Now we extend the capacity analysis to the multiuser MIMO system. To this end, Nt transmit antennas and Nr receive antennas are employed at the base station and each user, respectively. Thus, the link between the base station and any user constitutes a (Nt, Nr) MIMO system. The channel between the base station and user k can be characterized by a Nr× Nt matrix Hk = [h(k)ij ] where hij is the channel gain from the jth transmit antenna to the ith receive antenna. We assume that the links between each pair of transmit and receive antennas are subject to independent Nakagami fading. Therefore,|h(k)ij |2/σn2 are i.i.d. gamma distributed random variables as defined in (3.1)2. In the following, we consider four fading
2By letting the fading parameter m = (1 + K2)2/(1 + 2K2) [88], the considered MIMO channel model corresponds to that used in [37, 89] with aK-factor to represent a light of sight component.
mitigation based antenna schemes, including 1) selective transmission/selective combining (ST/SC); 2) maximum ratio transmission/maximum ratio combining (MRT/MRC); 3) se-lective transmission/maximum ratio combining (ST/MRC) and 4) space-time block codes (STBC). As remarked in Section 2.1.1, all of the four considered antenna schemes are capable of delivering full antenna diversity order over the MIMO channels. For a fair comparison, the total transmit power across all Nt antennas is constrained to the same level for all the antenna schemes.
3.3.1 ST/SC Scheme
Assume that orthogonal pilot signals transmit from Nt spatially separated antennas at the base station. By monitoring the pilot signals, each user can distinguish the link with the strongest SNR from NtNr possible transmit and receive antenna pairs at any time slot.
Collecting feedback from all K users, the base station can determine the target user and the associated transmit antenna radiating the best link quality. Under this policy, the scheduling rule can be mathematically expressed as
k∗ = arg max
k |h(k)ij |2 . (3.25)
Thus, we have flinkk∗ (γ) = fγmax(γ ; p, q, KNtNr) for the ST/SC scheme, where fγmax(γ ; · , · , · ) is defined in (3.10). Comparing the conditional PDF flinkk∗ (γ) associated with the ST/SC scheme and that of the SISO case, we obtain the system capacity with joint multiuser scheduling and the ST/SC antenna scheme in Nakagami fading channels as follows:
Cst-sc = Λ2
m, m
ρ, KNtNr
. (3.26)
Expression (3.26) can be interpreted from two folds. On the one hand, from a multiuser system point of view, the base station can see total KNr antennas at the receiving end rather than only Nr. The additional K(Nr− 1) virtual antennas from other users can be exploited to improve system capacity. On the other hand, the multiple antennas in the ST/SC scheme can be viewed as virtual users to increase the multiuser diversity order for
the considered scheduling algorithm. Thus, we can take a broader view to define the selection order S as the size of a set with S i.i.d. gamma random variables, which are provided by the multiple users and/or multiple antennas in the multiuser MIMO system. Consequently, the capacity improvement achieved by the ST/SC MIMO scheme with multiuser scheduling can be explained from the fact that the selection order is indeed increased from S = K to S = KNtNr.
3.3.2 MRT/MRC Scheme
In [32], the authors proposed an MRT/MRC scheme to deliver full antenna diversity over the MIMO channel, which was later analyzed in [33] under the Rayleigh fading channel in more detail. Given a known channel matrix, this method can maximize the received SNR by applying the specific beamforming weight wt at the transmitter and combining weight wr at the receiver. It was shown that by setting wt and wr to be the principle right and left singular vectors of the channel matrix respectively, the optimal received SNR can be attained with the effective output SNR γk = λmax(HHkHk)/σn2, where HHk is the transpose conjugate of Hk, and λmax(HHkHk) is the maximum eigenvalue of HHkHk. Based on the effective output SNR γk, the decision rule for the scheduler is
k∗ = arg max
k λmax(HHkHk) . (3.27)
Note that the effective output SNR γk is a random variable depending on different real-izations of the random channel matrix. Recently, the exact distribution of λmax(HHk Hk) for Rayleigh channels is given by [33]
fλmax(λ) =
Nr
i=1
(Nt+N r)i−2i2 j=Nt−Nr
dij ij+1λje−iλ/j ! , (3.28)
where dij is the associated coefficient determined by different combinations of Ntand Nr. As one can see from (3.28), it is not easy to derive the conditional PDF flinkk∗ (γ) directly based on (3.28). However, for generalized Nakagami fading channels, we can utilize the following
inequality [90]
Hk2F
min(Nt, Nr) ≤ λmax(HHkHk)≤ Hk2F (3.29) to obtain the lower and the upper bound of system capacity. Note that in (3.29) HkF
is the Frobenius matrix norm with Hk2F = Nr ca-pacity with the MRT/MRC scheme, respectively. By comparing these conditional PDFs with that of the SISO case, we reach
Λ2
where Cmrt-mrc is the system capacity achieved by the MRT/MRC scheme with multiuser scheduling.
