Fixed-rank Rayleigh Quotient Maximization by an MMPSK Sequence

Certain optimization problems in communication systems, such as limited-feedback constant-envelope beamforming or noncoherent $M$-ary phase-shift keying ($M$PSK) sequence detection, result in the maximization of a fixed-rank positive semidefinite quadratic form over the $M$PSK alphabet. This form is a special case of the Rayleigh quotient of a matrix and, in general, its maximization by an $M$PSK sequence is $\mathcal{NP}$-hard. However, if the rank of the matrix is not a function of its size, then the optimal solution can be computed with polynomial complexity in the matrix size. In this work, we develop a new technique to efficiently solve this problem by utilizing auxiliary continuous-valued angles and partitioning the resulting continuous space of solutions into a polynomial-size set of regions, each of which corresponds to a distinct $M$PSK sequence. The sequence that maximizes the Rayleigh quotient is shown to belong to this polynomial-size set of sequences, thus efficiently reducing the size of the feasible set from exponential to polynomial. Based on this analysis, we also develop an algorithm that constructs this set in polynomial time and show that it is fully parallelizable, memory efficient, and rank scalable. The proposed algorithm compares favorably with other solvers for this problem that have appeared recently in the literature.Comment: 15 pages, 12 figures, To appear in IEEE Transactions on Communication

Application of Expurgated PPM to Indoor Visible Light Communications - Part I: Single-User Systems

Visible light communications (VLC) in indoor environments suffer from the limited bandwidth of LEDs as well as from the inter-symbol interference (ISI) imposed by multipath. In this work, transmission schemes to improve the performance of indoor optical wireless communication (OWC) systems are introduced. Expurgated pulse-position modulation (EPPM) is proposed for this application since it can provide a wide range of peak to average power ratios (PAPR) needed for dimming of the indoor illumination. A correlation decoder used at the receiver is shown to be optimal for indoor VLC systems, which are shot noise and background-light limited. Interleaving applied on EPPM in order to decrease the ISI effect in dispersive VLC channels can significantly decrease the error probability. The proposed interleaving technique makes EPPM a better modulation option compared to PPM for VLC systems or any other dispersive OWC system. An overlapped EPPM pulse technique is proposed to increase the transmission rate when bandwidth-limited white LEDs are used as sources.Comment: Journal of Lightwave Technolog

Dispensing with channel estimation: differentially modulated cooperative wireless communications

As a benefit of bypassing the potentially excessive complexity and yet inaccurate channel estimation, differentially encoded modulation in conjunction with low-complexity noncoherent detection constitutes a viable candidate for user-cooperative systems, where estimating all the links by the relays is unrealistic. In order to stimulate further research on differentially modulated cooperative systems, a number of fundamental challenges encountered in their practical implementations are addressed, including the time-variant-channel-induced performance erosion, flexible cooperative protocol designs, resource allocation as well as its high-spectral-efficiency transceiver design. Our investigations demonstrate the quantitative benefits of cooperative wireless networks both from a pure capacity perspective as well as from a practical system design perspective

Sharp Bounds on Davenport-Schinzel Sequences of Every Order

One of the longest-standing open problems in computational geometry is to bound the lower envelope of $n$ univariate functions, each pair of which crosses at most $s$ times, for some fixed $s$. This problem is known to be equivalent to bounding the length of an order-$s$ Davenport-Schinzel sequence, namely a sequence over an $n$-letter alphabet that avoids alternating subsequences of the form $a \cdots b \cdots a \cdots b \cdots$ with length $s+2$. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since been applied to bounding the running times of geometric algorithms, data structures, and the combinatorial complexity of geometric arrangements. Let $\lambda_s(n)$ be the maximum length of an order-$s$ DS sequence over $n$ letters. What is $\lambda_s$ asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and Nivasch) when $s$ is even or $s\le 3$. However, since the work of Agarwal, Sharir, and Shor in the mid-1980s there has been a persistent gap in our understanding of the odd orders. In this work we effectively close the problem by establishing sharp bounds on Davenport-Schinzel sequences of every order $s$. Our results reveal that, contrary to one's intuition, $\lambda_s(n)$ behaves essentially like $\lambda_{s-1}(n)$ when $s$ is odd. This refutes conjectures due to Alon et al. (2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the Symposium on Computational Geometry, 201

A Scalable Model of Cerebellar Adaptive Timing and Sequencing: The Recurrent Slide and Latch (RSL) Model

From the dawn of modern neural network theory, the mammalian cerebellum has been a favored object of mathematical modeling studies. Early studies focused on the fan-out, convergence, thresholding, and learned weighting of perceptual-motor signals within the cerebellar cortex. This led in the proposals of Albus (1971; 1975) and Marr (1969) to the still viable idea that the granule cell stage in the cerebellar cortex performs a sparse expansive recoding of the time-varying input vector. This recoding reveals and emphasizes combinations (of input state variables) in a distributed representation that serves as a basis for the learned, state-dependent control actions engendered by cerebellar outputs to movement related centers. Although well-grounded as such, this perspective seriously underestimates the intelligence of the cerebellar cortex. Context and state information arises asynchronously due to the heterogeneity of sources that contribute signals to compose the cerebellar input vector. These sources include radically different sensory systems - vision, kinesthesia, touch, balance and audition - as well as many stages of the motor output channel. To make optimal use of available signals, the cerebellum must be able to sift the evolving state representation for the most reliable predictors of the need for control actions, and to use those predictors even if they appear only transiently and well in advance of the optimal time for initiating the control action. Such a cerebellar adaptive timing competence has recently been experimentally verified (Perrett, Ruiz, & Mauk, 1993). This paper proposes a modification to prior, population, models for cerebellar adaptive timing and sequencing. Since it replaces a population with a single clement, the proposed Recurrent Slide and Latch (RSL) model is in one sense maximally efficient, and therefore optimal from the perspective of scalability.Defense Advanced Research Projects Agency and the Office of Naval Research (N00014-92-J-1309, N00014-93-1-1364, N00014-95-1-0409)