Multi-Cell Cooperation for Future Wireless Systems 169 3. Centralized multi-cell based system We consider a multi-cell system based on the scenario defined in previous section where the BSs are transparently linked by optical fiber to a central unit. Thanks to the high speed backhaul, we can assume that all the information of all BSs, i.e., full CSI and data, belonging to the same super-cell are available at the JPU. Thus, to remove the multi-cell multiuser interference we can use a similar linear precoding algorithm designed for single cell based systems. The major difference between multi-cell and single cell systems is that the power constraints have to be considered on a per-BS basis instead. The proposed schemes are considered in two phases: singular value decomposition SVD based precoding and power allocation. 3.1 System model To build up the mathematical model we consider that user ,1, ,kk K = can receive up to k r N data symbols on subcarrier ,1, , c ll N = i.e., ,,1, ,, [] r k T kl k l kN l xx=…x and the global symbol vector, comprising all user symbol vectors, is 1, , =[ … ] TTT ll Kl xx x of size 1 r N × . The data symbol of user k on subcarrier l, is processed by the transmit precoder , tr k kl NN C × ∈W in JPU, before being transmitted over BSs antennas. These individual precoders together form the global transmit precoder matrix on subcarrier l , 1, , = ll Kl ⎡ ⎤ ⎣ ⎦ WW W of size tr NN× . Let the downlink transmit power over the t N distributed transmit antennas for user k and data symbol , 1, , k r ii N = on subcarrier l, be p k,i,l , with ,,1, ,, =… r k kl k l kN l pp ⎡ ⎤ ⎣ ⎦ p and the global power matrix [ ] { } 1, , =diag llKl P pp is of size rr NN× . Under the assumption of linear precoding, the signal transmitted by the JPU on subcarrier l is given by 1/2 = ll l l W Pxz and the global received signal vector on subcarrier l can be expressed by, 1/2 =+ lll ll l yHWP xn (1) where 1, , = T TT ll Kl ⎡⎤ ⎣⎦ HH H of size rt NN × is the global frequency flat fading MIMO channel on subcarrier l . The channel of user k is represented by , 1,, ,, ,,kl kl bkl Bkl ⎡⎤ = ⎣⎦ HH H H of size k rt NN × , and ,,bkl H of size kb rt NN× represents the channel between user k and BS ,1, ,bb B = on subcarrier l . The channel ,,bkl H can be decomposed as the product of the fast fading ,, c bkl H and slow fading ,bk ρ components, i.e., ,, ,, , = c bkl bkl bk ρ HH , where ,bk ρ represents the long-term power gain between BS b and user k and ,, c bkl H contains the fast fading coefficients with () 0,1CN entries. 1, , = T TT ll Kl ⎡⎤ ⎣⎦ nn n represents the global additive white Gaussian noise (AWGN) vector and ,,1, ,, r k T kl k l kN l nn ⎡ ⎤ =… ⎣ ⎦ n is the noise at the user k terminal on subcarrier l with zero mean and power 2 σ , i.e., 2 ,, E[ ]= r k H kl kl N σ nn I . The signal transmitted by the BS b on subcarrier l can be written as 1/2 ,, = bl bl l l W Pxz , where ,bl W of size b tr NN × represents the global precoder at BS b on subcarrier l . The average transmit power of BS b is then given by, Recent Advances in Wireless Communications and Networks 170 2 ,, ,, , 111 E ,, r kc N N K H b bkl bkl ii kil kil === ⎡⎤ ⎡ ⎤ = ⎣ ⎦ ⎣⎦ ∑∑∑ WW p z (2) where b z is the signal transmitted over the c N subcarriers and ,,bkl W of size bk tr NN× represents the precoder of user k on subcarrier l at BS b . 