Ezio Biglieri, Emanuele Grossi and Marco Lops (2012), "Random-Set Theory and Wireless Communications", Foundations and TrendsÂ® in Communications and Information Theory: Vol. 7: No. 4, pp 317-462. http://dx.doi.org/10.1561/0100000054

© 2012 E. Biglieri, E. Grossi and M. Lops

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**In this article:**

1. Solving Estimation Problems Where You Do Not Know the Number of Things You Do Not Know

2. An Engineering Introduction

3. The Complexity Problem and Some Solutions

4. Multiuser Detection

5. Channel Estimation

6. Concluding Remarks

A. Mathematical Aspects of Random-Set Theory

B. Relation to Point Processes

C. Relation to Dempsterâ€“Shafer Theory

Acronyms and notations

References

This monograph is devoted to random-set theory, which allows unordered collections of random elements, drawn from an arbitrary space, to be handled. After illustrating its foundations, we focus on Random Finite Sets, i.e., unordered collections of random cardinality of points from an arbitrary space, and show how this theory can be applied to a number of problems arising in wireless communication systems. Three of these problems are: (1) neighbor discovery in wireless networks, (2) multiuser detection in which the number of active users is unknown and time-varying, and (3) estimation of multipath channels where the number of paths is not known *a priori* and which are possibly time-varying. Standard solutions to these problems are intrinsically suboptimum as they proceed either by assuming a fixed number of vector components, or by first estimating this number and then the values taken on by the components. It is shown how random-set theory provides optimum solutions to all these problems. The complexity issue is also examined, and suboptimum solutions are presented and discussed.

1. Solving estimation problems where you do not know the number of things you do not know

2. An ingineering introduction

3. The complexity problem and some solutions

4. Multiuser detection

5. Channel estimation

6. Concluding remarks

A. Mathematical aspects of random-set theory

B. Relation to point processes

C. Relation to Dempster-Shafer theory

References

Acronyms and Notations

*Random-Set Theory and Wireless Communications* gives a concise and accessible treatment of this mathematical technique and
its application to wireless communications, concentrating on estimation problems where the quantities to be estimated are in a random number.

This monograph takes an engineering approach to the topic in which the theories of the most important techniques are explained in detail before
showing how they are applied specifically to two problems in wireless communications. The first half of the monograph introduces the theory of
Random Sets. In particular Random Finite Sets (RFS), Probability Hypothesis Density (PHD) filtering, and its "Cardinalized" form (CPHD) are
described in detail. Topics covered along the way include the definition of RFS probability density functions (pdfs), with extensions such as
Bayesian recursions, statistical Inference, and implementing reduced-complexity RFS estimators. The second half of
*Random-Set Theory and Wireless Communications* illustrates the application of RFS theory to two relevant problems of
Communication Theory: multiuser detection and channel estimation in dynamic environments.

*Random-Set Theory and Wireless Communications* is an important and comprehensive survey of how to use Random Set Theory in
the design of future wireless communication systems.