In the cases of SIMO (Nt = 1) and MISO (Nr = 1), the channel matrix is reduced to a rank-one vector. Under such conditions, the achieved system capacity in (3.30) is equal to the upper bound expression, i.e., Cmrc = Λ2
mNr, mρ, K
for the receive MRC and
Cmrt = Λ2
mNt, mρ, K
for the transmit MRT, respectively.
3.3.3 ST/MRC Scheme
Here, we study a hybrid scheme, which implements the ST at the transmitter and the MRC at the receiver over the MIMO channel [30]. With the MRC method utilized at the receiver, the effective SNR at the kthuser’s combiner output with respect to the jth transmit antenna can be written as γk=Nr
i=1|h(k)ij |2/σ2n [24]. Similarly, the base station gathers the effective SNR of all users and selects the target user according to the following criterion
k∗ = arg max scheme. Accordingly, the capacity of the multiuser scheduling system with the ST/MRC
scheme is described as follows:
Space-time block codes (STBC) pertain to another category of antenna schemes to provide antenna diversity gain without requiring prior channel knowledge at the transmitter. Here, we focus on the STBC with orthogonal structures, which was introduced in [28] and devel-oped more generally in [29]. Through coding over time as well as transmit antennas, the orthogonal STBC technique can also deliver full antenna diversity order by using simple linear processing at the receiver. From [55, 91], the effective SNR of user k at the output of the STBC decoder is given by γk = N1 selection rule for the base station scheduler is
k∗ = arg max the STBC method. Consequently, the capacity of the multiuser scheduling system with the orthogonal STBC can be expressed as
Cstbc = Λ2
Notice that the system capacity achieved with joint multiuser scheduling and the orthog-onal STBC will in general be lower than (3.34). The reason follows from the fact that only the STBC with full code rate can support the capacity promised by (3.34) while the full code rate STBC are available for a limited number of transmit antennas and signal con-stellations [29]. Nevertheless, we still can use (3.34) to evaluate the impact of applying the STBC scheme on top of the multiuser scheduling system since it represents an optimistic performance upper bound.
3.3.5 Discussions
Table 3.1 summarizes the system capacity achieved with all the aforementioned antenna diversity schemes in the considered multiuser scheduling system. Comparing Cst with
CscandCmrt withCmrc, one can find that the duality between the transmit and receive diversity methods. However, the practical considerations for implementing these antenna schemes in the multiuser scheduling system are quite different. First of all, the transmit methods have the potential advantage of relieving computation burdens for user terminals.
Moreover, the cost (benefit) of adding one more antenna at the base station for the transmit methods can be amortized (shared) by multiple users. Next, the transmit methods generally require additional pilot signals since the receiver would rely on them to estimate the SNR from each corresponding transmit antenna. Finally, the required amount of signalling in the feedback channel is usually larger for the transmit methods than the receive methods. For example, as compared with the receive SC method, using the transmit ST scheme requires the terminal to send back not only the SNR of the strongest channel, but also the signalling to indicate the best serving antenna. The transmit MRT scheme needs even a larger amount of overheads in the feedback channel because the channel magnitude and phase for all transmit antennas are mandatory for the base station to perform the MRT scheme correctly.