Asymptotic Optimality of the ZF Receiver

It is instructive to compare the result of this chapter using the zero-forcing with that of the previous chapter using the optimal receiver under the same max-min scheduling algorithm.

Fig. 5.1 have shown that the performance of the zero-forcing is far from optimal at the low SNR regime due to noise enhancement. However, in the multiuser MIMO system, the scheduling technique can virtually create a high SNR environment and thus reduce the occurrence of noise enhancement for the zero-forcing receiver. The following proposition explores the advantage of operating the zero-forcing receiver in the multiuser scheduling system.

Proposition 5.1 Let ˜C_opt^mi and ˜C_zf^mi be the achievable throughput for the optimal receiver and the zero-forcing receiver operating in the same multiuser MIMO system with the max-min scheduling policy. For any finite N , we have

(i) At the low SNR regime,

lim

K→∞

C˜_zf^mi

C˜_opt^mi = 1, for small ρ . (5.56) (ii) At the high SNR regime,

ρlim→∞

C˜_zf^mi

C˜_opt^mi = 1, for any finite K . (5.57)

Proof: Please refer to Appendix D.

Somewhat surprisingly, Proposition 5.1 indicates that the efficiency of the zero-forcing receiver in recovering the spatially multiplexed signals can approach that of the optimal receiver as the number of users goes to infinity. More generally, it also implies that in

1 20 40 60 80 100 120

Figure 5.4: Achievable throughput for the zero-forcing receiver in the multiuser MIMO system with various scheduling algorithms.

the multiuser MIMO environment with scheduling the receiver design requirement may be relaxed due to the effect of noise enhancement avoidance. Numerical comparisons of ˜C_zf^mi and C˜_opt^mi are provided in the next section.

5.6 Numerical Results

Figure 5.4 compares the achievable throughput of ˜Czf and Copt. Analogous to the definition of η₁ in (5.10), we define

η₂ = C˜zf

Copt

(5.58) to evaluate the efficiency of the zero-forcing receiver in the multiuser scheduling system to de-liver the multiplexing gain. In (5.58), ˜Czfcorresponding to the max-max, max-min, spatially-independent and spatially-greedy scheduling algorithms are derived in (5.27), (5.42), (5.48)

0 5 10 15 20 25 30 10⁻²

10⁻¹ 10⁰

ρ/γ_th (dB) Pout

K = 16, N=3

spatially−greedy scheduling spatially−indepedent scheduling

max−min scheduling

max−max scheduling round−robin scheduling

~

Figure 5.5: Outage probability for the zero-forcing receiver in the multiuser MIMO system with various scheduling algorithms.

and (5.52), respectively. Figs. 5.4-(a) and (b) provide the result of η₂ for a high SNR condi-tion at ρ = 10 dB and a low SNR condicondi-tion at ρ = −10 dB, respectively. One can observe that ˜Czf can be greater than Coptwith only a small number of K by using the spatially-greedy and spatially-independent scheduling algorithms. As for the scalar feedback scheduling, the achievable throughput ˜Czf can approach Copt with the increasing K for both the max-max and max-min scheduling algorithms. Furthermore, applying the max-min scheduling policy and operating the zero-forcing receiver at a high SNR condition can even result in a higher C˜zf than Copt with a mediate number of K.

Figure 5.5 presents the outage probability of ˜Pout with the various scheduling algorithms for K = 16 and N = 3. The outage probability subject to the round-robin scheduling is also plotted for comparison. It is shown that the vector feedback scheduling including the spatially-greedy and spatially-independent scheduling can significantly improve the outage

1 2 3 4 5 6 7 8 1.2

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

Number of antennas N Czf / Czf

K=32, ρ= 10 dB

max−max scheduling

max−min scheduling spatially−independent scheduling spatially−greedy scheduling

~

Figure 5.6: The impact of the number of antennas on the scheduling gain defined by ˜Czf/Czf. probability performance. As for the scalar feedback scheduling, the max-min scheduling can always yield a much better outage probability performance than the max-max scheduling.

Considering the outage probability and the achievable throughput as shown in Fig. 5.4, the max-min scheduling would be a more feasible strategy as compared with the max-max scheduling policy when only scalar feedback is available.

Figure 5.6 illustrates the impact of the number of antennas on the achievable scheduling gain defined by ˜Czf/Czf. Although the achievable link throughput of the zero-forcing receiver increases for all the studied scheduling algorithms as shown in Fig. 5.4, one can see from Fig.

5.6 that the scheduling gain reduces with the increasing N for the scalar feedback scheduling.

The phenomena of diminishing scheduling gain for the scalar feedback scheduling may be attributed to the insufficient feedback information as compared with the vector feedback scheduling.

1 20 40 60 80 100 120 0.6

0.65 0.7 0.75 0.8 0.85

N = 3, ρ = 10 dB

Number of users K η3 = Czf / Copt

spatially−independent scheduling

spatially−greedy scheduling max−min scheduling

max−max scheduling

~ ~

Figure 5.7: Comparison of the achievable throughput for the zero-forcing receiver and the optimal receiver operating in the same multiuser MIMO system.

In the final experiment, we compare the achievable throughput for the zero-forcing re-ceiver ( ˜Czf) and the optimal receiver ( ˜Copt) operating in the multiuser MIMO system under the same scheduling policy. Recall from (4.5) in Chapter 4 that the subchannel output SNR for the optimal receiver can be represented by λ_nρ/N . Therefore, by simply replacing γ_n in (5.18), (5.31), (5.45) and (5.50) with λ_nρ/N , ˜Copt subject to the max-max, max-min, spatially-independent and spatially-greedy scheduling algorithms can be obtained via com-puter simulations. Fig. 5.7 presents the comparison of ˜Czf and ˜Copt by plotting η₃ against the number of users, where η₃ is defined as

η₃ = C˜zf

C˜opt

. (5.59)

One can see that the achievable throughput for the zero-forcing receiver and the optimal receiver becomes closer with the increasing number of users in the multiuser scheduling sys-tem. This implies a nice cross-layer synergy that employing the simple zero-forcing receiver

on top of the multiuser scheduling system can deliver the remarkable multiplexing gain of the MIMO channel in a cost-effective manner.