3.2 Centralized precoder vectors In this section, we consider the SVD based precoding algorithm similar to the one proposed in (Yu et al., 2004). We assume that tr NN≥ . Briefly, we define ,kl H  as the following () - k rr t NN N× matrix, , 1, -1, 1, , T kl l k l k l Kl+ ⎡ ⎤ ⎣ ⎦ H=H H , H H  (3) If we denote rank of ,kl H  as ,kl L  then the null space of ,kl H  has dimension of , - k tkl r NL N≥  . The SVD of ,kl H  is partitioned as follows, (0) (1) ,,, ,, = H kl kl kl kl kl ⎡ ⎤ ⎦ ⎣ HUDVV  (4) where (0) ,kl V  holds the , - tkl NL  singular vectors in the null space of ,kl H  . The columns of (0) ,kl V  are candidate for user k precoding matrix ,kl W , causing zero gain at the other users, hence result in an effective SU-MIMO system. Since (0) ,kl V  potentially holds more precoders than the number of data streams user k can support, an optimal linear combination of these vectors must be found to build matrix ,kl W , which can have at most k r N columns. To do this, the following SVD is formed, (0) (0) (1) ,,, ,,, = H kl kl kl kl kl kl ⎡ ⎤ ⎦ ⎣ HV UD V V  (5) where ,kl D is ,,kl kl LL × and (1) ,kl V represents the ,kl L singular vectors with non-zero singular values. The , k kl r LN ≤ columns of the product (0) (1) ,,kl kl VV  represent precoders that further improve the performance subject to producing zero inter-user interference. The transmit precoder matrix will thus have the following form, (0) (1) (0) (1) 1/2 1/2 1, 1, , , llll l l Kl Kl ⎡⎤ == ⎣⎦ WVV VVP WP  (6) The global precoder matrix with power allocation, 1/ 2 1, , ll Kll ⎡⎤ = ⎣⎦ WW WP as computed above, block-diagonalizes the global equivalent channel l H , i.e., { } ,1, , , diag , , ll el eKl ⎡⎤ = ⎣⎦ HW H H … and the interference is completely removed considering perfect CSI. Let us define 1/ 2 ,, , , , , , = ekl klkl klklkl =HHWHWP of size kk rr NN × as the equivalent enhanced channel for user k on subcarrier l , where ,, =diag{ } kl kl P p is of size kk rr NN × . Rewriting equation (1) for this user, we have, ,,,,, =+ kl ekl kl kl yxnH (7) To estimate ,kl x , user k processes ,kl y by doing maximal ratio combining (MRC), and the soft decision variable , ˆ kl x is given by Multi-Cell Cooperation for Future Wireless Systems 171 ,,,,,,,,,,,, ˆ == + HH H kl ekl kl ekl ekl kl ekl kl xy xnHHHH (8) It should be mentioned that channel ,,ekl H can be easily estimated at UT k . It can be shown that, { } , ,, ,, ,1, ,1, , , , , diag , , rk r k H eklekl klkl kN lkNl pp λλ ⎡ ⎤ = ⎣ ⎦ …HH (9) where ,,kil λ is the i th singular value of matrix ,,kl kl HW. From equations (8) and (9) is easy to see that the instantaneous SNR of data symbol i of user k on subcarrier l can be written as ,, ,, ,, 2 SNR kil kil kil p λ σ = (10) From (10), assuming a M-ary QAM constellations, the instantaneous probability of error of data symbol i of user k on subcarrier l is given by (Proakis, 1995), ( ) ,,, ,,ekil kil PQSNR ψβ = (11) where () () 2 /2 () 1/ 2 t x Qx e dt π ∞ − = ∫ , ( ) 3/ 1M β = − and () ( ) 2 4/log 1 1/ M M ψ =−. 3.3 Power allocation strategies Once the multi-cell multiuser interference removed, the power loading elements of l P can be computed in order to minimize or maximize some metrics. Most of the proposed power allocation algorithms for precoded multi-cell based systems have been designed to maximize the sum rate, e.g., (Jing et al., 2008; Bjornson et al., 2010). In this paper, the criteria used to design power allocation are minimization of the average BER and sum of inverse of SNRs, which essentially lead to a redistribution of powers among users and therefore provide users fairness (which in practical cellular systems may be for the operators a goal as important as throughput maximization). The aim of these power allocation schemes is to improve the user’s fairness, namely inside each super-cell. A. Optimal minimum BER power allocation We minimize the instantaneous average probability under the per-BS power constraint tb P , i.e., ,, ,, , 111 , 1, , ,, r kc N N K H bkl bkl tb ii kil Pb B kil p === ⎡⎤ ≤= ⎣⎦ ∑∑∑ WW . Without loss of generality, we assume a 4-QAM constellation, and thus the optimal power allocation problem with per-BS power constraint can be formulated as, {} ,, ,, ,, ,, ,, , 111 2 111 ,, , 