A spectral projection method for transmission eigenvalues

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematic

Recursive integral method for transmission eigenvalues

Recently, a new eigenvalue problem, called the transmission eigenvalue problem, has attracted many researchers. The problem arose in inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing. The problem is numerically challenging because it is non-selfadjoint and nonlinear. In this paper, we propose a recursive integral method for computing transmission eigenvalues from a finite element discretization of the continuous problem. The method, which overcomes some difficulties of existing methods, is based on eigenprojectors of compact operators. It is self-correcting, can separate nearby eigenvalues, and does not require an initial approximation based on some a priori spectral information. These features make the method well suited for the transmission eigenvalue problem whose spectrum is complicated. Numerical examples show that the method is effective and robust.Comment: 18 pages, 8 figure

Initial results on an MMSE precoding and equalisation approach to MIMO PLC channels

This paper addresses some initial experiments using polynomial matrix decompositions to construct MMSE precoders and equalisers for MIMO power line communications (PLC) channels. The proposed scheme is based on a Wiener formulation based on polynomial matrices, and recent results to design and implement such systems with polynomial matrix tools. Applied to the MIMO PLC channel, the strong spectral dynamics of the PLC system together with the long impulse responses contained in the MIMO system result in problems, such that diagonlisation and spectral majorisation is mostly achieved in bands of high energy, while low-energy bands can resist any diagonalisation efforts. We introduce the subband approach in order to deal with this problem. A representative example using a simulated MIMO PLC channel is presented

Dual polarization nonlinear Fourier transform-based optical communication system

New services and applications are causing an exponential increase in internet traffic. In a few years, current fiber optic communication system infrastructure will not be able to meet this demand because fiber nonlinearity dramatically limits the information transmission rate. Eigenvalue communication could potentially overcome these limitations. It relies on a mathematical technique called "nonlinear Fourier transform (NFT)" to exploit the "hidden" linearity of the nonlinear Schr\"odinger equation as the master model for signal propagation in an optical fiber. We present here the theoretical tools describing the NFT for the Manakov system and report on experimental transmission results for dual polarization in fiber optic eigenvalue communications. A transmission of up to 373.5 km with bit error rate less than the hard-decision forward error correction threshold has been achieved. Our results demonstrate that dual-polarization NFT can work in practice and enable an increased spectral efficiency in NFT-based communication systems, which are currently based on single polarization channels

Towards Dual-functional Radar-Communication Systems: Optimal Waveform Design

We focus on a dual-functional multi-input-multi-output (MIMO) radar-communication (RadCom) system, where a single transmitter communicates with downlink cellular users and detects radar targets simultaneously. Several design criteria are considered for minimizing the downlink multi-user interference. First, we consider both the omnidirectional and directional beampattern design problems, where the closed-form globally optimal solutions are obtained. Based on these waveforms, we further consider a weighted optimization to enable a flexible trade-off between radar and communications performance and introduce a low-complexity algorithm. The computational costs of the above three designs are shown to be similar to the conventional zero-forcing (ZF) precoding. Moreover, to address the more practical constant modulus waveform design problem, we propose a branch-and-bound algorithm that obtains a globally optimal solution and derive its worst-case complexity as a function of the maximum iteration number. Finally, we assess the effectiveness of the proposed waveform design approaches by numerical results.Comment: 13 pages, 10 figures. 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