# THE downlink of multiuser multiple-input multiple-output

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5 DABBAGH AND LOVE: PRECODING FOR MULTIPLE ANTENNA GAUSSIAN BROADCAST CHANNELS 3841 right singular vectors of and is the set of vectors consisting a basis for the null space of. Note that, for, we let be the identity matrix. Since, for each the precoding matrix, is constrained to lie in the subspace spanned by achievable throughput of the system is given by, the (10) In order to determine the optimal set of precoding matrices given the set of users, a similar algorithm as in [21] can be followed; however, the basis for the null space of each user is now given by. Given the ordered set of users, let (11) Given the set of ordered users, let denote the sumrate capacity of the cooperative transmission scheme. In the following, the optimality of the SZFDPC scheme in the high-snr regime is established. Theorem 1: In the high-snr regime, given an arbitrary set of ordered users and assuming that and are of full row rank, the sum-rate capacity of the successive zero-forcing dirty-paper coding transmission scheme is asymptotically equal to the sum-rate capacity of the cooperative transmission scheme, i.e., (15) Proof: First note that when the users are allowed to cooperate, the effective multiuser channel is actually the single user point-to-point MIMO channel. If we let denote the rank of, the capacity of this channel is given by [23] (16) denote the optimal set of precoding matrices. Then (12) where and is the set of eigenvalues of the matrix is the solution of where for each singular value decomposition is obtained from the Similarly, the sum-rate capacity of the SZFDPC scheme is given by and is the matrix obtained by waterfilling over the singular values of the matrix (17)... (13) where and is the set of eigenvalues of the matrix is the solution of with power constraint. In general, multiuser diversity can be achieved if the base station maximizes the sum-rate capacity over different ordered sets of users. In this case, we need to consider different users orderings which can affect the achievable throughput because of the null space constraints; however, we can only consider sets of users with cardinality equal to due to the waterfilling algorithm which determines the optimal number of users that should be supported in any ordered set of users. Therefore, the sum-rate capacity of the SZFDPC transmission scheme with multiuser diversity is given by In order to show the asymptotic optimality of the SZFDPC transmission scheme, we need to show that the difference between the capacity expressions given in (16) and (17) goes to zero as. In order to do this, we will first show that This will imply that the geometric means given by (18) (14) In order to achieve multiuser diversity, we need to consider the different ordered sets as in (14). The problem with the maximization in (14) is that the number of different sets scales exponentially with the number of users in the system. Therefore, an efficient user selection technique is generally required especially when the number of users is large. are equal and similar argument as in [2] can be used to prove (15). In the following, we prove (18) by induction. Let us consider the matrix as given in (2), and let us write (19)

6 3842 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 From (19), it follows that Similarly, we have According to [12, p. 22] and assuming that the matrix is of full row rank and if we continue in the same way for each, we get Therefore, (18) follows immediately from the definition of. Now, similar to [2], let Now let (20) (21) then, for some, the waterfilling equation be the singular value decomposition of the matrix. Then, from (20) and (21) has solution, and has solution. Therefore, for, the sum-rate capacities of the cooperative and SZFDPC schemes are given by (22) Note that, if we let denote the set of right singular vectors corresponding to the first nonzero singular values of, i.e., and and in the limit, we get then from (22), we get Finally, since Let the sum-rate capacity of the optimal DPC transmission scheme be denoted by. The following corollary follows directly from Theorem 1 and from the fact that the sum-rate capacity of the optimal DPC scheme is both lower bounded and upper bounded by the sum-rate capacities of the SZFDPC and cooperative schemes, respectively. Corollary 1: In the high-snr regime From the above, we note that in the high-snr regime, the DPC, as well as the SZFDPC schemes, are optimal in the sense of achieving the cooperative channel sum-rate capacity. Also,

