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267
Large System Performance of Linear Multiuser Receivers in Multipath Fading Channels
 IEEE Trans. Inform. Theory
, 2000
"... A linear multiuser receiver for a particular user in a codedivision multipleaccess (CDMA) network gains potential benefits from knowledge of the channels of all users in the system. In fast multipath fading environments we cannot assume that the channel estimates are perfect and the inevitable cha ..."
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Cited by 68 (3 self)
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A linear multiuser receiver for a particular user in a codedivision multipleaccess (CDMA) network gains potential benefits from knowledge of the channels of all users in the system. In fast multipath fading environments we cannot assume that the channel estimates are perfect and the inevitable channel estimation errors will limit this potential gain. In this paper, we study the impact of channel estimation errors on the performance of linear multiuser receivers, as well as the channel estimation problem itself. Of particular interest are the scalability properties of the channel and data estimation algorithms: what happens to the performance as the system bandwidth and the number of users (and hence channels to estimate) grows? Our main results involve asymptotic expressions for the signaltointerference ratio of linear multiuser receivers in the limit of large processing gain, with the number of users divided by the processing gain held constant. We employ a random model for the spreading sequences and the limiting signaltointerference ratio expressions are independent of the actual signature sequences, depending only on the system loading and the channel statistics: background noise power, energy profile of resolvable multipaths, and channel coherence time. The effect of channel uncertainty on the performance of multiuser receivers is succinctly captured by the notion of effective interference.
A Random Matrix Model of Communication via Antenna Arrays
, 2001
"... A random matrix model is introduced that probabilistically describes the spatial and temporal multipath propagation between a transmitting and receiving antenna array with a limited number of scatterers for mobile radio and indoor environments. The model characterizes the channel by its richness d ..."
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Cited by 57 (7 self)
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A random matrix model is introduced that probabilistically describes the spatial and temporal multipath propagation between a transmitting and receiving antenna array with a limited number of scatterers for mobile radio and indoor environments. The model characterizes the channel by its richness delay profile which gives the number of scattering objects as a function of the path delay. Each delay is assigned the eigenvalue distribution of a random matrix that depends on the number of scatterers, receive antennas, and transmit antennas. The model allows to calculate signaltointerferenceandnoise ratios and channel capacities for large antenna arrays analytically and quantifies up to what extent rich scattering improves performance.
MIMO Channel Modelling and the Principle of Maximum Entropy
, 2004
"... In this paper , we devise theoretical grounds for constructing channel models for Multiinput Multioutput (MIMO) systems based on information theoretic tools. The paper provides a general method to derive a channel model which is consistent with one's state of knowledge. The framework we give her ..."
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Cited by 45 (25 self)
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In this paper , we devise theoretical grounds for constructing channel models for Multiinput Multioutput (MIMO) systems based on information theoretic tools. The paper provides a general method to derive a channel model which is consistent with one's state of knowledge. The framework we give here has already been fruitfully explored with success in the context of Bayesian spectrum analysis and parameter estimation. For each channel model, we conduct an asymptotic analysis (in the number of antennas) of the achievable transmission rate using tools from random matrix theory. A central limit theorem is provided on the asymptotic behavior of the mutual information and validated in the finite case by simulations. The results are both useful in terms of designing a system based on criteria such as quality of service and in optimizing transmissions in multiuser networks .
R.: Rcyclic families of matrices in free probability
 J. of Funct Anal
, 2000
"... We introduce the concept of “Rcyclic family ” of matrices with entries in a noncommutative probability space; the definition consists in asking that only the “cyclic” noncrossing cumulants of the entries of the matrices are allowed to be nonzero. Let A1,..., As be an Rcyclic family of d×d matric ..."
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Cited by 43 (0 self)
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We introduce the concept of “Rcyclic family ” of matrices with entries in a noncommutative probability space; the definition consists in asking that only the “cyclic” noncrossing cumulants of the entries of the matrices are allowed to be nonzero. Let A1,..., As be an Rcyclic family of d×d matrices over a noncommutative probability space (A, ϕ). We prove a convolutiontype formula for the explicit computation of the joint distribution of A1,...,As (considered in Md(A) with the natural state), in terms of the joint distribution (considered in the original space (A, ϕ)) of the entries of the s matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula. Moreover, let A1,...,As be a family of d × d matrices over a noncommutative probability space (A, ϕ), let D ⊂ Md(A) denote the algebra of scalar diagonal matrices, and let C be the subalgebra of Md(A) generated by {A1,...,As} ∪ D. We prove that the Rcyclicity of A1,..., As is equivalent to a property of C – namely that C is free from Md(C), with amalgamation over D.
On the Multiplication of Free NTuples of NonCommutative Random Variables
"... Let a 1 ; : : : ; an ; b 1 ; : : : ; b n be random variables in some (noncommutative) probability space, such that fa 1 ; : : : ; ang is free from fb 1 ; : : : ; b ng. We show how the joint distribution of the ntuple (a 1 b 1 ; : : : ; an b n ) can be described in terms of the joint distribution ..."
