4.3 Resource Allocation Algorithms for Product Rate Maximization
4.3.2 DP Based Algorithm for Product Rate Maximization
The DP based resource allocation algorithms developed in Chapter 3 and previous sec-tion have been proved to be capable of offering near-optimal performance. Using the same idea, we modify the corresponding cost function Jt as
Jt(Cst) = max set Cts(j : k). We use a modified algorithm similar to the OMPA algorithm to compute the maximal transmission rate for the single user case.
The DP-based WSRmax solution can be describe by using 4.15 to replace the ones of Table 3-1 in chapter 3.
4.3.3 Numerical Results and Discussions
Here, we take the MaxGain algorithm discussed in the last section and the DPRA algorithm under maxmin criterion for comparison. The second algorithm is denoted by DP based maxmin algorithm which uses DPRA process to maximum the minimal user rate at each level. In the other words, in DP based maxmin algorithm we just exploit the following objective function to replace the original one.
Jt(Cst) = max
In Fig. 4.4 under considering independent channel fading scenario, we can find the solution of DP based NBS has the largest fairness index. It is expected that DP based maxmin algorithm outperforms than the MaxGain algorithm.
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0 5 10 15 20 25 0.74
0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02
The fairness index
The number of users DP_NBS
DP_Maxmin MaxGain
Figure 4.4: Fairness index performance as a function of number of users; N = 64, P = 60, GNR = 0 dB.
Fig. 4.5 shows the performance in correlated channels. We assume the same the correlated channel model as that mentioned in the last section. From Figs. 4.4 and 4.5, we find that, for independent fading channels, fairness can be enhanced by adding the number of subcarriers. Simultaneously, it is shown in Fig. 4.5 that there is no noticeable difference in fairness performance between DP based NBS algorithm and the DP based maxmin algorithm. But in Fig. 4.5, the DPRA algorithm for NBS still yield the best system performance measured by the sum rate without degrading too much fairness performance.
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5 10 15 20 25
Figure 4.5: Fairness index versus number of users, (N = 128, GNR = 0 dB) for correlated and independent fading channels.
The sum rate of system
The number of users DP_NBS DP_Maxmin MaxGain
Figure 4.6: Sum rate performance versus number of users in a correlated channel; N = 128, GNR = 0 dB.
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Chapter 5
Resource Allocation for
MIMO-OFDMA Downlinks
In the downlink of a multiuser multiple-input multiple-output (MIMO) system, the multiple antennas at the base station (BS) allow for spatial multiplexing of transmissions to multiple users in the same time slot and frequency band. Due to their geographical dispersion, coordination among users is difficult, which makes the downlink of a multiuser system more challenging compared to single user MIMO systems.
From information theoretic point of view, the sum capacity achieving precoding or preprocessing technique is dirty paper coding (DPC) [18]. The DPC solution uses successive interference pre-cancellation approaches that employ complex encoding and decoding schemes. An intensive research effort is thus underway to devise suboptimal but more practical approaches to multiuser downlink signal processing. Beamforming or transmit pre-processing is a suboptimal and reduced complexity (compared to DPC) strategy, where each user stream is coded independently and multiplied by a beamform-ing weight vector for transmission through multiple antennas.
Proper design of the beamforming weight vectors allows the interference among dif-ferent streams to be minimized (or eliminated), thereby supporting multiple users si-multaneously. As a result, multiuser MIMO system substantially increases the system capacity by multiplexing users in the spatial domain.
The challenge is thus to design transmit and receive processing vectors such that
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space-division multiplexing is effectively achieved. Despite its suboptimality, for in-dependently fading channels linear beamforming has been shown to achieve the best trade-off between performance and complexity. In this chapter, we consider orthogonal linear precoding techniques to achieve orthogonal space division multiplexing (OSDM) in the downlink of multiuser MIMO systems, in which both base station as well as mobile stations employ multiple antennas. The orthogonal precoding allows transmission to the mobile users to be multiuser interference free. OSDM for multiuser MIMO systems has been proposed by several researchers [19] [20]. We consider two OSDM techniques that use a subspace projection methods to design precoding matrices: block diagonalization.