7 DABBAGH AND LOVE: PRECODING FOR MULTIPLE ANTENNA GAUSSIAN BROADCAST CHANNELS 3843 note that the above results are true with any user ordering. However, user ordering is important in the low-snr regime and as it will be later shown, in the low-snr regime, optimal user ordering is required for the SZFDPC scheme to achieve the sum-rate capacity of the optimal DPC scheme. IV. PRECODING WITH SUCCESSIVE ZERO-FORCING The DPC technique generally requires nonlinear operations and is currently considered very complex to implement in a practical communication system. In this section, we consider a linear transmission scheme that is based on partial interuser interference cancellation. The motivation for considering this kind of transmission scheme is two-fold. First, compared to other linear transmission schemes, our goal is to improve the performance of the system by increasing the achievable system throughput given a transmit power constraint. Second, given that zero-forcing of the interuser interference is required, we are interested in relaxing the constraint on the number of antennas at the base station versus the number of users receive antennas. This can increase the number of users that can be supported simultaneously and in return increase the achievable throughput. For example, let us consider the following case where the base station is equipped with four transmit antennas and there are three users in the system with two receive antennas each. Furthermore, it is assumed that the first and second users channels are highly correlated; however, all other pairs of users channels are independent. An example for such a channel can be if two of the three users are located close to the base station and the third user is far away. Now, if we consider the nulling constraint given in Section III, there exists a certain ordering of users where the base station can support the three users simultaneously; however, it is not possible in this case to support the three users in an interference free environment. Therefore, this is a simple case where optimization of the sum-rate capacity can be performed over the total number of users in the system. This kind of optimization can not be performed if complete interference cancellation is required. We will consider the following transmission scheme. Given an ordered set of users, the base station designs the precoding matrix of each user according to the null space constraint as in Section III, i.e., the precoding matrix, which corresponds to user in, is constrained to lie in the null space of. However, in contrast to the SZFDPC transmission scheme, the base station does not employ a DPC scheme and users design their receivers such that the remaining interference from other users transmitted signals is treated as a Gaussian noise that is independent of the desired signal. Because users precoding matrices are designed successively based on the null space of where complete interference cancellation is not required, and we do not employ any DPC technique for combating the remaining interference from other users, we refer to this scheme as successive zero-forcing (SZF). A. Sum Rate Capacity of SZF Given the above transmission scheme, we need to characterize the sum-rate capacity of the SZF scheme and to find a way of designing the optimal set of precoding matrices. Note that from the above constraint on the set of precoding matrices, the received signal for each is given by (23) Now, assuming that the set of vectors have Gaussian distribution and are independent, it follows that the interference term is also Gaussian. By treating this spatially correlated interference signal as noise, and by applying a whitening filter on the received signal, a minimum Euclidean distance decoding strategy is optimal [16], and the maximal achievable rate of each user is given by (24), shown at the bottom of the page, for a given ordered set of users and set of covariance matrices such that for each Hence, given the above rates, for each the achievable system throughput of the SZF scheme is given by (25) and when there are users in the system, the maximal achievable throughput with multiuser diversity is (26) In order to find the maximal achievable throughput and the optimal set of precoding matrices, we need to solve the optimization problem given in (26). Therefore, we need first to be able to solve the optimization problem in (25). In general, because of the nonconvexity of the rate equations in (24) in the set of covariance matrices, solving (25) optimally is a difficult problem. We note that from the rate equations in (24), the sumrate capacity optimization problem of the SZF scheme is similar to the sum-rate capacity optimization problem of the BC with added subspace constraints on the set of covariance matrices. In what follows, a suboptimal optimization algorithm that is based on the existing duality [27] between the BC and the MAC is proposed, and as it turns out, for several antenna system configurations, the achievable system throughput of the SZF scheme is strictly larger (for all SNR values) than the sum-rate capacity of the block-diagonalization scheme [21], where complete interuser interference cancellation is performed at the base station. (24)

8 3844 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 In [27], it was shown that any rate in the capacity region of the BC (achieved with DPC) is in the capacity region of the dual MAC (achieved with successive decoding) under a sum power constraint. Therefore, the sum-rate capacities of the BC and the dual MAC are also equal with (27) The sum-rate capacity of the dual MAC under a sum power constraint is given by (28), shown at the bottom of the page [27], [30], where is the set of covariance matrices for the MAC. Before we give the optimization algorithm for the SZF transmission scheme, in the following we give a short overview of the iterative waterfilling algorithm [14] that optimally solves the sum-rate capacity of the BC based on the above duality between the BC and the MAC. BC Iterative Waterfilling Algorithm: In order to solve the optimization problem in (28), an iterative waterfilling algorithm was proposed in [14]. An important feature of the algorithm is the linear complexity with the number of users supported simultaneously in the system. In fact, three algorithms were suggested with different linear complexities and rates of convergence. For the two-user case, the least complex algorithm converges for all channel realizations and with the fastest rate. For the more general case, a hybrid algorithm was suggested which trades off between rate of convergence and optimal convergence point. In this case, a good starting point for rapid convergence is found using the two-user algorithm, and then the iterative algorithm with the least complexity (for ) is used to find the optimal sum-rate capacity. The th iteration of the iterative waterfilling algorithm [14]. 1) Generate effective channels We note here that, initially, the covariance matrices are all initialized with the identity matrix, and for the first few iterations, the covariance matrices in step 3 are updated by without using the exponential filter. After solving (28) using the above optimization algorithm, we obtain the sum-rate capacity with the optimal set of covariance matrices for the MAC. In order to find the corresponding set of covariance matrices for the BC that is denoted by, we use the transformations provided in [27]. If we consider the simple case when, and let the th user BC covariance matrix The matrices and are obtained from the SVD of where is the square diagonal singular value matrix. In the following, the sum-rate capacity optimization algorithm for the SZF scheme is given. SZF Scheme Optimization Algorithm: Given the set of ordered users, the set of channel realizations, and the power constraint, do the following. 1) Find the optimal set of covariance matrices that solves (28) using the iterative waterfilling algorithm. 2) Use the MAC to BC covariance transformations (28) on the set, and let denote the transformed set of BC covariance matrices. 3) For, set 2) Perform waterfilling over an effective block diagonal channel matrix with power constraint and obtain 4) For : find by waterfilling over the effective channel matrix 3) Update the covariance matrices as (28)

