Jinming Wen

A sharp condition for exact support recovery of sparse signals with orthogonal matching pursuit

Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied in the literature. In this paper, we show that for any K-sparse signal x, if the sensing matrix A satisfies the restricted isometry property (RIP) of order K+1 with restricted isometry constant (RIC) δK+1 <; 1/√K+1, then under some constraint on the minimum magnitude of the nonzero elements of x, the OMP algorithm exactly recovers the support of x from the measurements y = Ax + v in K iterations, where v is the noise vector. This condition is sharp in terms of δK+1 since for any given positive integer K ≥ 2 and any 1/√K+1 ≤ t <; 1, there always exist a K-sparse x and a matrix A satisfying δK+1 = t for which OMP may fail to recover the signal x in K iterations. Moreover, the constraint on the minimum magnitude of the nonzero elements of x is weaker than existing results.

Effects of the LLL Reduction on the Success Probability of the Babai Point and on the Complexity of Sphere Decoding

A common method to estimate an unknown integer parameter vector in a linear model is to solve an integer least squares (ILS) problem. A typical approach to solving an ILS problem is sphere decoding. To make a sphere decoder faster, the well-known LLL reduction is often used as preprocessing. The Babai point produced by the Babai nearest plane algorithm is a suboptimal solution of the ILS problem. First, we prove that the success probability of the Babai point as a lower bound on the success probability of the ILS estimator is sharper than the lower bound given by Hassibi and Boyd [1]. Then, we show rigorously that applying the LLL reduction algorithm will increase the success probability of the Babai point and give some theoretical and numerical test results. We give examples to show that unlike LLLs column permutation strategy, two often used column permutation strategies SQRD and V-BLAST may decrease the success probability of the Babai point. Finally, we show rigorously that applying the LLL reduction algorithm will also reduce the computational complexity of sphere decoders, which is measured approximately by the number of nodes in the search tree in the literature.

An Efficient Algorithm for Optimally Solving a Shortest Vector Problem in Compute-and-Forward Design

We consider the problem of finding the optimal coefficient vector that maximizes the computation rate at a relay in the compute-and-forward scheme. Based on the idea of sphere decoding, we propose a highly efficient algorithm that finds the optimal coefficient vector. First, we derive a novel algorithm to transform the original quadratic form optimization problem into a shortest vector problem (SVP) using the Cholesky factorization. Instead of computing the Cholesky factor explicitly, the proposed algorithm realizes the Cholesky factorization with only O(n) flops by taking advantage of the structure of the Gram matrix in the quadratic form. Then, we propose some conditions that can be checked with O(n) flops, under which a unit vector is the optimal coefficient vector. Finally, by considering some useful properties of the optimal coefficient vector, we modify the Schnorr-Euchner search algorithm to solve the SVP. We show that the estimated average complexity of our new algorithm is O(n1.5p0.5) flops for independent identically distributed (i.i.d.) Gaussian channel entries with SNR P based on the Gaussian heuristic. Simulations show that our algorithm is not only much more efficient than the existing ones that give the optimal solution, but also faster than some best known suboptimal methods. Besides, we show that our algorithm can be readily adapted to output a list of L best candidate vectors for use in the compute-and-forward design. The estimated average complexity of the resultant list-output algorithm is O(n2.5p0.5 + n1.5p0.5 log(L) + nL) flops for i.i.d. Gaussian channel entries.

Joint Antenna Selection and Spatial Switching for Energy Efficient MIMO SWIPT System

