Beamforming for Downlink Multiuser MIMO Time‐Varying Channels Based on Generalized Eigenvector Perturbation

A beam design method based on signal‐to‐leakage‐plus‐noise ratio (SLNR) has been recently proposed as an effective scheme for multiuser multiple‐input multiple‐output downlink channels. It is shown that its solution, which maximizes the SLNR at a transmitter, can be simply obtained by the generalized eigenvectors corresponding to the dominant generalized eigenvalues of a pair of covariance matrices of a desired signal and interference leakage plus noise. Under time‐varying channels, however, generalized eigendecomposition is required at each time step to design the optimal beam, and its level of complexity is too high to implement in practical systems. To overcome this problem, a predictive beam design method updating the beams according to channel variation is proposed. To this end, the perturbed generalized eigenvectors, which can be obtained by a perturbation theory without any iteration, are used. The performance of the method in terms of SLNR is analyzed and verified using numerical results.     

Matching wavelet packets to Gaussian random processes

We consider the problem of approximating a set of arbitrary, discrete-time, Gaussian random processes by a single, representative wavelet-based, Gaussian process. We measure the similarity between the original processes and the wavelet-based process with the Bhattacharyya (1943) coefficient. By manipulating the Bhattacharyya coefficient, we reduce the task of defining the representative process to finding an optimal unitary matrix of wavelet-based eigenvectors, an associated diagonal matrix of eigenvalues, and a mean vector. The matching algorithm we derive maximizes the nonadditive Bhattacharyya coefficient in three steps: a migration algorithm that determines the best basis by searching through a wavelet packet tree for the optimal unitary matrix of wavelet-based eigenvectors; and two separate fixed-point algorithms that derive an appropriate set of eigenvalues and a mean vector. We illustrate the method with two different classes of processes: first-order Markov and bandlimited. The technique is also applied to the problem of robust terrain classification in polarimetric SAR images     

Optimal impulse response shortening for discrete multitonetransceivers

The authors have derived a new algorithm for the optimal shortening of a channel impulse response in discrete multitone (DMT) transceivers. This algorithm uses eigenvalues and eigenvectors to generate the coefficients of the shortening impulse response filter (SIRF). In comparison with the previous approach, this new algorithm can calculate the optimal settings of an SIRF with arbitrary length     

Generalized eigenvectors and fractionalization of offset DFTs and DCTs

The offset discrete Fourier transform (DFT) is a discrete transform with kernel exp[-j2/spl pi/(m-a)(n-b)/N]. It is more generalized and flexible than the original DFT and has very close relations with the discrete cosine transform (DCT) of type 4 (DCT-IV), DCT-VIII, discrete sine transform (DST)-IV, DST-VIII, and discrete Hartley transform (DHT)-IV. In this paper, we derive the eigenvectors/eigenvalues of the offset DFT, especially for the case where a+b is an integer. By convolution theorem, we can derive the close form eigenvector sets of the offset DFT when a+b is an integer. We also show the general form of the eigenvectors in this case. Then, we use the eigenvectors/eigenvalues of the offset DFT to derive the eigenvectors/eigenvalues of the DCT-IV, DCT-VIII, DST-IV, DST-VIII, and DHT-IV. After the eigenvectors/eigenvalues are derived, we can use the eigenvectors-decomposition method to derive the fractional operations of the offset DFT, DCT-IV, DCT-VIII, DST-IV, DST-VIII, and DHT-IV. These fractional operations are more flexible than the original ones and can be used for filter design, data compression, encryption, and watermarking, etc.     

Analysis of optical waveguide discontinuities

A novel approach for analyzing discontinuity problems in optical waveguides is presented. The method is a combination of the vector-finite-element method and the least-squares boundary residual method. The vector-H-field-finite-element method is capable of providing accurate eigenvalues and eigenvectors for a wide range of optical waveguide problems, including arbitrary shape, arbitrary index distribution, and anisotropic materials. The least-squares boundary residual method matches the continuity of tangential fields in the least-squares sense, taking into account many modes at the discontinuity plane, to give the general scattering matrix. A few results are presented to show the usefulness of the approach     

Closed-Form Orthogonal DFT Eigenvectors Generated by Complete Generalized Legendre Sequence

In this paper, we propose a new method for deriving the closed-form orthogonal discrete Fourier transform (DFT) eigenvectors of arbitrary length using the complete generalized Legendre sequence (CGLS). From the eigenvectors, we then develop a novel method for computing the DFT. By taking a specific eigendecomposition to the DFT matrix, after proper arrangement, we can derive a new fast DFT algorithm with systematic construction of an arbitrary length that reduces the number of multiplications needed as compared with the existing fast algorithm. Moreover, we can also use the proposed CGLS-like DFT eigenvectors to define a new type of the discrete fractional Fourier transform, which is efficient in implementation and effective for encryption and OFDM.      

