Theory of Compact Hausdorff Shape

In this paper, we aim to establish a new shape theory, compact Hausdorff shape (CH-shape) for general Hausdorff spaces. We use the "internal" method and direct system approach on the homotopy category of compact Hausdorff spaces. Such a construction can preserve most good properties of H-shape given by Rubin and Sanders. Most importantly, we can moreover develop the entire homology theory for CH-shape, including the exactness, dual to the consequence of Marde\v{s}i\'c and Segal.Comment: 21 page

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

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

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

Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation

We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, and define the H-shape cohomology index to develop the Morse equations. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and N\sm E need not to be a neighborhood of the compact invariant set. Moreover, in this new theory, the phase space is not required to be separable. We apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to obtain the existence of bounded full solutions

Attractors of Local Semiflows on Topological Spaces

In this paper we introduce a notion of an attractor for local semiflows on topological spaces, which in some cases seems to be more suitable than the existing ones in the literature. Based on this notion we develop a basic attractor theory on topological spaces under appropriate separation axioms. First, we discuss fundamental properties of attractors such as maximality and stability and establish some existence results. Then, we give a converse Lyapunov theorem. Finally, the Morse decomposition of attractors is also addressed.Comment: 22 page

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.Comment: 4 pages, 2 figures, accepted by IEEE Communications Letter

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

On the forward dynamical behavior of nonautonomous lattice dynamical systems

In this article, we study the forward dynamical behavior of nonautonomous lattice systems. We first construct a family of sets $\{\mathcal{A}_\varepsilon(\sigma)\}_{\sigma\in \Sigma}$ in arbitrary small neighborhood of a global attractor of the skew-product flow generated by a general nonautonomous lattice system, which is forward invariant and uniformly forward attracts any bounded subset of the phase space. Moreover, under some suitable conditions, we further construct a family of sets $\{\mathcal{B}_\varepsilon(\sigma)\}_{\sigma\in \Sigma}$ such that it uniformly forward exponentially attracts bounded subsets of the phase space. As an application, we study the discrete Gray-Scott model in detail and illustrate how to apply our abstract results to some concrete lattice system

On Relative Category and Morse Decompositions for Infinite-Dimensional Dynamical Systems

We employ the relative category to develop relations between the Wa\.zewski pair $(N,E)$ and the Morse decomposition of the maximal invariant set in \ol{N\sm E} for infinite-dimensional dynamical systems. Via these relations, we can detect connecting trajectories between Morse sets and obtain a dynamical-system version of critical point theorem with relative category