1, , 1 ,, min s.t. 0, 1, , , 1, , , 1, , r kc r kc kil k k N N K N H N K kil kil bkl bkl tb ii kil p rc kil kil r c p Pb B kil Q KN N pKiNlN p λ σ === === ⎧ ⎛⎞ ⎛⎞ ⎡⎤ ⎪ ≤= ⎣⎦ ⎜⎟ ⎜⎟ ⎨ ⎜⎟ ⎜⎟ ⎪ ⎝⎠ ⎝⎠ ≥= = = ⎩ ∑∑∑ ∑∑∑ WW k (12) Since the objective function is convex in ,,kil p , and the constraint functions are linear, this is a convex optimization problem. Therefore, it may be solved numerically by using for Recent Advances in Wireless Communications and Networks 172 example the interior-point method (Boyd & Vandenberghe, 2004). This scheme is referred as centralized per-BS optimal power allocation (Cent. per-BS OPA). B. Suboptimal power allocation approaches Since the complexity of the above scheme is too high, and thus it could not be of interest for real wireless systems, we also resort to less complex suboptimal solutions. The proposed strategy has two phases: first the power allocation is computed by assuming that all BSs of each super-cell can jointly pool their power, i.e., a TPC t P is imposed instead and the above optimization problem reduces to, {} ,, ,, ,, ,, , 111 2 111 ,, 1 ,, min s.t. 0, 1, , , 1, , , 1, , r kc r kc kil k k N N K N H N K kil kil kl kl t ii kil p rc kil kil r c p P kil Q KN N p Ki N l N p λ σ === === ⎧ ⎛⎞ ⎛⎞ ⎡⎤ ⎪ ≤ ⎣⎦ ⎜⎟ ⎜⎟ ⎨ ⎜⎟ ⎜⎟ ⎪ ⎝⎠ ⎝⎠ ≥= = = ⎩ ∑∑∑ ∑∑∑ WW k (13) with ,, ,, , 111 111 ,, rr kc kc NN NN KK H kl kl kil ii kil kil p kil p === === ⎡⎤ = ⎣⎦ ∑∑∑ ∑∑∑ WW , note that the k r N columns of ,kl w have unit norm. Using the Lagrange multipliers method (Haykin, 1996), the following cost function with μ Lagrange multiplier is minimized, ,, ,, ,1 , , 2 111 111 1 rr kc kc k NN NN KK kil kil ckilt rc kil kil p JQ p P KN N λ μ σ === === ⎛⎞ ⎛⎞ ⎜⎟ =+− ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑∑ ∑∑∑ (14) The powers ,,kil p can be determined by setting the partial derivatives of ,1c J to zero and as shown in (Holakouei et al., 2011), the solution is () ,, 2 2 ,, 0 2 24 ,, 8 kil k kil kil rc pW KN N λ σ λ πμ σ ⎛⎞ ⎜⎟ = ⎜⎟ ⎜⎟ ⎝⎠ (15) where 0 W stands for Lambert’s W function of index 0 (Corless et al., 1996). This function 0 ()Wx is an increasing function. It is positive for 0x > , and 0 (0) 0W = . Therefore, 2 μ can be determined iteratively to satisfy ,, 111 r kc N N K kil t kil p P = == = ∑∑∑ . The optimization problem of (13) is similar to the single cell power allocation optimization problem, where the users are allocated the same total multi-cell power, which may serve as a lower bound of the average BER for the multi-cell with per-BS power constraint. One solution based on Lambert W function that minimizes the instantaneous BER was also derived in the context of single user single cell MIMO systems (Rostaing et al., 2002). The second phase consists in scaling the power allocation matrix l P by a factor of β in order to satisfy the individual per-BS power constraints as discussed in (Zhang et al., 2009) which can be given by ,, ,, 1, , , 111 ,, max r kc tb N N K H bkl bkl bB ii kil P kil p β = === = ⎛⎞ ⎡ ⎤ ⎜⎟ ⎣ ⎦ ⎜⎟ ⎝⎠ ∑∑∑ WW (16) Multi-Cell Cooperation for Future Wireless Systems 173 This scaled power factor assures that the transmit power per-BS is less or equal to tb P . Note that this factor is less than one and thus the SNR given by (10) has a penalty of ( ) 10log dB β . This scheme is referred as centralized per-BS suboptimal iterative power allocation (Cent. per-BS SOIPA). Although this suboptimal solution significantly reduces the complexity relative to the optimal one, it still needs an iterative search. To further simplify we propose an alternative power allocation method based on minimizing the sum of inverse of SNRs, and a closed- form expression can be obtained. Note that minimizing the sum of inverse of SNRs is similar to the maximization of the harmonic mean of the SINRs