9 DABBAGH AND LOVE: PRECODING FOR MULTIPLE ANTENNA GAUSSIAN BROADCAST CHANNELS 3845 with power constraint ; set In the third step of the SZF optimization algorithm, we perform a projection of the optimal solution that is obtained by the iterative waterfilling algorithm, and then we project back to the original subspace. In this case, if the projection was unitary we would get the optimal waterfilling solution. Note that, in the above algorithm, denote the covariance matrices which include the subspace projections. This type of optimization is similar to solving a relaxed optimization problem, where the optimal solution can be obtained by solving the BC sum-rate capacity optimization problem, and then in order to satisfy the subspace constraint we perform the projection given in the third step. In the fourth step of the above optimization algorithm, we can design the covariance matrix optimally since the covariance matrices or equivalently the interference term contributed by users is already determined in steps, and since the achievable rates of users are not affected by modifying due to the subspace projection. Given the set of covariance matrices, the rate of the last user in is given by optimal transmission strategy for a base station with a single transmit antenna is always to transmit to the best user, but this was shown not to be the case [2], in general, when the base station has multiple transmit antennas. However, in certain cases, especially when the SNR is very low the optimal transmission strategy will be always to transmit to the best user as it will be later shown, and in this case the SZF scheme with optimal user ordering achieves the same sum-rate capacity of the SZFDPC and DPC schemes. B. SZF in the Low-SNR Regime In [2], it was shown that in the low-snr regime the ZFDPC scheme has a similar performance as the channel inversion scheme. It is, therefore, of interest to see if a similar relation exists between the SZF and the SZFDPC transmission schemes. Theorem 2: For any ordered set of users Proof: From (13) and (17), the sum-rate capacity of the SZFDPC scheme is where are the eigenvalues of and is the solution to (30) From the fact that for any two matrices and with suitable dimensions, and if we let For sufficiently small, only the dominant eigenvalue in (30) will be activated, and, therefore, the sum-rate capacity of the SZFDPC scheme is given by where (31) then the fourth step in the SZF optimization algorithm follows from In the maximization over the different ordered sets of users as given in (26), it is required to maximize the achievable throughput over Now consider the sum-rate capacity of the SZF scheme given in (25). The sum-rate capacity of the SZF scheme is first upper bounded by the sum-rate capacity of the SZFDPC scheme. Furthermore, from the optimization problem as given in (25), we note that the sum-rate capacity of the SZF scheme is also lower bounded by the capacity of the single-user channel with effective channel response for each. This is very easily seen if we let each of the covariance matrices be equal to zero, except for the th user. Hence (29) different sets. Also note that, from (25), it is clear that the SZF scheme always maximizes over different ordered sets of users that include users with full channel degrees of freedom (i.e., without any subspace constraint). In [24], it was shown that the Therefore, we have that in the low-snr regime the capacity of the SZF scheme is lower bounded and upper bounded by (31).