Computing the capacity of a MIMO fading channel under PSK signaling

We study the constrained capacity of a multiple-input multiple-output (MIMO) fading channel with a phase-shift keying (PSK) input alphabet and show a uniform prior distribution is capacity achieving. An expression for the capacity is derived which requires a single expectation and can be evaluated easily through simulation. The simulations are facilitated by analytical expressions for the eigenvalues and eigenvectors of a required covariance matrix. The derived expression is used to provide good approximations to the capacity at low signal-to-noise ratios (SNRs) as well as to compare the input-constrained MIMO capacity to the unconstrained MIMO capacity.     

矩阵特征分解二阶修正算法

本文研究了时变协方差矩阵特征分解的自适应更新问题,提出了矩阵特征分解二阶修正算法。首先将矩阵秩-l更新与矩阵一阶扰动问题联系起来;然后利用矩阵扰动理论将当前时刻的协方差矩阵特征值和特征向量展开为无穷级数形式,当扰动项趋于零时它们分别趋于前一时刻对应特征值和特征向量;将扰动级数二阶以后的所有项省略得到二阶修正算法;特别地研究了最小特征值重合时信号子空间的更新问题;在系统平稳和非平稳两种条件下分别进行仿真验证算法的性能,并且和一阶修正算法比较,仿真结果表明本文提出的方法具有更高的估计精度。     

A new theory for scattering of electromagnetic waves fromconducting or dielectric rough surfaces

The problem of electromagnetic scattering from a conducting or dielectric rough surface with arbitrary shape is studied. An exact solution, using a differential method, is provided for a plane wave with one-dimensional irregularity of the interface. The problem is reduced to the resolution of a linear system of partial differential equations with constant coefficients, and to the computation of eigenvalues and eigenvectors of a truncated infinite matrix. Numerical application is made to show the angular distribution of energy density in the case of an arbitrary profile of the scattering surface and its evolution when the nonperiodic profile tends to become periodic. The near field is computed on the interface and its enhancement in the illuminated region is observed. It increases with the height of the irregularity and with the frequency     

一种新的近场源三维参数联合估计算法

本文研究近场源距离、频率和到达角(DOA)三维参数的联合估计问题，并提出一种计算有效的新算法。该算法利用特征值及相应的特征矢量估计信号参数，不需要谱峰搜索且各参数自动配对。此外，新算法使用四阶累积量，因此适用于任意的加性高斯噪声环境。计算机仿真证实了所提算法的有效性。     

Multiple-antenna capacity in correlated Rayleigh fading with channel covariance information

We analyze a mobile multiple input multiple output wireless link with M transmit and N receive antennas operating in a spatially correlated Rayleigh flat fading environment. Only the correlations between the channel coefficients are assumed to be known at the transmitter and the receiver. The channel coefficients are correlated in space and uncorrelated in time from one coherence interval to another. These coefficients remain constant for a coherence interval of T symbol periods after which they change to another independent realization according to the spatial correlation model. For this system we characterize the structure of the input signal that achieves capacity. The capacity achieving transmit signal is expressed as the product of an isotropically distributed unitary matrix, an independent nonnegative diagonal matrix and a unitary matrix whose columns are the eigenvectors of the transmit fade covariance matrix. For the case where the number of transmit antennas M is larger than the channel coherence interval T, we show that the channel capacity is independent of the smallest M-T eigenvalues of the transmit fade covariance matrix. In contrast to the previously reported results for the spatially white fading model where adding more transmit antennas beyond the coherence interval length (M>T) does not increase capacity, we find that additional transmit antennas always increase capacity as long as their channel fading coefficients are spatially correlated with the other antennas. We show that for fast hopping or fast fading systems (T=1) with only channel covariance information available to the transmitter and receiver, transmit fade correlations are beneficial. Mathematically, we prove this by showing that capacity is a Schur-convex function of the vector of eigenvalues of the transmit fade correlation matrix. We also show that the maximum possible capacity gain due to transmitter fade correlations is 10logM dB.     