discussed in (Palomar, 2003). In this case, the optimization problem is written as, {} ,, 2 ,, , 111 ,, ,, 111 ,, ,, min s.t. 0, 1, , , 1, , , 1, , r kc r kc kil k N N K N H N K kl kl t ii kil p kil kil kil kil r c P kil p pKiNlN p σ λ === === ⎧ ⎛⎞ ⎡⎤ ⎪ ≤ ⎣⎦ ⎜⎟ ⎨ ⎜⎟ ⎪ ⎝⎠ ≥= = = ⎩ ∑∑∑ ∑∑∑ WW k (17) Since the objective function is convex in ,,kil p , and the constraint functions are linear, (17) is also a convex optimization problem. To solve it we follow the same suboptimal two phases approach as for the first problem. First, we impose a total power constraint and the following cost function, using again the Lagrangian multipliers method, is minimized, 2 ,2 , , ,, ,, 111 111 rr kc kc NN NN KK ckilt kil kil kil kil JpP p σ μ λ === === ⎛⎞ ⎜⎟ =+− ⎜⎟ ⎝⎠ ∑∑∑ ∑∑∑ (18) Now, setting the partial derivatives of ,2c J to zero and after some mathematical manipulations, the powers ,,kil p are given by, ,, ,, 111 ,, 1 r kc t kil N N K kil jnp j n p P p λ λ === = ∑∑∑ (19) The second phase consists in scaling the power allocation matrix l P by a factor of β , using (19) instead of (15), in order to satisfy the individual per-BS power constraints. This scheme is referred as centralized per-BS suboptimal closed-form power allocation (Cent. per-BS SOCPA). The above power allocation schemes can also be used, under minor modifications, for the case where the system is designed to achieve diversity gain instead of multiplexing gain. In diversity mode the same user data symbol is received on each receiver antenna, increasing the diversity order. Thus ,, , , , 1 1 rk k kil kN l r xx iN = =− and then the SNR is given by ,,, ,, 1 , 22 SNR r k N kl kil kl kl i kl p p λ α σ σ = == ∑ (20) and the power loading coefficient is computed only per user and subcarrier. In this case to compute the power allocation coefficients we should replace ,,kil λ by ,kl α and remove the script i in all equations. Recent Advances in Wireless Communications and Networks 174 4. Distributed multi-cell based system As discussed in section 2 due to limitations in terms of delay and capacity on backhaul network, it is necessary to reduce signalling overhead. For this purpose, in this section the precoders are designed in a distributed fashion, i.e., based on local CSI at each BS but we still consider data sharing and centralized power allocation techniques. 4.1 System model Assuming single antennas UTs and under the assumption of linear precoding, the signal transmitted by the BS b on sub-carrier l is given by, ,,,,,, 1 s, K bl bkl bkl kl k p = = ∑ x w (21) where p b,k,l represents the power allocated to UT k on sub-carrier l and BS b, 1 ,, t b N bkl × ∈w is the precoder of user k at BS b on sub-carrier l with unit norms, i.e., ,, 1, 1, , , 1, , , 1, , bkl c bBkKlN== = =w . The data symbol , s kl , with 2 , Es 1 kl ⎡⎤ = ⎢⎥ ⎣⎦ , is intended for UT k and is assumed to be available at all BSs. The average power transmitted by the BS b is then given by, 2 ,, 11 E c N K bbkl lk p == ⎡⎤ = ⎣⎦ ∑∑ x (22) where b x is the signal transmitted over the c N subcarriers. The received signal at the UT k on sub-carrier l , 11 , kl × ∈y , can be expressed by, ,,,,, 1 B H kl bkl bl kl b= =+ ∑ ynhx (23) where 1 ,, t b N bkl × ∈h represents the frequency flat fading channel between BS b and UT k on sub-carrier l and ( ) 2 , ~0, kl σ n CN is the noise. The channel ,,bkl h , as for the centralized approach, can be decomposed as the product of the fast fading ,, c bkl h and slow fading ,bk ρ components, i.e., ,, ,, , = c bkl bkl bk ρ hh , where ,bk ρ represents the long-term power gain between BS b and user k and ,, c bkl h contains the fast fading coefficients with ( ) 0,1CN entries. The antenna channels from BS b to user k , i.e. the components of ,, c bkl h , may be correlated but the links seen from different BSs to a given UT are assumed to be uncorrelated as the BSs of one super-cell are geographically separated. 