In this paper, we investigate joint antenna selection and spatial switching for quality-of-service-constrained energy efficiency (EE) optimization in a multiple-input multiple-output simultaneous wireless information and power transfer system. A practical linear power model taking into account the entire transmit–receive chain is accordingly utilized. The corresponding fractional-combinatorial and non-convex EE problem, involving joint optimization of eigenchannel assignment, power allocation, and active receive antenna set selection, subject to satisfying minimum sum-rate and power transfer constraints, is extremely difficult to solve directly. In order to tackle this, we separate the eigenchannel assignment and power allocation procedure with the antenna selection functionality. In particular, we first tackle the EE maximization problem under fixed receive antenna set using Dinkelbach-based convex programming, iterative joint eigenchannel assignment and power allocation, and low-complexity multi-objective optimization-based approach. On the other hand, the number of active receive antennas induces a tradeoff in the achievable sum-rate and power transfer versus the transmit-independent power consumption. We provide a fundamental study of the achievable EE with antenna selection and accordingly develop dynamic optimal exhaustive search and Frobenius-norm-based schemes. Simulation results confirm the theoretical findings and demonstrate that the proposed resource allocation algorithms can efficiently approach the optimal EE.

Compute-and-forward protocol design based on improved sphere decoding

We consider the compute-and-forward protocol design problem with the objective being maximizing the computation rate at a single relay, and propose an efficient method that finds the optimal solution based on sphere decoding. The problem can be transformed into a shortest vector problem (SVP), which can be solved in two steps. First, by fully exploiting the specific structure of the associated Gram matrix using the hyperbolic transformation, the Cholesky factor can be computed with only n2/2 + O(n) flops. Then, taking into account of some useful properties of the optimal solution, we modify the Schnorr-Euchner search algorithm to solve the SVP. Numerical results show that our proposed branch-and-bound method is much more efficient than the existing one that gives the optimal solution. Besides, compared with the suboptimal methods, our method offers the best performance at a cost lower than that of the LLL based method and similar to that of the quadratic programming relaxation method.

A linearithmic time algorithm for a shortest vector problem in compute-and-forward design

We modify the algorithm proposed by Sahraei et al. in 2015, resulting an algorithm with expected complexity of O(n log n) arithmetic operations to solve a special shortest vector problem arising in computer-and-forward design, where n is the dimension of the channel vector. This algorithm is more efficient than the best known algorithms with proved complexity.

Effects of Some Lattice Reductions on the Success Probability of the Zero-Forcing Decoder

Zero-forcing (ZF) decoder is a commonly used approximation solution of the integer least squares problem, which arises in communications and many other applications. Numerical simulations have shown that the LLL reduction can usually improve the success probability PZF of the ZF decoder. In this letter, we first rigorously show that both SQRD and V-BLAST, two commonly used lattice reductions, have no effect on PZF. Then, we show that LLL reduction can improve PZF when n = 2, we also analyze how the parameter δ in the LLL reduction affects the enhancement of PZF. Finally, an example is given which shows that the LLL reduction decrease PZF when n ≥ 3.

Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay

This paper is concerned with the stability of non-Fickian reaction-diffusion equations with a variable delay. It is shown that the perturbation of the energy function of the continuous problems decays exponentially, which provides a more accurate and convenient way to express the rate of decay of energy. Then, we prove that the proposed numerical methods are sufficient to preserve energy stability of the continuous problems. We end the paper with some numerical experiments on a biological model to confirm the theoretical results.

Stability and convergence of compact finite difference method for parabolic problems with delay

Abstract The compact finite difference method becomes more acceptable to approximate the diffusion operator than the central finite difference method since it gives a better convergence result in spatial direction without increasing the computational cost. In this paper, we apply the compact finite difference method and the linear θ-method to numerically solve a class of parabolic problems with delay. Stability of the fully discrete numerical scheme is investigated by using the spectral radius condition. When θ ∈ [ 0 , 1 2 ) , a sufficient and necessary condition is presented to show that the fully discrete numerical scheme is stable. When θ ∈ [ 1 2 , 1 ] , the fully discrete numerical method is proved to be unconditionally asymptotically stable. Moreover, convergence of the fully discrete scheme is studied. Finally, several numerical examples are presented to illustrate our theoretical results.

A modified KZ reduction algorithm

The Korkine-Zolotareff (KZ) reduction has been used in communications and cryptography. In this paper, we modify a very recent KZ reduction algorithm proposed by Zhang et al., resulting in a new algorithm, which can be much faster and more numerically reliable, especially when the basis matrix is ill conditioned.