) can be maximized directly by applying the Rayleigh-Ritz quotient theorem. This reduces to

[0106] $\begin{array}{cc}{\mathrm{SLNR}}_k=\frac{q_{k1}^H{M}_k{q}_{k1}}{q_{k1}^H{N}_k{q}_{k1}}\le {\lambda }_1\left(N_k^{-1}{M}_k\right).& \left(23\right)\end{array}$

[0107] It is worthwhile to note similarities between SLNR and SINR, which is actually the more relevant metric to optimize. In the expression for SINR for MS_k,

[0108] $\begin{array}{cc}\begin{array}{c}{\mathrm{SINR}}_k=\frac{P_k}{P_{\mathrm{Nk}}+\sum _{\underset{\left(j\ne k\right)}{j}}{P}_{L\_\mathrm{jk}}}\\ =\frac{\mathrm{Trace}\left[Q_k^H{M}_k{Q}_k\right]}{N_0{N}_R+\sum _{\underset{\left(j\ne k\right)}{j}}\mathrm{Trace}\left[\sum _{\left(b1,b2\right)}{\beta }_{\mathrm{jk}}^{\left(b1,b2\right)}{Q}_j^{\left(b1\right)H}{H}_k^{\left(b1\right)H}{H}_k^{\left(b2\right)}{Q}_j^{\left(b2\right)}\right]},\end{array}& \left(24\right)\end{array}$
the interference power term in the denominator sums over the different pre-coding matrices T_K of different MSs, while the channel realization term is the same. On the other hand, the leakage power of SLNR_k in Equation (18) sums over the different channels associated with different MSs, while the pre-coding matrix is the same. SINR and SLNR are random variables, as they are functions of the channel state, which is a random variable. It turns out that for two MSs, K=2, SINR_k and SLNR_k are identically distributed. The fact that the overall amount of power of the interference equals that of the power of interference leakage, suggests that the JLS method described above is still effective for inter-cell interference reduction.

[0109] The two above described embodiments provided closed-form solutions by optimizing metrics different from the sum-rate spectral efficiency metric in Equation (8). Most prior art for intra-cell interference reduction methods use iterative processes, based on a certain criterion, such as maximum sum rate, minimum signal-to-interference-plus-noise-ratio (SINR), minimum power, or iterative optimizations with gradient descend. However, in general, those processes suffer from non-convexity issues that result in suboptimum local solutions, instead of a global optimal solutions, and require a search for appropriate initial points, and are not designed for inter-cell interference.

[0110] To directly improve the spectral efficiency in our cooperative BS network, while keeping acceptable computational complexity, we describe an iterative method to optimize the target function of Equation (8), as shown in FIG. 3.

[0111] In step 1, determine the pre-coding matrices T₁, . . . , T_k for k=1, . . . , K with Equation (21) for all mobile stations K.

[0112] In step 2, each pre-coding matrix T_k is optimized independently, while keeping all other pre-coding matrices {T_j}_j≠k fixed.

[0113] In step 3, determine if a termination condition is reached, e.g., an increase in a value of the target function in Equation (8) is less than a pre-defined threshold, and otherwise repeat step 2.

[0114] Each step is a conventional singular value decomposition (SVD) on the pre-coding matrix, and a conventional water-filling power allocation on an equivalent matrix H_k—equ=Φ^−1/2H_k, with unit additive noise power.

[0115] Given the non-linear nature of the problem, the iterations are continued only as long as the target function increases, or the increase is less than a predetermined threshold. The starting point also plays an important role for the optimization. The iterations are initialized with the solution in Equation (21). The whole process can be interpreted as a “hill climbing” process, or a controlled iterative SVD (CISVD).

[0116] These steps guarantee that the process terminates. Compared with conventional random or exhaustive search process, our method intentionally optimizes one pre-coder in each step to improve the performance of the corresponding MS, while imposing a relatively low level of interference on other MSs.

[0117] When the timing advance is accurate, the joint BS pre-coding methods described above ensure that the desired signal components arrived synchronously. However, inaccurate timing advances are inevitable in practical cooperative base station MIMO networks, because of errors in delay estimation, station mobility, and inaccurate BS and MS synchronization.

