Abstract—We consider multiaccess multiple-input multipleoutput (MIMO) systems with finite rate feedback with the aim of understanding how to efficiently employ the given feedback resource to maximize the sum rate. A joint quantization and feedback strategy is proposed: the base station selects the strongest users, jointly quantizes their strongest eigen-channel vectors and broadcasts a common feedback to all the users. This joint strategy differs from an individual strategy in which quantization and feedback are performed independently across users, and it improves upon the individual strategy in the same way that vector quantization improves upon scalar quantization. To analyze the proposed strategy, the effect of user selection is described by extreme order statistics, while the effect of joint quantization is quantified through what we term “the composite Grassmann manifold”. The achievable sum rate is then estimated using random matrix theory providing an analytic benchmark for the performance. Index Terms—Beamforming, Grassmann manifold, limited feedback, MIMO, multiaccess channels.

This short note explains how to use ready-to-use components of symbolic software to convert between the free cumulants and the moments of measures without sophisticated programming. This allows quick access to low order moments of free convolutions of measures, which can be used to test whether a given probability measure is a free convolution of other measures.

Abstract. In this paper, we are interested in sequences of q-tuple of N ×N random matrices having a strong limiting distribution (i.e. given any non-commutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the q-tuple is enlarged with independent unitary Haar distributed random matrices. Besides, the limit of norms and traces in non-commutative polynomials in the enlarged family can be computed with reduced free product construction. This extends results of one author (C. M.) and of Haagerup and Thorbjørnsen. We also show that ap-tuple of independent orthogonal and symplectic Haar matrices have a strong limiting distribution, extending a recent result of Schultz.

Consider a wireless MIMO multi-hop channel with ns non-cooperating source antennas and nd fully cooperating destination antennas, as well as L clusters containing k non-cooperating relay antennas each. The source signal traverses all L clusters of relay antennas, before it reaches the destination. When relay antennas within the same cluster scale their received signals by the same constant before the retransmission, the equivalent channel matrix H relating the input signals at the source antennas to the output signals at the destination antennas is proportional to the product of channel matrices Hl, l = 1,...,L + 1, corresponding to the individual hops. We perform an asymptotic capacity analysis for this channel as follows: In a first instance we take the limits ns → ∞, nd → ∞ and k → ∞, but keep both ns/nd and k/nd fixed. Then, we take the limits L → ∞ and k/nd → ∞. Requiring that the Hl’s satisfy the conditions needed for the Marčenko-Pastur law, we prove that the capacity scales linearly in min{ns, nd}, as long as the ratio k/nd scales at least linearly in L. Moreover, we show that up to a noise penalty and a pre-log factor the capacity of a point-to-point MIMO channel is approached, when this scaling is slightly faster than linear. Conversely, almost all spatial degrees of freedom vanish for less than linear scaling. 2 I.