Index profiles: TE mode solutions

This letter presents a novel method to study one-dimensional waveguide problems in which the profiles of the refractive index are arbitrary. In the proposed method, distributions of both the refractive index and electric field are expanded into finite-term Fourier cosine series. A particular matrix is then formed by the Fourier spectral amplitudes that correspond to the refractive-index distribution. The eigenvalues and the eigenvectors of such a matrix correspond to the mode indices and the electric field distribution, respectively, in the waveguide considered herein. TE mode solutions are obtained for a step-index structure and an inhomogeneous structure. It is found that quite accurate results can be obtained by this proposed method     

Vector coding for partial response channels

A linear technique for combining equalization and coset codes on partial response channels with additive white Gaussian noise is developed. The technique, vector coding, uses a set of transmit filters or `vectors' to partition the channel into an independent set of parallel intersymbol interference (ISI)-free channels for any given finite (or infinite) block length. The optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length, and the gains of the individual channels are the eigenvalues. An optimal bit allocation and energy distribution, are derived for the set of parallel channels, under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions. Examples are presented that demonstrate performance advantages with respect to zero-forcing decision feedback methods that use the same coset code on the same partial response channel. Only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system     

The Uniform Correlation Matrix and its Application to Diversity

We consider a complex-valued L times L square matrix whose diagonal elements are unity, and lower and upper diagonal elements are the same, each lower diagonal element being equal to a (a ne 1) and each upper diagonal element being equal to b (b ne 1). We call this matrix the generalized semiuniform matrix, and denote it as M(a, b,L). For this matrix, we derive closed-form expressions for the characteristic polynomial, eigenvalues, eigenvectors, and inverse. Treating the non-real-valued uniform correlation matrix M(a, a*, L), where (middot)* denotes the complex conjugate and a ne a*, as a Hermitian generalized semiuniform matrix, we obtain the eigenvalues, eigenvectors, and inverse of M(a, a*, L) in closed form. We present applications of these results to the analysis of communication systems using diversity under correlated fading conditions     

Using perturbation theory to compute the morphological similarity of diffusion tensors

Computing the morphological similarity of diffusion tensors (DTs) at neighboring voxels within a DT image, or at corresponding locations across different DT images, is a fundamental and ubiquitous operation in the postprocessing of DT images. The morphological similarity of DTs typically has been computed using either the principal directions (PDs) of DTs (i.e., the direction along which water molecules diffuse preferentially) or their tensor elements. Although comparing PDs allows the similarity of one morphological feature of DTs to be visualized directly in eigenspace, this method takes into account only a single eigenvector, and it is therefore sensitive to the presence of noise in the images that can introduce error intothe estimation of that vector. Although comparing tensor elements, rather than PDs, is comparatively more robust to the effects of noise, the individual elements of a given tensor do not directly reflect the diffusion properties of water molecules. We propose a measure for computing the morphological similarity of DTs that uses both their eigenvalues and eigenvectors, and that also accounts for the noise levels present in DT images. Our measure presupposes that DTs in a homogeneous region within or across DT images are random perturbations of one another in the presence of noise. The similarity values that are computed using our method are smooth (in the sense that small changes in eigenvalues and eigenvectors cause only small changes in similarity), and they are symmetric when differences in eigenvalues and eigenvectors are also symmetric. In addition, our method does not presuppose that the corresponding eigenvectors across two DTs have been identified accurately, an assumption that is problematic in the presence of noise. Because we compute the similarity between DTs using their eigenspace components, our similarity measure relates directly to both the magnitude and the direction of the diffusion of water molecules. The favorable performance characteristics of our measure offer the prospect of substantially improving additional postprocessing operations that are commonly performed on DTI datasets, such as image segmentation, fiber tracking, noise filtering, and spatial normalization.     

平流层飞艇特征结构配置增稳设计

对飞艇非线性六自由度模型进行线性化,得到了飞艇的状态方程,并将其分为纵向和侧向两组.以飞艇非线性模型为基础对飞艇自然特性进行了分析,对飞艇的纵向运动及侧向运动的稳定性进行了分析仿真.运用基于线性矩阵不等式（LMI）的特征结构配置方法对平流层飞艇进行增稳控制律设计,不仅实现了特征值的配置,同时确定闭环系统特征值对应的特征向量和广义特征向量.     

