Homogeneous Field and WKB Approximation In Deformed Quantum Mechanics with Minimal Length

In the framework of the deformed quantum mechanics with minimal length, we consider the motion of a non-relativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepest descent, we obtain the asymptotic expansions of the wave function at large positive and negative arguments. We then employ the leading asymptotic expressions to derive the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a turning point. By the WKB connection formula, we prove the Bohr-Sommerfeld quantization rule up to $\mathcal{O}(\beta)$. We also show that, if the slope of the potential at a turning point is too steep, the WKB connection formula fall apart around the turning point.Comment: 31 pages; v2: a new subsection about applications of WKB approximation and references added, published versio

Greedy vector quantization

We investigate the greedy version of the $L^p$-optimal vector quantization problem for an $\mathbb{R}^d$-valued random vector $X\!\in L^p$. We show the existence of a sequence $(a_N)_{N\ge 1}$ such that $a_N$ minimizes $a\mapsto\big \|\min_{1\le i\le N-1}|X-a_i|\wedge |X-a|\big\|_{L^p}$ ($L^p$-mean quantization error at level $N$ induced by $(a_1,\ldots,a_{N-1},a)$). We show that this sequence produces $L^p$-rate optimal $N$-tuples $a^{(N)}=(a_1,\ldots,a_{_N})$ ($i.e.$ the $L^p$-mean quantization error at level $N$ induced by $a^{(N)}$ goes to $0$ at rate $N^{-\frac 1d}$). Greedy optimal sequences also satisfy, under natural additional assumptions, the distortion mismatch property: the $N$-tuples $a^{(N)}$ remain rate optimal with respect to the $L^q$-norms, $p\le q <p+d$. Finally, we propose optimization methods to compute greedy sequences, adapted from usual Lloyd's I and Competitive Learning Vector Quantization procedures, either in their deterministic (implementable when $d=1$) or stochastic versions.Comment: 31 pages, 4 figures, few typos corrected (now an extended version of an eponym paper to appear in Journal of Approximation

Joint Unitary Triangularization for MIMO Networks

This work considers communication networks where individual links can be described as MIMO channels. Unlike orthogonal modulation methods (such as the singular-value decomposition), we allow interference between sub-channels, which can be removed by the receivers via successive cancellation. The degrees of freedom earned by this relaxation are used for obtaining a basis which is simultaneously good for more than one link. Specifically, we derive necessary and sufficient conditions for shaping the ratio vector of sub-channel gains of two broadcast-channel receivers. We then apply this to two scenarios: First, in digital multicasting we present a practical capacity-achieving scheme which only uses scalar codes and linear processing. Then, we consider the joint source-channel problem of transmitting a Gaussian source over a two-user MIMO channel, where we show the existence of non-trivial cases, where the optimal distortion pair (which for high signal-to-noise ratios equals the optimal point-to-point distortions of the individual users) may be achieved by employing a hybrid digital-analog scheme over the induced equivalent channel. These scenarios demonstrate the advantage of choosing a modulation basis based upon multiple links in the network, thus we coin the approach "network modulation".Comment: Submitted to IEEE Tran. Signal Processing. Revised versio

Artificial-Noise-Aided Secure Multi-Antenna Transmission with Limited Feedback

We present an optimized secure multi-antenna transmission approach based on artificial-noise-aided beamforming, with limited feedback from a desired single-antenna receiver. To deal with beamformer quantization errors as well as unknown eavesdropper channel characteristics, our approach is aimed at maximizing throughput under dual performance constraints - a connection outage constraint on the desired communication channel and a secrecy outage constraint to guard against eavesdropping. We propose an adaptive transmission strategy that judiciously selects the wiretap coding parameters, as well as the power allocation between the artificial noise and the information signal. This optimized solution reveals several important differences with respect to solutions designed previously under the assumption of perfect feedback. We also investigate the problem of how to most efficiently utilize the feedback bits. The simulation results indicate that a good design strategy is to use approximately 20% of these bits to quantize the channel gain information, with the remainder to quantize the channel direction, and this allocation is largely insensitive to the secrecy outage constraint imposed. In addition, we find that 8 feedback bits per transmit antenna is sufficient to achieve approximately 90% of the throughput attainable with perfect feedback.Comment: to appear in IEEE Transactions on Wireless Communication

Analysis of Basis Pursuit Via Capacity Sets

Finding the sparsest solution $\alpha$ for an under-determined linear system of equations $D\alpha=s$ is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of $\alpha$ that allow its recovery using L1-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of $D$ called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given $D$, called the "capacity sets". We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of $D\alpha=s$