Maximum Likelihood Criterion Based Metrics

The optimum ML criterion is known to have high computational complexity and poor scaling with a growth in the number of users. However, it gives the best performance of all decoding algorithms. Whether or not ML decoding is actually employed, the theoretical performance of ML processing can still be used as a scheduling metric since good ML performance is likely to be linked with good performance of other sub-optimal approaches. Hence, we propose a group scheduling metric based on frame error rate (FER) and individual scheduling metrics based on the SER of each user.

Group Scheduling Metrics Based on FER

The frame defined in this chapter is the N × 1 symbol vector transmitted by all the MSs. If the

decoded vector, sssk, is different (in the sense that any element of the vector is different from the

corresponding element of the transmitted vector) from the transmitted vector sssi, an error occurs.

Following the same procedure for the SER in Sec. 7.3.2, the FER can be upper bounded to give

FER ≤ M−NX

i

X

k6=i

Pe(sssi → sssk) , (8.24)

where Pe(sssi → sssk) is the pair wise error probability as defined in Sec. 7.2. In Chapter 7,

Pe(sssi → sssk) has been calculated in closed form for both Rayleigh and Rician fading scenarios.

However, these results are not particularly compact and may impose some unwanted computational burden in scheduling. Therefore, in this section we exploit the simple high SNR approximations for the scheduling metric derivations. This permits us to define the following group scheduling

metric for Rayleigh fading environment as Mg = M−N X i X k6=i αik, (8.25)

where αik=Qn_l=1R λl, M is the number of constellation points, and the λl terms are defined as

1 λl

= cccH

ikDDDlcccik, (8.26)

with the variables in (8.26) defined in Sec. 7.2. Both summations run over all possible MN

transmit vectors. For a Rician fading environment, the corresponding metric is

Mg = M−N X i X k6=i ˜ αik, (8.27) where ˜αik = Qnr=1R ˜λrexp  −˜λr|Σr|2 

and the constants are defined in Sec. 7.2. The double summation in (8.25) and (8.27) can be regarded as an expectation operation over the discrete con-

stellation points. Therefore, it is clear that Mg is a function of the elements of the power profile

matrix, PPP .

Individual Scheduling Metrics Based on SER

Individual scheduling metrics based on SER can be calculated following the same procedure as in Sec. 7.3.2. In the SER upper bound expression, the PEP is replaced by high SER approximations of PEP which give compact tractable expressions. These high SNR approximations are found in Sec. 7.3.1. For the sake of completeness, we provide the final result here. For Rayleigh fading,

mk= M−N M X m=1 X i X k αik, (8.28)

and for Rician fading, mk= M−N M X m=1 X i X k ˜ αik. (8.29)

Note that the summations in (8.28) and (8.29) have a slightly different domain than the FER based scheduling metrics discussed in Sec. 7.3.2.

Note that the summations that appear in the group and individual metrics grow exponentially with the number the of users. Therefore, with larger numbers of users and BSs, ML criterion based scheduling metrics may become computationally inefficient. However, they are simple and efficient for small systems.

8.4

Summary

In this chapter, we have considered the idea of using long term CSI for user scheduling in a macro- diversity MU MIMO-MAC channel. In particular, CDI based user scheduling may be beneficial in CoMP systems where a limited backhaul interconnection network is present and the use of instanta- neous CSI is restricted by delays and communication overheads. Despite the analytical difficulty in deriving scheduling metrics using CDI, in this chapter we have proposed user scheduling methods based on ergodic capacity, linear multiuser receivers and maximum likelihood decoding. Fur- thermore, we have derived several systematic group scheduling metrics and individual scheduling metrics.

Conclusions and Future Work

Broadly speaking, the main focus of this thesis is the analytical investigation of fundamental performance measures for the macrodiversity MIMO-MAC. This investigation considers finite sized systems with completely arbitrary average link gain profiles, in contrast to simplified channel matrix assumptions, such as the classical Wyner model, widely used in network MIMO. Therefore, the results in this thesis may have applications in other areas where similar multivariate statistical problem arise. The analytical results can be useful in many different applications, such as:

• In contrast to previous ad-hoc approximations, our results reveal a direct functional link be- tween fundamental performance measures and the average link gains. In some systems, the results give a remarkably simple, yet sophisticated relationship, while in other systems, the re- sults may be rather complex. These functions gives system designers a greater range of options to optimize the performance of macrodiversity links.

• Analytical results can be used for efficient computation rather than using inefficient Monte- Carlo simulations which can greatly reduce the system design time. The exact results presented in the thesis can be used instead of simulation based methods for faster calculation of perfor- mance measures, even though analytical expressions are lengthy.

• If there is a requirement to further examine the statistical behavior over random average link

gains (i.e., system level simulations involving averaging over the PPP matrix), our results save

hours of computation time. If Monte-Carlo simulation is used for averaging over both fast fading and slow fading, then this is highly inefficient. In contrast, the analytical results can be used for averaging over fast fading and simulation based methods can be used to average over slow fading.

• The analytical work fills a gap in our knowledge of wireless communication. The results are very attractive on some fronts, as far as their practical applications are concerned, while on other fronts, they turn out to be complex, but intriguing. We believe this will inspire future research for better, simpler and faster results in this area of communication and beyond. More specific comments on the analytical methodologies and the results achieved in this research are summarized below followed by a brief description of potential future research.

9.1

Summary of Analysis and Key Results