4.2 Distributed precoder vectors As discussed above, to design the distributed precoder vector we assume that the BSs have only knowledge of local CSI, i.e., BS b knows the instantaneous channel vectors ,, ,, bkl kl∀h , reducing the feedback load over the backhaul network as compared with the full centralized precoding approach. We consider a zero forcing transmission scheme with the phase of the received signal at each UT aligned. From (21) and (23) the received signal at UT k on sub- carrier l can be decomposed in, Multi-Cell Cooperation for Future Wireless Systems 175  , ,, ,, ,, , ,, ,, ,, , , 111, S BBK HH kl bkl bkl bkl kl bkl b j lb j l j lkl bbjjk Noise Desired ignal Multiuser Multicell Interference ps ps ===≠ =+ + ∑∑∑   ynhw h w (24) where ,,bkl w is a unit-norm zero forcing vector orthogonal to 1K − channel vectors, { } ,, H bjl j k≠ h . Such precoding vectors always exist because we assume that the number of antennas at each BS is higher or equal to the number of single antenna UTs, i.e. b t NK≥ . Note that here K is the number of users that share the same set of resources. Considering an OFDMA based system, the total number of users can be significantly larger than K, since different set of resources can be shared by different set of users. By using such precoding vectors, the multi-cell interference is cancelled and each data symbol on each subcarrier is only transmitted to its intended UT. Also, for any precoding vector ,,bkl w in the null space of { } ,, H bjl j k≠ h , ,, ,, j bkl bkl e ϕ =ww is also in the null space of { } ,, H bjl j k ≠ h . Thus, we can choose the precoding vectors such that the terms ,, ,, H bkl bkl hw all have zeros phases, i.e., ( ) ,, ,, ()0, ,, H bkl bkl bkl∠=∀hw . These precoding vectors can be easily computed, so if ,,bkl W is found to lie in the null space of { } ,, H bjl j k ≠ h , the final precoding vector ,, , 1, , , bkl bB=w 1, , , 1, , c kKlN== , with the phase of the received signal at each UT aligned, is given by, ( ) ,, ,, ,, ,, ,, ,, bkl bkl H H bkl bkl bkl H bkl = h w h W W W (25) where ( ) 1 ,, tt bb NNK bkl × −+ ∈W holds the ( ) 1 b t NK − + singular vectors in the null space of { } ,, H bjl j k≠ h . For the case where b t NK = , only one vector lies in the null space of { } ,, H bjl j k≠ h , but for tb NK> more than one vector lie in the null space of { } ,, H bjl j k ≠ h . In this latter case, the final ,,bkl w vector is a linear combination of the ( ) 1 b t NK − + possible solutions. The equivalent channel between BS b and UT k , on sub-carrier l can be expressed as, ( ) ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, bkl bkl bkl H H bkl eq HH H bkl bkl bkl bkl bkl bkl H bkl === W WW W h hw h h h h (26) From (26) we can observe that the equivalent channel, ,, e q bkl h , is a positive real number. By using the precoding vectors defined in (25) and considering (26), the received signal in (24) reduces to, ,,,,, ,, 1 B eq kl bkl kl kl bkl b ps = =+ ∑ yhn (27) Recent Advances in Wireless Communications and Networks 176 It should be mentioned that at the UT, to allow high order modulations, only the ,, ,, e q bkl bkl p h coefficients are needed to be estimated instead of all the complex coefficients of the channel, leading to a low complexity UT design. Since the ( ) 1 b t NK−+ components of ,, ,, H bkl bkl h W are i.i.d. Gaussian variables, () 2 ,, eq bkl h is a chi-square random variable with ( ) 21 b t NK − + degrees of freedom. Once the ,, e q bkl h variables are independent, each user is expected to achieve a diversity order of ( ) 1 b t BN K−+ (assuming that all channels have the same average power, i.e., , , ( , ) bk bk ρ ρ =∀ and ,, 1, ( , , ) bkl p bkl = ∀ ). Also, because the received signals from different BSs have the same phase, they are added coherently at the UTs, and thus an additional antenna gain is achieved. 