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Cited by 42 (15 self)
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Let a 1 ; : : : ; an ; b 1 ; : : : ; b n be random variables in some (noncommutative) probability space, such that fa 1 ; : : : ; ang is free from fb 1 ; : : : ; b ng. We show how the joint distribution of the ntuple (a 1 b 1 ; : : : ; an b n ) can be described in terms of the joint distributions of (a 1 ; : : : ; an ) and (b 1 ; : : : ; b n ), by using the combinatorics of the ndimensional Rtransform. We point out a few applications that can be easily derived from our result, concerning the leftandright translation with a semicircular element (see Sections 1.61.10) and the compression with a projection (see Sections 1.111.14) of an ntuple of noncommutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper. Research done while this author was on leave at the Fields Institute, Waterloo, and the Queen's University, Kingston, holding a Fellowship of NSERC, Canada. y Supported by a Heisenberg ...
Resource Pooling and Effective Bandwidths in CDMA Networks with Multiuser Receivers and Spatial Diversity
 IEEE Trans. Inform. Theory
, 1999
"... Much of the performance analysis on multiuser receivers for directsequence codedivision multipleaccess (CDMA) systems is focused on worst case nearfar scenarios. The user capacity of powercontrolled networks with multiuser receivers are less wellunderstood. In [1], it was shown that under som ..."
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Cited by 42 (3 self)
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Much of the performance analysis on multiuser receivers for directsequence codedivision multipleaccess (CDMA) systems is focused on worst case nearfar scenarios. The user capacity of powercontrolled networks with multiuser receivers are less wellunderstood. In [1], it was shown that under some conditions, the user capacity of an uplink powercontrolled CDMA cell for several important linear receivers can be very simply characterized via a notion of effective bandwidth. In the present paper, we show that these results extend to the case of antenna arrays. We consider a CDMA system consisting of users transmitting to an antenna array with a multiuser receiver, and obtain the limiting signaltointerference (SIR) performance in a large system using random spreading sequences. Using this result, we show that the SIR requirements of all the users can be met if and only if the sum of the effective bandwidths of the users is less than the total number of degrees of freedom in the system. The effective bandwidth of a user depends only on its own requirement. Our results show that the total number of degrees of freedom of the whole system is the product of the spreading gain and the number of antennas. In the case when the fading distributions to the antennas are identical, we show that a curious phenomenon of "resource pooling" arises: the multiantenna system behaves like a system with only one antenna but with the processing gain the product of the processing gain of the original system and the number of antennas, and the received power of each user the sum of the received powers at the individual antennas.
Kerov’s central limit theorem for the Plancherel measure on Young diagrams
, 2003
"... Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim 2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boun ..."
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Cited by 41 (7 self)
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Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim 2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan– Shepp 1977, Vershik–Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999.
Noncommutative martingale inequalities
, 1997
"... We prove the analogue of the classical BurkholderGundy inequalites for noncommutative martingales. As applications we give a characterization for an ItoClifford integral to be an Lpmartingale via its integrand, and then extend the ItoClifford integral theory in L2, developed by Barnett, Streater ..."
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Cited by 41 (9 self)
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We prove the analogue of the classical BurkholderGundy inequalites for noncommutative martingales. As applications we give a characterization for an ItoClifford integral to be an Lpmartingale via its integrand, and then extend the ItoClifford integral theory in L2, developed by Barnett, Streater and Wilde, to Lp for all 1 < p < ∞. We include an appendix on the noncommutative analogue of the classical Fefferman duality between H¹ and BMO.
Free semigroupoid algebras
, 2003
"... Abstract. Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a structure theory for the weak operator topology closed a ..."
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Cited by 36 (14 self)
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Abstract. Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a structure theory for the weak operator topology closed algebras generated by these representations, which we call free semigroupoid algebras. We characterize semisimplicity in terms of the graph and show explicitly in the case of finite graphs how the Jacobson radical is determined. We provide a diverse collection of examples including; algebras with free behaviour, and examples which can be represented as matrix function algebras. We show how these algebras can be presented and decomposed in terms of amalgamated free products. We determine the commutant, consider invariant subspaces, obtain a Beurling theorem for them, conduct an eigenvalue analysis, give an elementary proof of reflexivity, and discuss hyperreflexivity. Our main theorem shows the graph to be a complete unitary invariant for the algebra. This classification theorem makes use of an analysis of unitarily implemented automorphisms. We give a graphtheoretic description of when these algebras are partly free, in the sense that they contain a copy of a free semigroup algebra. 1.
Stochastic Calculus With Respect To Free Brownian Motion And Analysis On Wigner Space
, 1998
"... . We define stochastic integrals with respect to free Brownian motion, and show that they satisfy BurkholderGundy type inequalities in operator norm. We prove also a version of Ito's predictable representation theorem, as well as product form and functional form of Ito's formula. Finally we develop ..."
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Cited by 35 (3 self)
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. We define stochastic integrals with respect to free Brownian motion, and show that they satisfy BurkholderGundy type inequalities in operator norm. We prove also a version of Ito's predictable representation theorem, as well as product form and functional form of Ito's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space. Introduction In this paper we develop a stochastic integration theory with respect to the free Brownian motion. A strong motivation for undertaking this work was provided by two sources. On one hand the stochastic quantization approach to Master Fields, as described in [D], requires the development of a stochastic calculus with respect to free Brownian motion, in order to be implemented in a mathematically rigourous way. On the other hand, the theory of free entropy developped by D. Voiculescu suggests the study of "free" Gibbs states, whose definition is analogous to the classical Gibbs st...