6.3 Downlink Beamforming
7.1.2 Throughput Enhancement Based Antenna Techniques for Multiuser
Chapter 4 and [117] demonstrated the advantage of using multiuser diversity to enhance the degraded link quality of the diversity-deficient spatial multiplexing MIMO system. In particular, a novel SWNSF scheduling algorithm is proposed for the multiuser MIMO system with low-rate feedback channels. The SWNSF scheduling is a fair scheduling policy for near-far users and requires only scalar feedback. Through a tractable eigenvalue analysis, it is shown that the SWNSF scheduling can enhance the receive SNR of all subchannels for any selected user so that better link reliability (and thus coverage extension) and higher link throughput (and thus system capacity improvement) are achieved. It is also observed that a large number of antennas could attenuate the scheduling gain due to the effect of channel damping .
7.1.3 Throughput Enhancement Based Antenna Techniques for Multiuser Scheduling Systems with Zero-Forcing Receivers
Chapter 5 and [118] studied the performance of the zero-forcing receiver operating in the multiuser MIMO system with various scheduling policies. The motivation is to exploit the inherent property of poor channel avoidance from a multiuser scheduling system to overcome the drawback of noise enhancement for the zero-forcing receiver. It is shown that the cross-layer cooperation between the simple zero-forcing receiver and the scheduling technique can achieve the full theoretical capacity of the MIMO system. Moreover, it is shown 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 operating under the same scheduling algorithms. An important implication from this result is that the multiuser diversity gain can be used to simultaneously improve system performances as well as reduce receiver implementation complexity. Another important observation from comparing the performance of implementing the scalar feedback and vector feedback scheduling is that the amount of feedback information plays a curial role to enhance the downlink performance of the multiuser MIMO system.
7.1.4 Interference Suppression Based Antenna Techniques for TDD/CDMA Systems
Chapter 6 and [119] investigated the effect of beamforming techniques from the perspective of suppressing the opposite direction interference in TDD/CDMA systems. Four antenna beamforming schemes are studied to alleviate the impact of the strong opposite direction interference from adjacent cells. With only uplink beamforming, it is shown that the MVDR beamformer (Scheme II) instead of the conventional beamforming method (Scheme I) should be adopted since the beam-steering can not effectively suppress the opposite direction inter-ference. However, by exploiting the reciprocal property of TDD channels and the synergy of combining the downlink transmit and uplink receive beamforming, the low-cost Scheme III which adopts the beam-steering method in both the downlink transmission and uplink
reception can also provide satisfactory performance.
7.2 Suggestions For Future Research
The research in this dissertation covers a wide variety of multiple antenna techniques for their applications in contemporary multiuser scheduling and TDD/CDMA wireless systems with the network perspective emphasis. Other research topics related to or inspired by this research are addressed as follows.
• The impact of imperfections
As a first step to explore the relationship of the antenna technique with the multiuser scheduling and TDD/CDMA communication systems, this dissertation has made ideal assumptions in system modelling. For instant, it is assumed that the base station can always have the correct and instantaneous feedback information to perform scheduling.
Also the perfect DOA information and channel reciprocity of TDD channels are as-sumed for antenna beamforming. Considering the inevitable channel estimation error or feedback delay in practical systems, it is worth further investigating the impact of channel uncertainty on the resulting system performance.
• Optimal scheduling design with feedback rate/delay constraints
It has been known that in a point-to-point MIMO system the full channel side infor-mation at the transmitter can only add to limited capacity gain (especially at the high SNR condition) as compared with the case of no channel side information. However, in the multiuser MIMO system with scheduling, it is shown in this dissertation that the amount of feedback plays a curial role to enhance the downlink capacity. The difference mainly comes from the fact that in the multiuser MIMO scenario the receive antennas among multiple users can not “really” cooperate so that the joint decoding among them is not possible. Since the higher feedback rate consumes more reverse link capacity, how to design an optimal scheduling algorithm to enhance the downlink
capacity as much as possible while maintaining a low-rate feedback turns out to be an important research topic.
Another important research topic is the optimal scheduling design for the data traffic with delay constraints. This dissertation has implicitly assumed that all the users can wait for services with no delay constraints. Under such circumstances, the mean delay time for any serviced user would increase with the number of users in the system. For the data traffic with stringent delay constraints, how to design an optimal scheduling strategy to meet the delay constraints while enhancing the system performance still remains an important research direction.
• Antenna beamforming for the multiuser scheduling system
From the complementary diversity-multiplexing point of view, this dissertation has demonstrated the superiority of the multiplexing-based antenna schemes over the diversity-based antenna schemes as applied in the multiuser scheduling system. How-ever, the issue of using the beamforming technique for the multiuser scheduling system has not been discussed in the dissertation. A recent study [115] revealed that it is pos-sible for the beamforming technique to deliver degree-of-freedom gains in the MIMO system under certain propagation conditions. Consequently, the pro and con study of combining the antenna beamforming technique with the multiuser scheduling system still remains an open issue.