Eigenvalues and eigenvectors of general gyroelectric media

Explicit and relatively simple expressions for eigenvalues and guided (propagating) eigenvectors of a general gyroelectric medium, where the preferred guided wave direction, zˆ, is parallel to the gyrotropic axis and anisotropy is confined to a plane transverse to z, are given. Some special cases of interest, namely, Hermitian, symmetric (biaxial), and uniaxial permittivity tensors, are also considered. The natural, or optic, coordinate basis is used to derive the source-free eigenvectors and to explicitly reveal the polarization states of those eigenvectors. Also under this basis, the evolution of eigenvalues and eigenvectors as off-diagonal terms of the permittivity tensor uniformly vanish, a transition from the biaxial to the uniaxial case, is discussed     

Revealing stable and unstable modes of denoisers through nonlinear eigenvalue analysis

In this paper, we propose to analyze stable and unstable modes of black-box image denoisers through nonlinear eigenvalue analysis. We aim to find input images for which the denoiser output is proportional to the input. We treat this as a generalized nonlinear eigenproblem. Potential implications are wide, as most image processing algorithms can be viewed as black-box operators. We introduce a generalized nonlinear power-method to solve eigenproblems for such operators. This allows us to reveal stable modes of the denoiser: optimal inputs, achieving superior PSNR in noise removal. Analogously to the linear case, such stable modes show coarse structures and correspond to large eigenvalues. We also provide a method to generate unstable modes, which the denoiser suppresses strongly, which are textural with small eigenvalues. We validate the method using total-variation (TV) and demonstrate it on the EPLL (Zoran–Weiss) and the Non-local means denoisers. Finally, we suggest an encryption–decryption application.     

Optimal Transmission with Imperfect Channel State Information at the Transmit Antenna Array

We study the optimal transmission strategy of a multiple-inputsingle-output wireless communication link. The receiver has perfectchannel state information while the transmitter hasonly long-term channel state information in terms of the channelcovariance matrix. It was recently shown that the optimal eigenvectors of the transmitcovariance matrix correspond with the eigenvalues of the channelcovariance matrix. However, the optimal eigenvalues are difficult tocompute. We study the properties of these optimal capacity achieving eigenvalues, and present a necessary and sufficient condition for theoptimal eigenvalues of the transmit covariance matrix. Furthermore, we develop a necessary and sufficient condition forachieving capacity when transmitting in all directions. We compare thecapacity gain of an optimal diversity system with a system which works with beamforming, and we derive an upperbound. We answer the main questions regarding the system design using the developed results. Additionally, we show inwhich way the multiplexing gain can be computed in case the channel covariancematrix is given. We compute the maximum number of required paralleldata streams, and we define a multiplexing function inorder to obtain a measure for the available multiplexinggain. Furthermore, we show that the capacity gain is small considering theadditional complexity at the receiver. We illustrate allresults by numerical simulations.     

A Unified SNR-Dependent Analysis of Space–Time Codes in Fading Channels

The pairwise error probability (PEP) for multiple- input multiple-output (MIMO) radio interfaces is investigated by means of a novel formulation based on compound matrices. The proposed approach is suitable for any MIMO system having average upper-bounded PEP written as [det( I + gamma A)]^-zeta, where A is a Hermitian matrix, zeta an integer number, and gamma the signal-to-noise ratio (SNR); that bound frequently results in MIMO single- and multicarrier transmissions. It is shown that the minimization of the bounded PEP should consider the whole set of nonzero compound matrices of A. In particular, the SNR of interest marks the compound matrix that mainly drives the system performance. Both diversity advantage and coding gain are given as continuous functions of the variable gamma, hence, their asymptotic behaviors are taken as important case of studies. The interaction effects between channel code and propagation environment are also discussed. It is shown how the eigenvectors and eigenvalues of the autocorrelation channel matrix may be considered for code design. It is also proved the maximization of the code rank is not always a necessary requirement for performance improvement being its optimal value fixed by the channel structure and SNR of interest. Finally, the analysis is applied to space-time trellis-coded transmissions over spatially correlated slow Rayleigh-fading channels.