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Jian Song

Publications and source records attributed to Jian Song.

At least 181 records · Page 10Linked to original sources
arXiv2019

Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise

In this article, we consider the stochastic wave equation on $\mathbb{R}_{+} \times \mathbb{R}$, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structures are given by locally integrable functions $γ$ (in time) and $f$ (in space), which are the Fourier transforms of tempered measures $ν$ on $\mathbb{R}$, respectively $μ$ on $\mathbb{R}$. Our main result shows that the law of the solution $u(t,x)$ of this equation is absolutely continuous with respect to the Lebesgue measure.

math.PR↗
arXiv2018

A High-Accuracy Adaptive Beam Training Algorithm for MmWave Communication

In millimeter wave communications, beam training is an effective way to achieve beam alignment. Traditional beam training method allocates training resources equally to each beam in the pre-designed beam training codebook. The performance of this method is far from satisfactory, because different beams have different beamforming gain, and thus some beams are relatively more difficult to be distinguished from the optimal beam than the others. In this paper, we pro- pose a new beam training algorithm which adaptively allocates training resources to each beam. Specifically, the proposed algorithm allocates more training symbols to the beams with relatively higher beamforming gain, while uses less resources to distinguish the beams with relatively lower beamforming gain. Through theoretical analysis and numerical simulations, we show that in practical situations the proposed adaptive algorithm asymptotically outperforms the traditional method in terms of beam training accuracy. Moreover, simulations also show that this relative performance behavior holds in non-asymptotic regime.

cs.IT↗
arXiv2018

Schauder estimates for equations with cone metrics, II

This is the continuation of our paper \cite{GS}, to study the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, we prove the short-time existence of the conical Kähler-Ricci flow with conical singularities along a divisor with simple normal crossings.

math.DG↗
arXiv2018

Joint Transceiver Optimization for Wireless Communication PHY with Convolutional Neural Network

Deep Learning has a wide application in the area of natural language processing and image processing due to its strong ability of generalization. In this paper, we propose a novel neural network structure for jointly optimizing the transmitter and receiver in communication physical layer under fading channels. We build up a convolutional autoencoder to simultaneously conduct the role of modulation, equalization and demodulation. The proposed system is able to design different mapping scheme from input bit sequences of arbitrary length to constellation symbols according to different channel environments. The simulation results show that the performance of neural network based system is superior to traditional modulation and equalization methods in terms of time complexity and bit error rate (BER) under fading channels. The proposed system can also be combined with other coding techniques to further improve the performance. Furthermore, the proposed system network is more robust to channel variation than traditional communication methods.

eess.SP↗
arXiv2018

Hölder continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise

In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang's condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in $L^p(Ω)$. Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, with optimal exponents, and under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in Hu, Huang, Nualart and Tindel (2015), and Song (2017) under more restrictive conditions, and with sub-optimal exponents for Hölder continuity.

math.PR↗
arXiv2018

Compactness of Kahler-Ricci solitons on Fano manifolds

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence, any sequence in $\mathcal{KR}(n)$ converge in the Gromov-Hausdorff topology to a Kähler-Ricci soliton on an $n$-dimensional $\mathbb{Q}$-Fano variety with log terminal singularities.

math.DG↗
arXiv2018

Limit theorems for functionals of two independent Gaussian processes

Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of moments, in which Fourier analysis, the chaining argument introduced in \cite{nx1} and a paring technique are employed.

math.PR↗
arXiv2017

Nonlinear Feynman-Kac formulae for SPDEs with space-time noise

We study a class of backward doubly stochastic differential equations (BDSDEs) involving martingales with spatial parameters, and show that they provide probabilistic interpretations (Feynman-Kac formulae) for certain semilinear stochastic partial differential equations (SPDEs) with space-time noise. As an application of the Feynman-Kac formulae, random periodic solutions and stationary solutions to certain SPDEs are obtained.

math.PR↗
arXiv2017

Extensions of the I-MMSE Relation

Unveiling a fundamental link between information theory and estimation theory, the I-MMSE relation by Guo, Shamai and Verdu~\cite{gu05}, together with its numerous extensions, has great theoretical significance and various practical applications. On the other hand, its influences to date have been restricted to channels without feedback or memory, due to the absence of its extensions to such channels. In this paper, we propose extensions of the I-MMSE relation to discrete-time and continuous-time Gaussian channels with feedback and/or memory. Our approach is based on a very simple observation, which can be applied to other scenarios, such as a simple and direct proof of the classical de Bruijn's identity.

cs.IT↗
arXiv2017

Spatial Modulation for More Spatial Multiplexing: RF-Chain-Limited Generalized Spatial Modulation Aided MmWave MIMO with Hybrid Precoding

The application of hybrid precoding in millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems has been proved effective for reducing the number of radio frequency (RF) chains. However, the maximum number of independent data streams is conventionally restricted by the number of RF chains, which leads to limiting the spatial multiplexing gain. To further improve the achievable spectral efficiency (SE), in this paper we propose a novel generalized spatial modulation (GenSM) aided mmWave MIMO system to convey an extra data stream via the index of the active antennas group, while no extra RF chain is required. Moreover, we also propose a hybrid analog and digital precoding scheme for SE maximization. More specifically, a closed-form lower bound is firstly derived to quantify the achievable SE of the proposed system. By utilizing this lower bound as the cost function, a two-step algorithm is proposed to optimize the hybrid precoder. The proposed algorithm not only utilizes the concavity of the cost function over the digital power allocation vector, but also invokes the convex $\ell_\infty$ relaxation to handle the non-convex constraint imposed by analog precoding. Finally, the proposed scheme is shown via simulations to outperform state-of-the-art mmWave MIMO schemes in terms of achievable SE.