[0118] The timing advance error or jitter of BS_b in sending the signal for MS_k is denoted by J_k^(b). Therefore, BS_b advances the time when the signal x_k^(b)(m) for MS_k is transmitted by an time interval offset
Δ{tilde over (τ)}_k^(b)=τ_k^(b)−τ_k^(bk)+J_k^(b)=Δτ_k^(b)+J_k^(b).  (25)

[0119] The joint and marginal statistics of the jitters are assumed to be known at the cooperative base stations. Consequently, the delay offset at MS_k due to the signal transmitted by BS_b for MS_j is
{tilde over (τ)}_jk^(b)=Δτ_k^(b)−Δ{tilde over (τ)}_j^(b).  (26)

[0120] In addition to the MUI term, J_k in Equation (10.1), inaccurate timing advances also result in inter-symbol interference (ISI). By applying Equation (25), Equation (1) is modified to

[0121] $\begin{array}{cc}\begin{array}{c}{y}_k\left(m\right)=\sum _b{\gamma }_k^{\left(b\right)}{H}_k^{\left(b\right)}{T}_k^{\left(b\right)}{s}_k\left(m\right)+\sum _b{\alpha }_k^{\left(b\right)}{H}_k^{\left(b\right)}{T}_k^{\left(b\right)}{s}_k\left(m_k^{\left(b\right)}\right)+\\ \sum _{\underset{\left(j\ne k\right)}{j}}\sum _b{H}_k^{\left(b\right)}{T}_j^{\left(b\right)}{\tilde{i}}_{\mathrm{jk}}^{\left(b\right)}+n_k\\ =H_k{\Lambda }_k{T}_k{s}_k+O_k+J_k+n_k,\end{array}& \left(27\right)\end{array}$
where

[0122] ${\gamma }_k^{\left(b\right)}=\rho \left(-J_k^{\left(b\right)}\right)\le 1,O_k=\sum _b{\alpha }_k^{\left(b\right)}{H}_k^{\left(b\right)}{T}_k^{\left(b\right)}{s}_k\left(m_k^{\left(b\right)}\right)$
is the ISI term with α_k^(b) and m_k^(b) given by

[0123] $\left\{{\alpha }_k^{\left(b\right)},m_k^{\left(b\right)}\right\}=\{\begin{array}{cc}\left\{\rho \left(T_S-J_k^{\left(b\right)}\right),m+1\right\},& \mathrm{if}{J}_k^{\left(b\right)}>0\\ \left\{\rho \left(-T_S-J_k^{\left(b\right)}\right),m-1\right\},& \mathrm{if}{J}_k^{\left(b\right)}<0\\ \left\{0,m\right\},& \mathrm{if}{J}_k^{\left(b\right)}=0.\end{array}$

[0124] The block diagonal matrix Λ_k=blockdiag{γ_k⁽¹⁾I_NT . . . γ_k^(B)I_NT} can be interpreted as a power degradation matrix due to inaccurate signal synchronization. It is equal to I_NTB for an accurate timing advance.

[0125] From Equation (27), the information rate of MS k is:
R_k=log|I+Φ_k⁻¹H_kΛ_kT_kT_k^HΛ_k^HH_k^H|,  (28)
where the covariance of the noise plus interference term now becomes:

[0126] ${\Phi }_k=N_{0}I+\sum _{\left(b1,b2\right)}{\alpha }_k^{\left(b1\right)}{\alpha }_k^{\left(b2\right)}{H}_k^{\left(b1\right)}{T}_k^{\left(b1\right)}{T}_k^{\left(b2\right)H}{H}_k^{\left(b2\right)H}1\left(\mathrm{sgn}\left(J_k^{\left(b1\right)}\right)-\mathrm{sgn}\left(J_k^{\left(b2\right)}\right)\right)+\sum _{\underset{\left(j\ne k\right)}{j}}\sum _{\left(b1,b2\right)}{\tilde{\beta }}_{\mathrm{jk}}^{\left(b1,b2\right)}{H}_k^{\left(b1\right)}{T}_j^{\left(b1\right)}{T}_j^{\left(b2\right)H}{H}_k^{\left(b2\right)H}.$

[0127] Here the indicator function 1 ( . . . ) equals one if the input argument is zero, and zero otherwise, and sgn(x) is the function

[0128] $\mathrm{sgn}\left(x\right)=\{\begin{array}{cc}1,& x\ge 0\\ -1,& x<0.\end{array}$

[0129] The asynchronous interference coefficients

[0130] $\left\{{\tilde{\beta }}_{\mathrm{jk}}^{\left(b1,b2\right)}\right\}$
due to the inaccurate timing advances are now determined by

[0131] $\left\{{\tilde{\tau }}_{\mathrm{jk}}^{\left(b\right)}\right\}$
in Equation (26), in the same way as τ_jk^(b) was determined in Equation (5).