• Cross-layer optimization issues
This dissertation has exemplified a nice cross-layer cooperation that leverages the com-bination of the scheduling technique in the MAC layer with the spatial multiplexing MIMO technique in the physical layer. As we remarked in the introduction, it is possible to find more good examples of cross-layer cooperations by using the network perspective methodology. One possible research topic following this philosophy is the joint design of the DCA technique and the antenna beamforming technique for the TDD/CDMA system. Due to the huge calculation in solving the DCA optimization
problem, [18] asserted that it is almost impossible to resolve the opposite direction interference issue by using the DCA method. On the other hand, this dissertation has demonstrated that using the beamforming technique can effectively cancel the number of strong opposite direction interference. Therefore, the cross-layer design of DCA and antenna beamforming techniques may open up a possibility to simultaneously reduce the complexity of the DCA algorithm and lower the implementation requirement for antenna beamforming.
Appendix A
Derivation of Equation (3.16)
For integer values of p, the PDF of (3.10) can be written as [94]
fγmax(γ ; p, q, K) = K
where the coefficient aki is defined in (3.17). Substituting (A.1) into (3.14) and using the integral [86]
Note that in (A.2) and (A.3), Γ(·, ·) is the incomplete gamma function, defined by
Γ(a, x) =
Finally, applying the identity [82, Eq. 6.5.19]
Γ(−n, x) = (−1)n
to (A.3), we obtain the expression (3.16).
Appendix B
Proof of Proposition 4.3
Proof of (i): By applying the variable transformation (4.28) to (4.13) and using the result of Proposition 4.2, the joint PDF of Sk,1,· · · , Sk,N can be written as
Thus, Sk,i and λk,1 are independent.
Proof of (ii): Since the spacing Sk,i are independent of λk,1, we have ˜λk,i = Sk,i+ ˜λk,1 where Sk,i and ˜λk,1 are also independent. Thus, it is followed that
φ˜λ
where we have used the integral identity [100, eq. 3.312]
Combining (B.4) and (B.3) yields the result of (4.29).
Proof of (iii): Since the appropriate derivative of the Laplace transform evaluated at its argument ω = 0 gives rise to moments, this proof of (4.30) is completed by
E[˜λk,i] = − dφ˜λ
Proof of (iv): We first establish the following identity N matrix. Then (4.31) is achieved by using (B.7) and (4.27)
N
Appendix C
Derivation of Equation (4.43)
We derive the upper bound for the link capacity under the SWNSF scheduling when ρk is large in this Appendix. Starting from (4.41), we have
C˜k = E from the arithmetic-geometric inequality, and (d) from (4.39) together with the Jensen’s inequality.
Appendix D
Proof of Proposition 5.1
Proof of (i): Starting from (5.39), we can write N
where the (D.1) follows from direct calculations of (5.34) and (5.38). At the low SNR regime with ρ 1, ˜Czfmi can be expressed as
C˜zfmi ≈ N N=1
E[ ˜γn:Nmi ] . (D.3)
Combining (D.3) and the result (4.39) of Chapter 4, we complete the proof by
lim Proof of (ii): To begin with, we establish the upper bound and lower bound for the ˜Czfmi and ˜Coptmi at the high SNR regime. First, the upper bound for the ˜Coptmi for large ρ has been given in (4.43), which is
C˜optmi ≤ N logρ
Second, we can obtain an upper bound for ˜Czfmi at the high SNR regime as follows:
where (a) follows from the large ρ approximation along with the Jensen’s inequality and (b) from direct calculations of (5.39). Third, a lower bound for ˜Coptmi can be simply given by
C˜optmi > Coptmi
where (D.8) uses the large ρ approximation to the Taylor expansion of (5.54).
Combining (D.5), (D.6), (D.7) and (D.8), we can obtain 1 + L2 independent of ρ, the proof is completed by taking the limit of ρ in (D.9) to infinity.
Appendix E
Derivation of Equation (6.30)
Applying the matrix inversion lemma [90] to (6.22), we get
Φ−1x = 1 Pr
Φ−1k − Φ−1k akaHkΦ−1k 1 + aHkΦ−1k ak
. (E.1)
Substituting (E.1) into (6.26) yields
wmv = Φ−1x ak aHk Φ−1x ak
= Φ−1k ak− Φ−1k akaHkΦ−1k ak/(1 + aHkΦ−1k ak) aHk Φ−1k ak− aHk Φ−1k akaHkΦ−1k ak/(1 + aHkΦ−1k ak)
= Φ−1k ak
1− aHkΦ−1k ak/(1 + aHkΦ−1k ak) aHk Φ−1k ak
1− aHkΦ−1k ak/(1 + aHkΦ−1k ak)
= Φ−1k ak
aHk Φ−1k ak . (E.2)
Note that the interference-plus-noise power in the denominator of (6.28) can be expressed as
(aHkΦ−1x ak)−1− Pr = PrwmvH Φkwmv . (E.3) As a result, applying (E.2) and (E.3) to (6.28) gives the result of (6.30).
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