cs.IT↗
arXiv2017

Some results on optimal stopping problems for one-dimensional regular diffusions

For a type of employee stock option (ESO) and an American put option with a barrier, we obtain closed-form formulae for the value functions and provide a complete characterization for optimal stopping/continuation regions. Some comparison principles for the critical levels and the value functions are given. This work is inspired by the characterization of the value functions for general one-dimensional regular diffusion processes developed in \cite{DK03} by Dayanik and Karatzas.

math.PR↗
arXiv2017

Hyperbolic Anderson Model with space-time homogeneous Gaussian noise

In this article, we study the stochastic wave equation in arbitrary spatial dimension $d$, with a multiplicative term of the form $σ(u)=u$, also known in the literature as the Hyperbolic Anderson Model. This equation is perturbed by a general Gaussian noise, which is homogeneous in both space and time. We prove the existence and uniqueness of a solution of this equation (in the Skorohod sense) and the Hölder continuity of its sample paths, under the same respective conditions on the spatial spectral measure of the noise as in the case of the white noise in time, regardless of the temporal covariance function of the noise.

math.PR↗
arXiv2017

Degeneration of Kahler-Einstein manifolds of negative scalar curvature

Let $π: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact Kähler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique Kähler-Einstein metric on $\mathcal{X}_t= π^{-1}(t)$. We show that as $t\rightarrow 0$, $(\mathcal{X}_t, g_t)$ converges in pointed Gromov-Hausdorff topology to a unique finite disjoint union of complete metric length spaces $\coprod_{α=1}^\mathcal{A} (Y_α, d_α)$ without loss of volume. Each $(Y_α, d_α)$ is a smooth open Kähler-Einstein manifold of complex dimension n outside its closed singular set of Hausdorff dimension no greater than $2n-4$. Furthermore, $\coprod_{α=1}^\mathcal{A} Y_α$ is a quasi-projective variety isomorphic to $\mathcal{X}_0 \setminus LCS(\mathcal{X}_0)$, where $\mathcal{X}_0$ is a projective semi-log canonical model and $LCS(\mathcal{X}_0)$ is the non-log terminal locus of $\mathcal{X}_0$. This is the first step of our approach toward compactification of the analytic geometric moduli space of Kähler-Einstein manifolds of negative scalar curvature.

math.DG↗
arXiv2017

Geometric estimates for complex Monge-Ampere equations

We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein metrics on Kahler manifolds of nonnegative Kodaira dimensions.

math.DG↗
arXiv2017

On the Achievable Spectral Efficiency of Spatial Modulation Aided Downlink Non-Orthogonal Multiple Access

In this paper, a novel spatial modulation aided non-orthogonal multiple access (SM-NOMA) system is proposed. We use mutual information (MI) to characterize the achievable spectral efficiency (SE) of the proposed SM-NOMA system. Due to the finite-alphabet space-domain inputs employed by SM, the expression of the corresponding MI lacks a closed-form formulation. Hence, a lower bound is proposed to quantify the MI of the SM-NOMA system. Furthermore, its asymptotic property is also theoretically investigated in both low and high signal-to-noise ratio (SNR) regions. The SE performance and its analysis of our proposed SM-NOMA system are confirmed by simulation results.

cs.IT↗
arXiv2017

Spectral-Efficient Analog Precoding for Generalized Spatial Modulation Aided MmWave MIMO

Generalized spatial modulation (GenSM) aided millimeter wave (mmWave) multiple-input multiple-output (MIMO) has recently received substantial academic attention. However, due to the insufficient exploitation of the transmitter's knowledge of the channel state information (CSI), the achievable rates of state-of-the-art GenSM-aided mmWave MIMO systems are far from being optimal. Against this background, a novel analog precoding scheme is proposed in this paper to improve the spectral efficiency (SE) of conventional GenSM-aided mmWave MIMOs. More specifically, we firstly manage to lower-bound the achievable SE of GenSM-aided mmWave MIMO with a closed-form expression. Secondly, by exploiting this lower bound as a cost function, a low-complexity iterative algorithm is proposed to design the analog precoder for SE maximization. Finally, numerical simulations are conducted to substantiate the superior performance of the proposed design with respect to state-of-the-art GenSM-aided mmWave MIMO schemes.

cs.IT↗
arXiv2017

Generalized Spatial Modulation Aided MmWave MIMO with Sub-Connected Hybrid Precoding Scheme

Due to the high cost and low energy efficiency of the dedicated radio frequency (RF) chains, the number of RF chains in a millimeter wave (mmWave) multiple-input multiple-output (MIMO) system is usually limited from a practical point of view. In this case, the maximum number of independent data streams is also restricted by the number of RF chains, which consequently leads to limiting the potentially attainable spatial multiplexing gain. In order to address this issue, in this paper, a novel generalized spatial modulation (GenSM) aided mmWave MIMO system is proposed, which enables the transmission of an extra data stream via the index of the active antennas group and requires no extra RF chain. Moreover, a two-step algorithm is also proposed to optimize the hybrid precoder design with respect to spectral efficiency (SE) maximization. Finally, numerical simulation results demonstrate the superior SE performance achieved by the proposed scheme.

cs.IT↗