Capacity Analysis

Now we precede to derive the link capacity and system capacity with the SWNSF scheduling.

We define the system capacity as the sum of the link capacity delivered to each user on average. Therefore, the system capacity with the SWNSF scheduling can be written as

C^SWNSF =

where p_k= 1/K follows Proposition 4.1 and C˜_k= is the link capacity of user k subject to the SWNSF scheduling. When user k is at the low SNR regime, (4.41) can be approximated by using (4.31), that is,

C˜_k ρ_k

When user k is at high SNR regime, ˜C_k in (4.41) can be upper bounded by

where the derivation is in Appendix C.

For comparison, we also express the system capacity with the RR scheduling as

C^RR =

where C_k is the link capacity of user k under the RR scheduling. Again, since the RR algorithm dose not alter the statistics of the receive SNR for any target user, Ck in (4.44) is the same as (4.5). In [103], a closed-form expression for C_k is given as

C_k = e^N/ρk

where E_r(·) is the exponential integral function of order r, defined as [82]

E_r(z) =

 _∞

1

e^−ztt^−rdt . (4.46)

Next, we evaluate the capacity gain of ˜C_k over C_k when applying the SWNSF scheduling.

4.4.5 Numerical Example

Here, we give some numerical examples to demonstrate the capacity benefits brought by the SWNSF scheduling. We divide the total K users equally into five groups. The users in the same group have the same value of average SNR ρ. The values of ρ from the first group to the fifth group are set to [30, 20, 10, 0, −10] dB, respectively. In the following, we apply the low SNR approximation of (4.42) and the high SNR upper bound of (4.43) to evaluate the link capacity of the users in groups{4, 5} and that in groups {1, 2, 3}, respectively.

Figure 4.3 shows the capacity improvement accomplished by the SWNSF scheduling for the users in different groups. The performance metric is the ratio of the link capacity achieved by the SWNSF scheduling to that by the RR scheduling, i.e. ˜C_k/C_k. We assume K = 50 and N = 3 in this example. As one can see from Fig. 4.3, the SWNSF scheduling

1 2 3 4 5 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

User group

SWNSF scheduling gain

K = 50, N = 3

Simulation Analysis

Figure 4.3: Capacity improvement with the SWNSF scheduling for the users in different user groups.

can provide 15% to 38% capacity gain over the RR scheduling for the users in the groups 1 to 5, respectively. This implies that the SWNSF scheduling tends to bring about higher capacity gain for the users with lower ρ.

Figure 4.4 shows the average capacity gain resulting from the SWNSF scheduling with different numbers of users in the system. The performance metric shown in the y-axis is the average capacity gain defined by _K¹ _K

k=1 C˜_k

C_k. One can see from Fig. 4.4 that the capacity gain increases as K increases. Furthermore, the average capacity gain achieved by the SWNSF scheduling diminishes due to the effect of channel damping when a larger number of antenna elements is employed.

1 10 20 30 40 50 60 70 80 90 100 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7

Number of users K

Average scheduling gain

Analysis Simulation

N=3 N=2

Figure 4.4: Average capacity gain resulting from the SWNSF scheduling with different numbers of users in the system.

4.5 Chapter Summary

The whole idea of this chapter is to promote a novel notion of using the multiuser diversity:

one can devote the physical antennas to gaining capacity with the multiplexing-based MIMO scheme while taking advantage of multiuser diversity to compensate the degraded link qual-ity for the multiplexing-based MIMO system. To be specific, we have proposed a SWNSF scheduling algorithm, which requires only scalar feedback, to demonstrate the merit of com-bining the multiplexing-based antenna technique with the multiuser scheduling system from both the perspectives of cell coverage extension and system capacity improvement.

