We consider a linear time-invariant channel with given transfer function, whose output is corrupted by additive white Gaussian noise and random erasures. Determining the capacity of this channel requires obtaining the asymptotic spectral distribution of a submatrix of a nonnegative definite Toeplitz matrix obtained by retaining each column/row independently and with identical probability. This is a new problem in random matrix theory that does not follow from previously known results. We find explicitly the required asymptotic eigenvalue distribution in terms of a fixed-point equation, that yields the eta-transform of the distribution. This results represents the first known connection between the well-known theory of large Toeplitz matrices (Grenander-Szego) and random matrices. Also, we shall stress an appealing formal analogy with free-probability and the S-transform, even though freeness does not generally apply to this problem. This may suggest that freeness is a too restrictive condition for the S-transform to apply. Furthermore, we find the optimal input spectrum that achieves capacity and show that this is given by the waterfilling solution as in the case of no erasures, but computed for a scaled SNR that takes into account the presence of erasures. We find simple and easily computable upper and lower bounds to capacity, and we characterize the effect of erasures on the key quantities that determine the high-SNR and low-SNR regimes of spectral efficiency versus Eb/N0.
Applications of these results are, for example, the capacity of Gaussian channels with impulsive noise, in the limit of very large impulse power, the capacity of a Wyner-model cellular system with centralized processing, where the base stations are connected to the central processor through unreliable links that can be either ``on'' or ``off'' with a certain probability.
(joint work with Sergio Verdu, Antonia Tulino and Shlomo Shamai)
BiographyGiuseppe Caire was born in Torino, Italy, in 1965. He received the B.Sc. in Electrical Engineering from Politecnico di Torino (Italy), in 1990, the M.Sc. in Electrical Engineering from Princeton University in 1992 and the Ph.D. from Politecnico di Torino in 1994. He was a recipient of the AEI G.Someda Scholarship in 1991, has been with the European Space Agency (ESTEC, Noordwijk, The Netherlands) from May 1994 to February 1995, was a recipient of the COTRAO Scholarship in 1996 and of a CNR Scholarship in 1997.
He has been visiting Princeton University in summer 1997 and Sydney University in summer 2000. He has been Assistant Professor in Telecommunications at the Politecnico di Torino, Associate Professor at the University of Parma, Italy, Professor with the Department of Mobile Communications at the Eurecom Institute, Sophia-Antipolis, France, and he is now professor with the EE Department of the Viterbi School of Engineering, University of Southern California, Los Angeles, CA.
He served as Associate Editor for the IEEE Transactions on Communications in 1998-2001 and as Associate Editor for the IEEE Transactions on Information Theory in 2001-2003. He received the Jack Neubauer Best System Paper Award from the IEEE Vehicular Technology Society in 2003, and the Joint IT/Comsoc Best Paper Award in 2004. Since November 2004 he is member of the Board of Governors of the IEEE Information Theory Society and is Fellow of IEEE.
His current interests are in the field of communications theory, information theory and coding theory with particular focus on wireless applications.