4.3 Power allocation strategies In this section the same three criteria considered for the centralized approach are used to design the power allocation. However, it should be emphasised that for this scenario only the equivalent channels, i.e., ,, e q bkl h , are needed to be known at the JPU. A. Optimal minimum BER power allocation From (27) the instantaneous SNR of user k on sub-carrier l can be written as, 2 ,, ,, 1 , 2 SNR B eq bkl bkl b kl p σ = ⎛⎞ ⎜⎟ ⎝⎠ = ∑ h (28) The instantaneous probability of error for user k is obtained in similar way in section 3. We minimize the instantaneous average probability under the per-BS power constraint b t P , i.e., ,, 11 , 1, , c b N K bkl t lk p Pb B == ≤= ∑∑ . By assuming a 4-QAM constellation, the optimal power allocation problem with per-BS power constraint can be formulated as, {} ,, ,, ,, ,, 1 11 11 ,, , 1, , 1 min s.t. 0, 1, , , 1, , , 1, , c c b bkl B eq N K bkl bkl N K bkl t b lk p c lk bkl c p pPb B Q KN p bB KlN σ = == == ⎛⎞ ⎛⎞ ⎧ ⎜⎟ ⎜⎟ ≤= ⎪ ⎜⎟ ⎜⎟ ⎨ ⎜⎟ ⎜⎟ ⎪ ⎜⎟ ⎜⎟ ≥= = = ⎜⎟ ⎩ ⎜⎟ ⎝⎠ ⎝⎠ ∑ ∑∑ ∑∑ h k (29) In this distributed approach, the objective function is convex in p b,k,l , and the constraint functions are linear this is also a convex optimization problem. Therefore, it may be also solved numerically by using for example the interior-point method. This scheme is referred as distributed per-BS optimal power allocation (Dist. per-BS DOPA). In this section, the distributed term is referred to the precoder vectors since the power allocation is also computed in a centralized manner. B. Suboptimal power allocation approaches As for the centralized approach, the complexity of the above scheme is too high, and thus it is not of interest for real wireless systems, we also resort to less complex suboptimal solutions. The proposed strategy has two phases: first the power allocation is computed by assuming that all BSs of each super-cell can jointly pool their power, i.e., a TPC P t is imposed instead and the above optimization problem reduces to, Multi-Cell Cooperation for Future Wireless Systems 177 {} ,, ,, ,, ,, 1 11 1 11 ,, 1 min s.t. 0, 1, , , 1, , , 1, , c c bkl B eq N BK bkl bkl N K bkl t b blk p c lk bkl c p pP Q KN p bB KlN σ = == = == ⎛⎞ ⎛⎞ ⎧ ⎜⎟ ⎜⎟ ≤ ⎪ ⎜⎟ ⎜⎟ ⎨ ⎜⎟ ⎜⎟ ⎪ ⎜⎟ ⎜⎟ ≥= = = ⎜⎟ ⎩ ⎜⎟ ⎝⎠ ⎝⎠ ∑ ∑∑∑ ∑∑ h k (30) with 1 b B tt b PP = = ∑ and using the Lagrange multipliers method, the following cost function with μ Lagrange multiplier is minimized, ,, ,, 1 ,1 , , 11 111 1 cc B eq bkl bkl NN KBK b dbklt c lk blk p JQ p P KN μ σ = == === ⎛⎞ ⎜⎟ ⎛⎞ ⎜⎟ =+− ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎜⎟ ⎜⎟ ⎝⎠ ∑ ∑∑ ∑∑∑ h (31) The powers ,, , ( , , ) bkl p bkl ∀ can be determined by setting the partial derivatives of ,1d J to zero and as shown in (Silva et al., 2011) the solution is, () () () 2 2 2 2 ,, ,, 1 ,, 0 22224 2 ,, 1 8 B eq eq ikl bkl i bkl B c eq ikl i pW NK σ πμ σ = = ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ = ⎜⎟ ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑ ∑ h h h (32) Therefore, 2 μ can be determined iteratively, using constraint ,, 11 1 c N BK bkl t blk p P = == = ∑∑∑ . The second phase consists of replacing 2 μ by 2 , 1, , b bB μ = in (32), and then computing iteratively different 2 b μ to satisfy the individual per-BS power constraints instead, i.e., 2 b μ are computed to satisfy, ,, 11 ,, , 1, , 0, 1, , , 1, , , 1, , c b N K bkl t lk bkl c pPb B p bB KlN == ⎧ ≤= ⎪ ⎨ ⎪ ≥= = = ⎩ ∑∑ k (33) This suboptimal scheme is referred as distributed per-BS sub-optimal iterative power allocation (Dist. per-BS SOIPA). Although this suboptimal solution significantly reduces