Table 4.1 presents a quantitative sketch of how the proposed SWNSF scheduling en-hances the coverage and capacity of the multiuser MIMO system. We normalize both the cell radius and link capacity for the SISO system to unity. The cell radius and link ca-pacity for K = 20, 50 and N = 2, 3 are shown in Table 4.1 in comparison with the SISO

Table 4.1: Coverage and capacity enhancements with the SWNSF scheduling for K = 20, 50 and N = 1, 2, 3

K = 20 K = 50

Multiuser MIMO systems Relative Cell Radius

Relative Capacity Gain

Relative Cell Radius

Relative Capacity Gain

N = 1 SISO 1 1 1 1

N = 2 MIMO with

RR scheduling

0.71 1.94 0.71 1.94

N = 2 MIMO with

SWNSF scheduling 1.52 2.74 1.65 2.94

N = 3 MIMO with

RR scheduling 0.58 2.89 0.58 2.89

N = 3 MIMO with

SWNSF scheduling 1.24 3.54 1.35 3.72

case. From the coverage performance aspect, the large number of antennas for the spatial multiplexing MIMO system can result in coverage shrinkage because of link reliability degra-dation. However, the SWNSF scheduling can virtually extend coverage by improving the receive SNR quality. From the capacity performance aspect, one can see that, as expected, the capacity of the spatial multiplexing MIMO system grows linearly with the number of antennas. Moreover, applying the SWNSF scheduling can further improve the capacity for the MIMO system. In the next chapter, we continue to explore the advantage of applying the multiplexing-based antenna technique for the multiuser MIMO system.

Chapter 5

Throughput Enhancement Based Antenna Techniques for Multiuser

Scheduling Systems with Zero-Forcing Receivers

In this chapter, we revisit the topic of combining the multiplexing-based antenna technique with the multiuser scheduling system. In contrast with Chapter 4 where the optimal receiver is assumed, this chapter considers the zero-forcing receiver. Although the zero-forcing re-ceiver is known to suffer from noise enhancement in a point-to-point (or single-user) MIMO system, the inherent property of poor channel quality avoidance from multiuser scheduling provides a natural way to overcome this drawback. To probe the merit of such cross-layer cooperation, this chapter introduces an analytical framework to evaluate the performance of the zero-forcing receiver operating in the multiuser MIMO system with scheduling. Fur-thermore, a number of scheduling algorithms that require different amounts of feedback information will be studied to assess the impact of feedback on the resulting performance.

5.1 The Zero-Forcing Receiver for a Single-User MIMO System

To begin with, we consider a point-to-point (single-user) MIMO system consisting of N_t transmit and N_r receive antennas. Let x and y be N_t× 1 and Nr× 1 signal vectors at the transmitter and the receiver, respectively. Let n denote the N_r× 1 spatially white noise

with the covariance matrix E[nn^H] = σ²I_Nr, where (·)^H is the complex conjugate transpose operation . Then, the link between x and y is modelled by

y = Gx + n =√

gHx + n , (5.1)

where √

g depicts the large-scale (local average) behavior of the channel gain, and the N_r× Nt normalized channel matrix H characterizes the channel fading. We consider the flat Rayleigh fading such that every entry of H is independent, circular-symmetric complex Gaussian random variables CN (0, 1). The total transmit power across all Nt antennas is constrained by E[x^Hx]≤ Pt.

Assume that at the transmitter the spatial multiplexing with equal power allocation method is applied to send Nt parallel data streams over the MIMO channel. Meanwhile, at the receiver the zero-forcing algorithm is used to recover the received spatially multiplexed signals. By multiplying y with H^†/√g, where H^† = (H^HH)⁻¹H^H is the pseudo-inverse of H, the received signal vector is decoded as

ˆ

x = H^†y/√

g = x + H^†n/√

g . (5.2)

The output SNR at the n-th subchannel after the zero-forcing processing in (5.2) can be expressed by [104] and [105], we know that γ_n in (5.3) is a Chi-squared distributed random variable with 2(N_r−Nt+1) degrees of freedom and all{γn}^Nn=1^t are independent and identically distributed (i.i.d.). Thus, the probability density function (PDF) of γ_n can be written as

f_γn(γ_n) = N_te^−Ntγn/ρ

Note that the assumption of N_r ≥ Nt has been implicitly made for the spatial multiplexing MIMO system. Next, we utilize (5.4) to derive the achievable throughput and the outage probability for the zero-forcing receiver.