Eigenvalues and eigenvectors of covariance matrices for signalsclosely spaced in frequency

The eigenstructures of common covariance matrices are identified for the general case of M closely spaced signals. It is shown that the largest signal-space eigenvalue is relatively insensitive to signal separation. By contrast, the ith largest eigenvalue is proportional to δω^2(i-1) or δω^4(i-1), where δω is a measure of signal separation. Therefore, matrix conditioning degrades rapidly as signal separation is reduced. It is also shown that the limiting eigenvectors have remarkably simple structures. The results are very general, and apply to planar far-field direction-finding problems involving almost arbitrary scenarios, and also to time-series analysis of sinusoids, exponentials, and other signals     

Eigenvalues and eigenvectors of generalized DFT, generalized DHT,DCT-IV and DST-IV matrices

In this paper, the eigenvalues and eigenvectors of the generalized discrete Fourier transform (GDFT), the generalized discrete Hartley transform (GDHT), the type-IV discrete cosine transform (DCT-IV), and the type-IV discrete sine transform (DST-IV) matrices are investigated in a unified framework. First, the eigenvalues and their multiplicities of the GDFT matrix are determined, and the theory of commuting matrices is applied to find the real, symmetric, orthogonal eigenvectors set that constitutes the discrete counterpart of Hermite Gaussian function. Then, the results of the GDFT matrix and the relationships among these four unitary transforms are used to find the eigenproperties of the GDHT, DCT-IV, and DST-IV matrices. Finally, the fractional versions of these four transforms are defined, and an image watermarking scheme is proposed to demonstrate the effectiveness of fractional transforms     

Excitation of electromagnetic waves in a gyroelectric cylinder

The coupling of electromagnetic waves originating from a dipole to an infinite gyroelectric cylinder is treated analytically. The anisotropy axis of the gyrotropic medium is assumed to coincide with the cylinder axis. An electric dipole type primary excitation, with an arbitrary axis of orientation, is considered and the dyadic Green's function is derived for the gyroelectric cylinder. The field inside the anisotropic medium is expressed as a superposition of transverse magnetic (TM) and transverse electric (TE) type waves and a purely longitudinal wave. Excitation of guided modes along the gyroelectric guide and radiation patterns of dipoles radiating in the proximity of the gyroelectric cylinder are analyzed and computed for several cases.     

Splitting axially heterogeneous modes in microwave gyromagnetic and gyroelectric resonators

Theory of calculating eigenmodes spectrum for gyrotropic resonators when using scalar potentials formalism is developed. Equations for calculating splitting of resonant frequencies of gyromagnetic and gyroelectric cylinder resonators in magnetic field are derived. Theoretical results are compared to the experimentally obtained figures for microwave gyromagnetic resonators made of barium hexaferrite. Example of a resonator made of indium antimonide demonstrates the possibility of using magnetized semiconductor resonators cooled to liquid nitrogen temperature to achieve the same characteristics offered by resonators made of magnetically firm hexaferrites when designing unidirectional microwave devices.     

Dispersion relation for general anisotropic media

The dispersion relation is derived for an anisotropic medium with arbitrary tensor permittivity and permeability. Particular cases and applications to layered media are discussed.     

Subsurface communication between dipoles in general media

In his book, Dipole Radiation in the Presence of a Conducting Half-Space, Baños analyzes the problem of communicating between two points in the ocean by means of infinitesimal dipoles. His general formulas involve complex integrals which are reduced to simple approximate expressions in special cases that correspond to limited ranges of the numerous parameters. A numerical study of the general problem for antennas in seawater was made by Siegel and King and compared with Baños' special cases. In this paper the numerical approach is extended to other than conducting half-spaces. The three cylindrical components of the electric field of a horizontal electric dipole in seawater, lake water, and dry earth have been computed and intercompared as functions of the frequency and of the distance between the source and the receiver. The applicability of Baños' special formulas is examined. The results are discussed in their bearing on the design of directive systems for lateral-wave and direct transmission.     

图和线图的邻接谱及拉普拉斯谱的关系

图谱理论是图论研究的重要领域之一,也是非常活跃的研究方向。实践表明,对特征值的计算十分复杂,但可以研究不同定义的谱之间的关系确定特征值的上下界。通过利用对称矩阵和半正定矩阵的一些性质,讨论简单无向图G及其线图Gl的邻接谱之间的一些关系,推广已有的结果。同时也讨论图的邻接谱和拉普拉斯谱之间的关系。对判定某些图的邻接谱和拉普拉斯谱的范围具有一定的指导作用。     

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.     

Eigenvalues and troposcatter multipath analysis

Performance predictions for troposcatter channels with multipath are developed using the eigenvalues in the Karhunen-Loeve representation of the tropo signal. Flat Rayleigh fading predictions, in the presence of jamming, are easily extended to multipath channels with this technique     

Duality transformations for general bi-isotropic (nonreciprocalchiral) media

The transformation is defined to leave a given bi-isotropic medium invariant, whence it is self-dual in this transformation. It is shown that duality transformations always exist in pairs, labeled as left-hand and right-hand transformations. Self-dual fields are seen to be generalizations of the wave fields E± applied in the analysis of reciprocal chiral media. It is demonstrated that plane wave propagation and reflection problems in bi-isotropic media can be solved easily in terms of self-dual field decompositions. Nonreciprocity is seen to give rise to effects like polarization rotation in reflection, which cannot be interpreted in terms of reciprocal chiral media     