10 3846 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 Theorem 3: In the low-snr regime, the sum-rate capacity of the SZF scheme with optimal user ordering is asymptotically equal to the sum-rate capacity of the optimal DPC scheme Proof: From Theorem 2, since the sum-rate capacities of the SZF and SZFDPC schemes are asymptotically equal for any ordered set, we can equivalently consider the sum-rate capacity of the SZFDPC scheme. The sum-rate capacity of the optimal DPC scheme is given by the sum-rate capacity of the dual MAC. In this case, given the ordered set of users V. DISCUSSION In general, because of the suboptimality of the SZF optimization algorithm proposed in the previous section, it might be possible to improve over the performance of this algorithm by considering different constrained optimization techniques where a locally or globally optimum point could be found. However, the sum-rate capacity optimization problem of the SZF scheme can in general be written as (32) Now similar to the proof in Theorem 2, in the low-snr regime, the capacity of the DPC scheme is given by the eigenvalue which corresponds to the channel with the largest singular value. From (31), given the ordered set of users, since for each we have that the capacity of the SZFDPC scheme is upper bounded by the capacity given by the channel with the maximal eigenvalue. Therefore, by performing optimal user ordering, where we do not have any subspace constraint for the first user in we get the above result. Therefore, we can conclude that in the low-snr regime the SZF transmission scheme is optimal when user ordering is performed and is asymptotically equal to the SZFDPC scheme for any set of ordered users. Furthermore, with optimal user ordering the sum-rate capacity of the SZFDPC scheme is asymptotically equal to the sum-rate capacity of the DPC scheme. As mentioned earlier, as a linear transmission scheme, the SZF method improves the achievable system throughput. This will be shown later in the simulation results section where we apply the SZF optimization algorithm. In a system satisfying the transmit-receive antenna constraint, it is also possible to compare the throughput of the SZF scheme with other linear transmission schemes such as the BD scheme proposed in [21] where precoding at the base station is performed for interference cancellation. In this case, we will show that complete cancellation of the interuser interference at the base station is a suboptimal transmission strategy, and by letting the users work under limited interference, an improved system throughput is achievable. Another important issue is the ability of supporting multiple number of users simultaneously. In general, multiuser precoding-based techniques where interference cancellation is required at the base station are strongly limited by the antenna system configuration, by the rank of each user s channel, and by the correlation between different users channels. In this case, it will be shown that, in some cases, the SZF scheme can simultaneously support an increased number of users while having a gain in the system throughput. This form of the SZF optimization problem is similar to the standard form of the BC sum-rate capacity optimization problem, except that in this case we have in addition a set of subspace constraints given by (32). As mentioned above, by ignoring the set of subspace constraints, this problem can be solved optimally using the duality property between the BC and the MAC. The set of subspace constraints, however, is linear in the optimization matrix variables, and, therefore, it imposes a new set of linear equalities. We note that the above form of the SZF optimization problem requires an increased number of optimization variables. Let denote the number of column vectors in as defined in Section III. Since each covariance matrix must be positive semi-definite and Hermitian, each can be generated from a corresponding length vector in the following way. For each, 1) Generate the matrix from. 2) Set. Therefore, the actual number of optimization variables should be less than what is given in the above general form, and especially for users ordered last in, the number of optimization variables significantly decreases with. VI. SIMULATION RESULTS In this section, simulation results for the above proposed transmission schemes are provided. Dirty-Paper Coding-Based Transmission Schemes: In this part of the simulation results, the SZFDPC transmission scheme proposed in Section III is simulated, where we compare the proposed transmission scheme with the other DPC-based transmission schemes. In particular, we consider the optimal DPC and the QR or ZFDPC transmission schemes. In Fig. 1, a plot of the average sum-rate capacity versus the SNR is provided for a system with two users equipped with two