[0132] The same timing inaccuracy in Equation (25) makes the cooperative BSs have inaccurate timing advance estimation in Equation (26), which leads to errors in the determination of

[0133] $\left\{{\tilde{\beta }}_{\mathrm{jk}}^{\left(b1,b2\right)}\right\}.$

[0134] As we can see from Equations (27) and (28), the timing advance inaccuracy degrades performance due to the power degradation, term Λ_k, the erroneous estimation of

[0135] $\left\{{\tilde{\beta }}_{\mathrm{jk}}^{\left(b1,b2\right)}\right\},$
and the additional ISI term O_k. While the exact value of the jitter is unknown, its statistics can be determined and exploited by the cooperating BSs to reduce the performance degradation, as follows.

[0136] Forms of the JWF and JLS methods that incorporate the timing advance inaccuracy in addition to asynchronous interference leakage are derived in Appendix B.

[0137] For JWF, the joint pre-coding matrix of MS_k is:
T_k=[C′_k+κ_kI]⁻¹ Λ_kH_k^H,  (29)
where Λ_k=blockdiag{ γ_k⁽¹⁾I_NT . . . γ_k^(B)I_NT}. The matrix C′_k, which is similar in form to matrix C in Equation (14), has submatrices

[0138] $\begin{array}{cc}{C}_k^{\prime \left(b1b2\right)}=\{\begin{array}{cc}\sum _{\underset{\left(j\ne k\right)}{j}}{\stackrel{\stackrel{\_}{~}}{\beta }}_{\mathrm{kj}}^{\left(b1b2\right)}{H}_j^{\left(b1\right)H}{H}_j^{\left(b2\right)}+\stackrel{\_}{{\alpha }_k^{\left(b1\right)}{\alpha }_k^{\left(b2\right)}}{P}_k^{\left(b1b2\right)}{H}_k^{\left(b1\right)H}{H}_k^{\left(b2\right)},& \mathrm{for}b1\ne b2\\ \sum _{j=1}^K{\stackrel{\stackrel{\_}{~}}{\beta }}_{\mathrm{kj}}^{\left(bb\right)}{H}_j^{\left(b\right)H}{H}_j^{\left(b\right)}+\left(\stackrel{\_}{a_k^{\left(b\right)2}}+\stackrel{\_}{{\gamma }_k^{\left(b\right)2}}\right)H_k^{\left(b\right)H}{H}_k^{\left(b\right)},& \mathrm{for}b1=b2=b,\end{array}\mathrm{where}{\stackrel{\_}{\gamma }}_k^{\left(b\right)}=E_{J_k^{\left(b\right)}}\left\{\rho \left(-J_k^{\left(b\right)}\right)\right\},{\stackrel{\_}{\alpha }}_k^{\left(b\right)}=\mathrm{Pr}\left(J_k^{\left(b\right)}\ge 0\right)E_{J_k^{\left(b\right)}}\left[\rho \left(T_S-J_k^{\left(b\right)}\right)\right]+\mathrm{Pr}\left(J_k^{\left(b\right)}<0\right)E_{J_k^{\left(b\right)}}\left[\rho \left(-T_S-J_k^{\left(b\right)}\right)\right],P_k^{\left(b1b2\right)}=\mathrm{Pr}\left\{\mathrm{sgn}\left(J_k^{\left(b1\right)}\right)=\mathrm{sgn}\left(J_k^{\left(b2\right)}\right)\right\},\stackrel{\_}{{\alpha }_k^{\left(b1\right)}{\alpha }_k^{\left(b2\right)}}=E_{J_k^{\left(b1\right)}{J}_k^{\left(b2\right)}}\left[{\alpha }_k^{\left(b1\right)}{\alpha }_k^{\left(b2\right)}\right],\stackrel{\_}{a_k^{\left(b\right)2}}=E_{J_k^{\left(b\right)}}\left[a_k^{\left(b\right)2}\right],\stackrel{\_}{{\gamma }_k^{\left(b\right)2}}=E_{J_k^{\left(b\right)}}\left[{\gamma }_k^{\left(b\right)2}\right],\mathrm{and}{\stackrel{\stackrel{\_}{~}}{\beta }}_{\mathrm{kj}}^{\left(b1,b2\right)}=E_{J_k^{\left(b\right)}}\left[{\tilde{\beta }}_{\mathrm{jk}}^{\left(b1,b2\right)}\right].& \left(30\right)\end{array}$