the complexity relative to the optimal one, it still needs an iterative search. To further simplify we also propose for the distributed scenario, an alternative power allocation method based on minimizing the sum of inverse of SNRs. In this case, the optimization problem is written as, {} ,, 2 ,, 11 2 11 ,, ,, ,, 1 ,1, , min s.t. 0, 1, , , 1, , , 1, , c c b bkl N K N K bkl t lk p B lk eq bkl c bkl bkl b pPb B p bB KlN p σ == == = ⎛⎞ ⎜⎟ ⎧ ⎜⎟ ≤= ⎪ ⎜⎟ ⎨ ⎛⎞ ⎜⎟ ⎪ ≥= = = ⎩ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑ ∑∑ ∑ k h (34) Recent Advances in Wireless Communications and Networks 178 The objective function is convex in ,,bkl p , and the constraint functions are linear, (34) is also a convex optimization problem. To solve it we follow the same suboptimal two phases approach as for the first problem. First, we impose a total power constraint and the following cost function, using again the Lagrangian multipliers method, is minimized, 2 ,2 , , 2 11 111 ,, ,, 1 cc NN KBK dbklt B lk blk eq bkl bkl b JpP p σ μ == === = ⎛⎞ =+− ⎜⎟ ⎜⎟ ⎛⎞ ⎝⎠ ⎜⎟ ⎝⎠ ∑∑ ∑∑∑ ∑ h (35) Now, setting the partial derivatives of ,2d J to zero and after some mathematical manipulations, the powers ,,bkl p can be shown to be given by, ( ) () 2 ,, ,, 3 2 ,, 1 eq bkl bkl B eq ikl i p β = = ⎛⎞ ⎜⎟ ⎝⎠ ∑ h h (36) where 2 / β μσ = . As for the first approach, (36) can be re-written by replacing β by , 1, , b bB β = , which are computed to satisfy the individual per-BS power constraints and the closed-form solution achieved is then given by, ( ) () () () 2 ,, ,, 2 3 2 ,, ,, 3 111 2 ,, 1 b c eq t bkl bkl eq N BK bjp eq ikl B ipj eq ijp i P p === = = ⎛⎞ ⎜⎟ ⎝⎠ ⎛⎞ ⎜⎟ ⎝⎠ ∑∑∑ ∑ h h h h (37) This second suboptimal scheme is referred as distributed per-BS closed-form power allocation (Dist. per-BS SOCPA). The precoder vectors are designed by assuming that BSs have only knowledge of local CSI. However, since we consider a centralized power allocation, to compute all powers the ,, , eq bkl ∀hb,k,l coefficients should be available at the joint processing unit (JPU). In the distributed multi-cell system each BS should send a real vector of size c KN to the JPU. Note that in the centralized approach discussed in section 3, each BS should send to the JPU a complex vector of size b tc NKN , i.e. 2 b t N more information. Although, in this section single antenna UTs were assumed, the formulation can be straightforwardly extended for multiple antenna UTs just by considering each antenna as a single antenna UT. The main difference is that the long term channel power will be the same for all antennas belonging to the same UT. 5. Results and discussions 5.1 Simulation parameters In order to evaluate the proposed centralized and distributed multi-cell cooperation schemes, we assume ITU pedestrian channel model B (Guidelines IMT2000, 1997), with the [...]... admission control and so on As a part of resouce management, joint call admission control tightly interacts with vertical handoff and QoS provisioning schemes in integrated WLAN and 3G cellular networks 192 Recent Advances in Wireless Communications and Networks 2.2 Vertical handoff In integrated networks, there are two types of handoff: intra-technology handoff and intertechnology handoff (Lampropoulos... describes concepts, architecture and vertical handoffs in integrated WLAN and cellular networks 2.1 Architecture of integrated WLAN and 3G cellular networks Driven by the anywhere and anytime mobile service concept, it is expected that 4G wireless networks will be heterogeneous, integrating different networks to provide seamless Internet access for mobile users The integrated WLAN and 3G cellular network takes... Rostaing, P.; Berder, O.; Burel, G & Collin, L (2002) Minimum BER diagonal precoder for MIMO digital