On the transport capacity of Gaussian multiple access and broadcast channels

We study the transport capacity of the Gaussian multiple access channel (MAC), which consists of multiple transmitters and a single receiver, and the Gaussian broadcast channel (BC), which consists of a single transmitter and multiple receivers. The transport capacity is defined as the sum, over all transmitters (for the MAC) or receivers (for the BC), of the product of the data rate with a reward r(x) which is a function of the distance x that the data travels. In the case of the MAC, assuming that the sum of the transmit powers is upper bounded, we calculate in closed form the optimal power allocation among the transmitters, that maximizes the transport capacity, using Karush-Kuhn-Tucker (KKT) conditions. We also derive asymptotic expressions for the optimal power allocation, that hold as the number of transmitters approaches infinity, using the most-rapid-approach method of the calculus of variations. In the case of the BC, we calculate in closed form the optimal allocation of the transmit power among the signals to the different receivers, both for a finite number of receivers and for the case of asymptotically many receivers, using our results for the MAC together with duality arguments. Our results can be used to gain intuition and develop good design principles in a variety of settings. For example, they apply to the uplink and downlink channel of cellular networks, and also to sensor networks which consist of multiple sensors that communicate with a single central station. Work was carried out while all authors were with the Telecommunications Research Center Vienna (ftw.), and supported by K plus funding for the ftw. project I0 “Signal and Information Processing.” Parts of this work have appeared, in preliminary form, in [1,2,3], Gautam A. Gupta holds a joint B.S./M.S. degree in mathematics and computing at the Department of Mathematics of the Indian Institute of Technology at New Delhi. During the summer of 2003, he attended a summer course on Probability and Statistical Mechanics organized by the Scoula Normale Superiore, in Pisa, Italy. During the summers of 2004 and 2005 he worked at the Telecommunications Research Center Vienna (ftw.) as a summer intern. During the spring of 2006, he was a visitor at the Norwegian University of Science and Technology, working toward his M. S. Thesis. Stavros Toumpis received the Diploma in electrical and computer engineering from the National Technical University of Athens, Greece, in 1997, the M.S. degrees in electrical engineering and mathematics from Stanford University, CA, in 1999 and 2002, respectively, and the Ph.D. degree in electrical engineering, also from Stanford, in 2003. From 1998 to 1999, he worked as a Research Assistant for the Mars Global Surveyor Radio Science Team, providing operational support. From 2000 to 2003, he was a Member of the Wireless Systems Laboratory, at Stanford University. From 2003 to 2005, he was a Senior Researcher with the Telecommunications Research Center Vienna (ftw.), in Vienna, Austria. Since 2005, he is a Lecturer at the Department of Electrical and Computer Engineering of the University of Cyprus. His research is on wireless ad hoc networks, with emphasis on their capacity, the effects of mobility on their performance, medium access control, and information theoretic issues. Jossy Sayir received his Dipl. El.-Ing. degree from the ETH Zurich in 1991. From 1991 to 1993, he worked as a development engineer for Motorola Communications in Tel Aviv, Israel, contributing to the design of the first digital mobile radio system ever produced by Motorola. He returned to ETH from 1993 to 1999, getting his PhD in 1999 under the supervision of Prof. J.L. Massey. The title of his thesis is “On Coding by Probability Transformation.” Since 2000, he has been employed at the Telecommunications Research Center (ftw) in Vienna, Austria, as a senior researcher. His research interests include iterative decoding methods, joint source and channel coding, numerical capacity computation algorithms, Markov sources, and wireless ad hoc and sensor networks. Since July 2002, he manages part of the strategic research activities at Ftw and supervises a group of researchers. He has taught courses on Turbo and related codes at Vienna University of Technology and at the University of Aalborg, Denmark. He has served on the organization committees of several international conferences and workshops. Ralf R. Müller was born in Schwabach, Germany, 1970. He received the Dipl.-Ing. and Dr.Ing. degree with distinction from University of Erlangen-Nuremberg in 1996 and 1999, respectively. From 2000 to 2004, he was with Forschungszentrum Telekommunikation Wien (Vienna Telecommunications Research Center) in Vienna, Austria. Since 2005 he has been a full professor at the Department of Electronics and Telecommunications at the Norwegian University of Science and Technology (NTNU) in Trondheim, Norway. He held visiting appointments at Princeton University, U.S.A., Institute Eurecom, France, The University of Melbourne, Australia, and The National University of Singapore and was an adjunct professor at Vienna University of Technology. Dr. Müller received the Leonard G. Abraham Prize (jointly with Sergio S. Verdú) from the IEEE Communications Society and the Johann-Philipp-Reis Prize (jointly with Robert Fischer). He was also presented an award by the Vodafone Foundation for Mobile Communications and two more awards from the German Information Technology Society (ITG). Dr. Müller is currently serving as an associate editor for the IEEE Transactions on Information Theory.