11 DABBAGH AND LOVE: PRECODING FOR MULTIPLE ANTENNA GAUSSIAN BROADCAST CHANNELS 3847 Fig. 1. Dirty-paper coding-based schemes: Sum-rate capacity with arbitrary user ordering, K =2; M =4; and N = N =2. Fig. 2. Dirty-paper coding-based schemes: Sum-rate capacity with arbitrary user ordering, K = 2; M = 8; and N = N =4. receive antennas each and a base station with four transmit antennas. In this case, the channels of all users are generated independently with independent and identically distributed (i.i.d.) complex Gaussian elements. First, the performance of the SZFDPC scheme is shown, and it is compared with the ZFDPC scheme and the DPC (optimal noncooperative) transmission scheme. Furthermore, in Fig. 1, we plot the sum-rate capacity of the optimal cooperative transmission scheme. In this case, we do not perform optimal user ordering, and by Monte- Carlo simulation the average sum-rate capacity is computed. As it can be seen from Fig. 1, and in accordance with Theorem 1, the sum-rate capacity of the SZFDPC scheme is asymptotically optimal in the high-snr regime, where it approaches the sum-rate capacities of the cooperative and the DPC transmission schemes. In the ZFDPC transmission scheme that is based on the QR decomposition of the channel, each receive antenna is treated as an independent user, and as given in Section II, DPC is performed after the multiplication by a unitary precoding matrix. By comparing the ZFDPC and the SZFDPC transmission schemes, we can see that the achievable throughput is improved for all values of the SNR and the improvement is greater for a medium range of SNR values. In Fig. 2, a similar simulation as before is performed; however, in this case, we consider a system with two users but with four receive antennas each and a base station equipped with eight transmit antennas. Similar to the previous simulation, we have the same conclusion regarding the asymptotic optimality of the proposed scheme in the high-snr regime. However, if we compare the performance of the SZFDPC scheme with the ZFDPC scheme, we can see that in this case the improvement in the system throughput is even larger than before. This is an intuitive result because of the fact that in the ZFDPC scheme different users antennas are treated as independent users. Therefore, changing the antenna configuration in the system such that we increase the degrees of freedom in each user s channel should improve the performance of the SZFDPC scheme relative to the ZFDPC scheme. Fig. 3. Dirty-paper coding-based schemes: Sum-rate capacity with arbitrary user ordering, K =3;M =9; and N = N = N =3. In Fig. 3, we perform a simulation with three users equipped with three receive antennas each and a base station equipped with nine transmit antennas. Similar conclusions can be also made here, where we can see the improvement in the system throughput when the SZFDPC is considered, and the asymptotic optimality in the high-snr regime. In Fig. 4, we plot the sum-rate capacities of the SZFDPC and the ZFDPC transmission schemes as a function of the number of users for an SNR equal to 5 db. In this case, we also fix the number of receive antennas of each user to two and we let. From Fig. 4, we can see the improvement in the achievable throughput for all values of and notice the linear increase in capacity with the number of users in the system. SZF Versus Linear-Based Transmission Schemes: In the following, the SZF transmission scheme from Section IV is simulated, and a comparison of this scheme with the BD,

12 3848 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 Fig. 4. Dirty-paper coding-based schemes: Sum-rate capacity with arbitrary user ordering, M =2K, and N =2for all i. Fig. 5. Achievable throughput of the SZF transmission scheme, K =2;M = 4; and N = N =2. the channel inversion, and the regularized channel inversion schemes is made. We note here that, in both the channel inversion and regularized channel inversion schemes, each user s receive antenna is treated as an independent user and in this case, as given in (4), the total number of receive antennas should be less or equal to the number of transmit antennas. In general, optimal user ordering is performed for each channel realization, and we consider the cases when complete interuser interference cancellation can be performed and when only partial interuser interference cancellation is possible. In Fig. 5, a simulation of a communication system with two users with two receive antennas each and a base station with four transmit antennas is performed. In this case, it is also assumed that the different users channels are generated independently with i.i.d. Rayleigh fading elements. For the SZF method, we apply the precoding algorithm given in Section IV where we compute the covariance matrices of the different users and the corresponding achievable rates. The achievable throughput is plotted in Fig. 5. In the same figure, we can also see the sum-rate capacity of the BD transmission scheme and the sum-rate capacity which corresponds to the channel inversion and regularized channel inversion schemes. As mentioned above, in this case for each channel realization we perform optimal user ordering. As we can see from Fig. 5, with the SZF transmission scheme and for all values of the SNR, the achievable system throughput has improved compared to the BD scheme and the improvement is much more significant compared to the channel inversion scheme. In Fig. 5, we also plot the sum-rate capacity of the DPC scheme. As we can see from the figure, and as given in Theorem 3, the SZF transmission scheme is asymptotically optimal in the low-snr regime where it approaches the DPC sum-rate capacity. A similar simulation was performed for a system with three users with two receive antennas each and a base station with six transmit antennas. The results are shown in Fig. 6, and the improvement over the BD and channel inversion schemes is Fig. 6. Achievable throughput of the SZF transmission scheme, K =3;M = 6; and N = N = N =2. clear. Therefore, we can conclude that a complete interuser interference cancellation is not necessary and an improved system throughput can be achieved using the SZF method with only partial interference cancellation. In Fig. 7, we consider a system with an asymmetric number of users receive antennas. We assume that there are two users in the system. However, one of the users has two receive antennas and the other has four receive antennas. The base station is equipped with four transmit antennas. In this asymmetric case when the channels are of full rank the BD transmission scheme can not support the two users simultaneously. Using the SZF scheme, the base station can support the two users simultaneously if the user with the two receive antennas is the first user in. As it can be seen from Fig. 7, the achievable system throughput is larger for all values of the SNR when we compare the SZF scheme with the maximal achievable single user throughput (optimized over the two users). Hence, with the

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