Matrix Completion-Based Channel Estimation for MmWave Communication Systems With Array-Inherent Impairments

Hybrid massive MIMO structures with reduced hardware complexity and power consumption have been widely studied as a potential candidate for millimeter wave (mmWave) communications. Channel estimators that require knowledge of the array response, such as those using compressive sensing (CS) methods, may suffer from performance degradation when array-inherent impairments bring unknown phase errors and gain errors to the antenna elements. In this paper, we design matrix completion (MC)-based channel estimation schemes which are robust against the array-inherent impairments. We first design an open-loop training scheme that can sample entries from the effective channel matrix randomly and is compatible with the phase shifter-based hybrid system. Leveraging the low-rank property of the effective channel matrix, we then design a channel estimator based on the generalized conditional gradient (GCG) framework and the alternating minimization (AltMin) approach. The resulting estimator is immune to array-inherent impairments and can be implemented to systems with any array shapes for its independence of the array response. In addition, we extend our design to sample a transformed channel matrix following the concept of inductive matrix completion (IMC), which can be solved efficiently using our proposed estimator and achieve similar performance with a lower requirement of the dynamic range of the transmission power per antenna. Numerical results demonstrate the advantages of our proposed MC-based channel estimators in terms of estimation performance, computational complexity and robustness against array-inherent impairments over the orthogonal matching pursuit (OMP)-based CS channel estimator.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Equal Entries in Totally Positive Matrices

We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) $n\textrm{-by-}n$ matrix is $\Theta(n^{4/3})$ (resp. $\Theta(n^{3/2}$)). Relationships with point-line incidences in the plane, Bruhat order of permutations, and $TP$ completability are also presented. We also examine the number and positionings of equal $2\textrm{-by-}2$ minors in a $2\textrm{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\textrm{-by-}2$ minors and outerplanar graphs.Comment: 15 page

Endomorphisms of spaces of virtual vectors fixed by a discrete group

Consider a unitary representation $\pi$ of a discrete group $G$, which, when restricted to an almost normal subgroup $\Gamma\subseteq G$, is of type II. We analyze the associated unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ on the Hilbert space of "virtual" $\Gamma_0$-invariant vectors, where $\Gamma_0$ runs over a suitable class of finite index subgroups of $\Gamma$. The unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ is uniquely determined by the requirement that the Hecke operators, for all $\Gamma_0$, are the "block matrix coefficients" of $\overline{\pi}^{\rm{p}}$. If $\pi|_\Gamma$ is an integer multiple of the regular representation, there exists a subspace $L$ of the Hilbert space of the representation $\pi$, acting as a fundamental domain for $\Gamma$. In this case, the space of $\Gamma$-invariant vectors is identified with $L$. When $\pi|_\Gamma$ is not an integer multiple of the regular representation, (e.g. if $G=PGL(2,\mathbb Z[\frac{1}{p}])$, $\Gamma$ is the modular group, $\pi$ belongs to the discrete series of representations of $PSL(2,\mathbb R)$, and the $\Gamma$-invariant vectors are the cusp forms) we assume that $\pi$ is the restriction to a subspace $H_0$ of a larger unitary representation having a subspace $L$ as above. The operator angle between the projection $P_L$ onto $L$ (typically the characteristic function of the fundamental domain) and the projection $P_0$ onto the subspace $H_0$ (typically a Bergman projection onto a space of analytic functions), is the analogue of the space of $\Gamma$- invariant vectors. We prove that the character of the unitary representation $\overline{\pi}^{\rm{p}}$ is uniquely determined by the character of the representation $\pi$.Comment: The exposition has been improved and a normalization constant has been addressed. The result allows a direct computation for the characters of the unitary representation on spaces of invariant vectors (for example automorphic forms) in terms of the characters of the representation to which the fixed vectors are associated (e.g discrete series of PSL(2, R) for automorphic forms

On Spectral Triples in Quantum Gravity II

A semifinite spectral triple for an algebra canonically associated to canonical quantum gravity is constructed. The algebra is generated by based loops in a triangulation and its barycentric subdivisions. The underlying space can be seen as a gauge fixing of the unconstrained state space of Loop Quantum Gravity. This paper is the second of two papers on the subject.Comment: 43 pages, 1 figur