transmissions, Elsevier Signal Processing, vol 82, no 10, pp 1 477 -1480, 2002 186 Recent Advances in Wireless Communications and Networks Somekh, O.; Zaidel, B & Shamai, S (20 07) Sum rate characterization of joint multiple cellsite processing, IEEE Transaction of Information Theory, vol 53, no 12, pp 4 473 -... there are 196 Recent Advances in Wireless Communications and Networks several drawbacks of this policy method, such as high latencies to fetch context information during the candidate access point classification procedure, and no optimization policy is defined for resource allocation in integrated networks Nework health t monitor Network capacities BER Call blocking probabilities Call dropping probability... by considering congestion scenarios in both the WLAN and UMTS cellular networks Specifically, a system cost function is derived and 1 97 Joint Call Admission Control in Integrated Wireless LAN and 3G Cellular Networks minimized by admitting passive vertical handoffs with a probability, and it is proven that there exists at least one optimal value for the target passive handoff probability In this way,... Linear Precoding, IEEE Transaction on Wireless Communication, vol 8, no 4, pp 1910-1921, 2009 Zhang, R (2010) Cooperative multi-cell block diagonalization with per-base-station power constraints, in Proceeding of IEEE WCNC, 2010 Part 2 Upper Layers 9 Joint Call Admission Control in Integrated Wireless LAN and 3G Cellular Networks Chunming Liu, Chi Zhou, Niki Pissinou and S Kami Makki T-Mobile, Illinois... priority for resource than new voice and data call requests, while WLANs only support coarse packet-level access without considering handoffs priorities So in Joint Call Admission Control in Integrated Wireless LAN and 3G Cellular Networks 193 integrated WLAN and 3G cellular networks, seamless vertical handoffs and call admission control must be considered as dependent and joint mechanisms to ensure both... wireless/ optical architecture, In Proceeding of IEEE WCNC’10, 2010 Foschini, G J & Gans, M J (1998) On limits of wireless communications in a fading environment when using multiple antenna, Wireless Personal Communication Magazine, vol 6, no 3, pp 311-335, 1998 Foschini, G J et al (2006) Coordinating multiple antenna cellular networks to achieve enormous spectral efficiency, in IEEE Proceedings on Communications, ... authentication in the WLAN (Liu & Zhou, 2005a) In contrast to high cost of Tight Coupling, the Loose Coupling is an IP-based mechanism, and approach separates the data paths in the 802.11 WLAN and 3G cellular networks (Liu, 2006) The 802.11 WLAN gateway routers connect to the Internet, and all data traffic is 191 Joint Call Admission Control in Integrated Wireless LAN and 3G Cellular Networks Internet GGSN... al., 2011) The intra-technology handoff is traditional Horizontal Handoff (HHO) in which mobile terminals handoff between two adjacent base stations or access points using same access technology In contrast, inter-technology handoff is called Vertical Handoff (VHO), and happens when mobile terminals roam between two networks with different access technologies, for example, between WLAN and 3G UMTS network . it may be solved numerically by using for Recent Advances in Wireless Communications and Networks 172 example the interior-point method (Boyd & Vandenberghe, 2004). This scheme is. using the precoding vectors defined in (25) and considering (26), the received signal in (24) reduces to, ,,,,, ,, 1 B eq kl bkl kl kl bkl b ps = =+ ∑ yhn ( 27) Recent Advances in Wireless. = ⎩ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑ ∑∑ ∑ k h (34) Recent Advances in Wireless Communications and Networks 178 The objective function is convex in ,,bkl p , and the constraint functions are linear, (34) is also a