With block diagonalization, each user’s precoding matrix is designed such that the trans-mitted signal of that user lies in the null space of all other remaining users’ channels, and hence multiuser interference is pre-eliminated. With the sum power constraint, block diagonalization takes the sum rate maximization approach, which tends to select the strong users often causing unfairness among users. Hence, minimizing transmit power while achieving desired quality of service for users may be interesting in practice. In this thesis, we will propose a low-complexity suboptimal algorithm to solve this problem.
Due to the null space dimensional requirements of block diagonalization, the numbers of users supported in the same time/frequency slot are limited for a given number of transmit antennas. Therefore this technique should be combined with radio resource management so that a multiuser diversity gain can be achieved. Multiuser diversity arises when a large number of users with independently fading channels are present, and hence it is likely that a user or multiple users experience high channel gain in any given time/frequency slot. In addition ,to reduce the hardware complexity at the mobile units, there is no receive antenna selection when users are selected for transmission in any given time/frequency slot, with which all antennas are selected for reception.
In [21], the authors proposed a user selection for single carrier case in MISO systems.
However, due to the coupling of all users’ channels in an OFDMA network, optimal
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selection of user subsets involves exhaustive search through all combinations of active users, which is computationally very complex for systems with a large number of users and frequency/time slots. Hence, we propose simplified user selection resource alloca-tion algorithm for block diagonalizaalloca-tion. The rest of the Chapter is organized as follows.
Section 2.1 describes the system model. The brief review of block diagonalization tech-nique is shown in section 5.2. Section 5.3 describes the problem formulation. Finally, we propose our low-complexity suboptimal algorithm and some numerical results for discussion.
5.1 System Description
In this section, we provide a general description of a typical cellular-based MIMO-OFDMA system. Fig. 5.1 shows a downlink multiuser MIMO system in which a base station transmits data to K users over N subcarriers. The BS is equipped with NT
antennas and the jth user terminal has nj antennas. The total number of receive an-tennas is thus given by NR = PK
j=1nj. We also assume that BS has perfect channel state information (CSI) of all active system users. The CSI may be obtained via chan-nel reciprocity for time division duplex (TDD) systems or through feedback links. The issue of inaccuracy of CSI at the transmitter as well as the emerging partial CSI feed-back schemes are some of the practically important issues for multiuser MIMO downlink transmission but will not be addressed.
We denote by Hj,i ∈ Cnj×NT the downlink channel matrix of the jth user. A flay Rayleigh fading channel is assumed so that the elements of Hj,i, j = 1, 2, · · · , K and i = 1, 2, · · · , N can be modelled as independent and identically distributed (i.i.d.) complex zero-mean Gaussian random variables with variance of 0.5 per dimension. Hence, for the ith subcarrier, the received signal of the jth user can be expressed as
yj,i = Hj,i Ki
X
k=1
Wk,ixk,i+ nj,i for ∀j ∈ Ωi (5.1)
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User 1’s
Figure 5.1: Block diagram of a multiuser MIMO system.
where nj,i∈ Cnj×NT denotes zero mean additive Gaussian noise with E{nj,inHj,i} = σ2Inj. Ki = |Ωi| represents the number of users who simultaneously communicate over the ith subcarrier. After linear processing at the receiver, the received signal can be reformulated as
where Uj,i∈ CNT×ni is the receive process matrix of user j over the ith subcarrier.
We focus on the scenario of multiuser interference free environments. Based on the above assumptions, Wk,i should be designed such that
Hj,iWk,i = 0 for all k 6= j and k, j ∈ Ωi (5.3)
As a result, the multiuser MIMO system with Ki users over the ith subcarrier can be decomposed into Ki parallel single-user MIMO system as follows.
yj,i = Hj,iWk,ixk,i+ nj,i (5.5)