5.7 Chapter Summary

In this chapter, we have presented an analytical framework to evaluate the performance of the zero-forcing receiver operating in the multiuser MIMO system with a variety of scheduling algorithms. Using the order statistics technique, we derive closed-form expressions for the achievable throughput and outage probability that facilitate efficient numerical comparisons among the various scheduling strategies. Our results indicate that the cross-layer coopera-tion between the simple zero-forcing receiver and scheduling technique can achieve the full capacity of the MIMO system. Furthermore, it is demonstrated that with the increasing number of users in the multiuser MIMO system, the efficiency of the zero-forcing receiver in recovering the spatially multiplexed data can approach that of the optimal receiver. This result further endorses the strength of combining the multiplexing-based antenna scheme with the multiuser scheduling system.

Chapter 6

Interference Suppression Based

Antenna Techniques for TDD/CDMA Systems

This chapter shifts the focus to the TDD/CDMA system. As introduced in Section 2.3, the TDD system is flexible to deliver asymmetric traffic services by allocating different numbers of time slots for uplink and downlink transmissions. However, in a TDD/CDMA system, asymmetric traffic may result in severe opposite direction interference because downlink transmitted signals from neighboring base stations may interfere with the uplink received signals of the home cell. In this chapter, we investigate the effect of antenna beamforming techniques from the perspective of suppressing the opposite direction interference. Particu-larly, four antenna beamforming schemes will be considered in this chapter, including

• The uplink receive beam-steering method is employed at base stations (Scheme I);

• The uplink receive MVDR beamformer is employed at base stations (Scheme II);

• The beam-steering method is jointly applied in both the downlink transmission and uplink reception at all base stations (Scheme III);

• The downlink transmit beam-steering and the uplink receive MVDR beamformer are jointly employed at all base stations (Scheme IV).

Cell 1 Cell 2

Cell 3

Cell 4

Cell 5 Cell 6

Cell 0

Figure 6.1: An example to illustrate the interference scenario in the TDD/CDMA system, where Bod = {2, 4, 6} represents the set of the neighboring cells generating the opposite direction interference and Bsd ={1, 3, 5} represents the cells generating the same direction interference.

6.1 System Model

Consider a TDD/CDMA cellular system with seven cells as shown in Fig. 6.1, where the home cell is indexed with j = 0 and six adjacent cells are labelled with 1 to 6. Assume that cell 0 in the center is in the uplink mode during a particular time slot t_o. Let Bod and Bsd

denote the set of the neighboring cells during time slot t_o operating in the downlink mode and those operating in the uplink mode, respectively. Figure 6.1 illustrates an example with Bod = {2, 4, 6} and Bsd = {1, 3, 5}. In this example, downlink transmissions of cells 2, 4, and 6 will cause the opposite direction interference (i.e., the base stations to base stations interference) to the uplink receiving signals of cell 0, while cells 1, 3, and 5 result in the same direction interference (i.e., the mobile stations to base stations interference).

In our model, we consider propagation loss and log-normal distributed shadowing. Then the link gain G(r, α) between the transmitter and receiver is described as

G(r, α) = κ₀r^−µ10^α/10 , (6.1)

where r is the propagation distance, κ₀ is a constant, µ is the path loss exponent and α is a normal distributed random variable with zero mean and standard deviation of σ_s dB. Let P_t be the total transmit power of a base station, and d_j the distance from cell j (j ∈ Bod) to the home cell. Then the total opposite direction interference introduced by the adjacent cells is equal to

I_od= 

j∈Bod

P_tG(d_j, α_j) . (6.2)

Assume that uplink power control is ideally executed so that the received signal power of each mobile user is maintained at a constant level Pr at base stations. Then the same direction interference introduced by mobile k_j of Bsd (j ∈ Bsd) is equal to

I_kj = P_rG(r₀, α₀)/G(r_kj, α_kj) respectively. For ease of notation, let

ξ_kj =

r_kj r₀

µ

10^{(α0−αkj)/10}. (6.4)

Note that the term (α₀ − αkj) in (6.4) can be represented by another normal distributed random variable with a modified standard deviation [76]. Let K_j denote the number of active mobile users in cell j (j ∈ Bsd) that are in their uplink transmission cycles during a particular time slot t_o. Then from (6.3) and (6.4), the total same direction interference introduced by adjacent cells can be expressed by

I_sd = 

In addition to the opposite direction and the same direction interfering signals, there still exists the intracell interference in the TDD/CDMA system, denoted as I_ic. Since power

control is assumed to be ideal, the received signal power of all users in a cell will be maintained at a constant level P_r. Thus, I_ic can be expressed as

I_ic= P_r(K₀− 1) , (6.6)

where K₀ is the number of active mobile users in the home cell that are transmitting uplink signals in time slot t_o. Thus, based on the definitions of I_od, I_sd, and I_ic corresponding to (6.2), (6.5) and (6.6), respectively, the uplink received bit energy-to-interference density ratio γ_k for a target mobile k in the home cell can be written as

γ_k = LP_r

I_od+ I_sd+ I_ic+ σ²_n , (6.7) where L is the processing gain and σ_n² is the white thermal noise power. In the next sec-tion, we will further derive the expression of γ_k with consideration of the effect of antenna beamforming.