Solutions of Maxwell's equations for general nonperiodic waves inlossy media

Solutions of Maxwell's equations in lossy media for signals excited by a general applied source at the boundary plane are given. The excitation at the boundary plane can be through either electric or magnetic functions of any general time variation. No additional terms need be added to Maxwell's equations to obtain the solutions. Excitations by an electric step, exponential, and finite duration sinusoidal; functions of time are given as examples     

A general expression of dyadic Green's functions in radiallymultilayered chiral media

A general expression of spectral-domain dyadic Green's function (DGF) is presented for defining the electromagnetic radiation fields in spherically arbitrary multilayered and chiral media. Without any loss of the generality, each of the radial multilayers could be the chiral layer with different permittivity, permeability, and chirality admittance, while both distribution and location of current sources are assumed to be arbitrary. The DGF is composed of the unbounded DGF and the scattering DGF, based on the method of scattering superposition. The scattering DGF in each layer is constructed in terms of the modified and normalized spherical vector wave functions. The coefficients of the scattering DGFs are derived and expressed in terms of the equivalent reflection and transmission coefficients, by applying boundary conditions satisfied by the coefficient matrices     

Electromagnetic fields of elementary dipole antennas embedded instratified general gyrotropic media

Using matrix methods in conjunction with Fourier transformation techniques, the field excited by arbitrarily oriented elementary electric or magnetic dipole sources in the presence of a uniaxial or biaxial stratified gyrotropic medium is obtained in the form of a full dyadic Green's function. Both the permittivity and permeability tensors, being completely unrestricted, assume their most general forms in each layer separately. The singular behavior of the solution in the close vicinity of the source-point is properly taken into account by separate, suitably selected dyadic delta function terms. Using suitably selected upward or downward wave amplitude matrices, the field inside any of the layers is determined     

Eigenvalues of LC ladder networks

The system matrix of a resistance-terminated LC ladder network can be derived directly from the scattering-parameter formulation as well as from the wave-chain matrix technique. The two resulting matrices are different in form but have the same eigenvalues, and are thus related by similarity transformations.     

A general approach to extend Berenger's absorbing boundarycondition to anisotropic and dispersive media

Previous studies have focused on the extension of the Berenger's (1994) absorbing boundary condition (ABC)-the perfectly matched layer (PML)-to anisotropic and dispersive media. We describe how the PML can be extended for anisotropic and/or dispersive media (or even bianisotropic media) with little analytical effort, in a simple and general conceptual setting, by using an analytical continuation of the coordinate space of the Maxwell's equations to a complex variables domain. Using this approach, there is no conceptual distinction between the derivation of PML for more complex media and the derivation for isotropic, dispersionless media     

A general method for FDTD modeling of wave propagation in arbitraryfrequency-dispersive media

A general formulation is presented for finite-difference time-domain (FDTD) modeling of wave propagation in arbitrary frequency-dispersive media. Two algorithmic approaches are outlined for incorporating dispersion into the FDTD time-stepping equations. The first employs a frequency-dependent complex permittivity (denoted Form-1), and the second employs a frequency-dependent complex conductivity (denoted Form-2). A Pade representation is used in Z-transform space to represent the frequency-dependent permittivity (Form-1) or conductivity (Form-2). This is a generalization over several previous methods employing either Debye, Lorentz, or Drude models. The coefficients of the Pade model may be obtained through an optimization process, leading directly to a finite-difference representation of the dispersion relation, without introducing discretization error. Stability criteria for the dispersive FDTD algorithms are given. We show that several previously developed dispersive FDTD algorithms can be cast as special cases of our more general framework. Simulation results are presented for a one-dimensional (1-D) air/muscle example considered previously in the literature and a three-dimensional (3-D) radiation problem in dispersive, lossy soil using measured soil data     

A simple, nearly perfectly matched layer for general electromagnetic media

A new implementation of the perfectly matched layer (PML) absorbing boundary condition is presented. This formulation is designed such that the partial differential equations in the PML are identical to those in the regular medium for any linear electromagnetic material. This makes this method particularly simple to implement, especially in dispersive and anisotropic materials. We call this method the nearly perfectly matched layer (NPML) because it employs variable changes that are not strictly exact when the PML conductivity is spatially variant. Comparisons with the convolutional PML in a Lorentz dielectric show that the NPML is as effective an absorber as exact PML formulations.     

Eigenvalues of a dielectric-coated conducting cone

The general problem of radiation/scattering from a dielectric coated semi-infinite conical structure excited by an arbitrary surface current distribution on the dielectric layer is formulated. Since the angular eigenfunction expansion is not suitable for this problem, the radial eigenfunction expansion is employed. The boundary value method is applied to obtain the fields in the form of infinite double series over the appropriate eigenfunctions in terms of spherical Hankel and associated Legendre functions. The conical dielectric shell may be lossy or lossless and the series solution generally involves complex eigenvalues which are calculated numerically. Using a small conducting sphere at the tip of the cone, the singularity of the Hankel functions at the origin is overcome, thus permitting the use of the orthogonality relations of Sommerfeld's complex-order wave functions to solve the problem and construct sets of infinite simultaneous linear equations which are presented in matrix form.     

On the generalized eigenvectors of a class of moment matrices

An investigation is made of the eigenstructure of a class of lower triangular moment matrices that arose in the context of finding the forced response of IIR filters to typical excitations. It is found that the Jordan matrix can have at most two types of Jordan blocks. The modal matrix is shown to have a peculiar structure where the progenitors in the column partitions corresponding to the Jordan blocks have a certain pattern