Orthogonal functionals of the Poisson process

In analogy to the orthogonal functionals of the Brownian-motion process developed by Wiener, ltô, and others, a theory of the orthogonal functionals of the Poisson process is presented making use of the concept of multivariate orthogonal polynomials. Following a brief discussion of Charlier polynomials of a single variable, multivariate Charlier polynomials are introduced. An explicit representation as well as an orthogonality property are given. A multiple stochastic integral of a multivariate function with respect to the Poisson process, called the multiple Poisson-Wiener integral, is defined using the multivariate Charlier polynomials. A multiple Poisson-Wiener integral, which gives a polynomial functional of the Poisson process, is orthogonal to any other of different degree. Several explicit forms are given for the sake of application. It is shown that any nonlinear functional of the Poisson process with finite variance can be developed in terms of these orthogonal functionals, corresponding to the Cameron-Martin theorem in the case of the Brownian-motion process. Finally, some possible applications to nonlinear problems associated with the Poisson process are briefly discussed.     

The correlation function of Gaussian noise passed through nonlinear devices

This paper is concerned with the output autocorrelation functionR^{y}of Gaussian noise passed through a nonlinear device. An attempt is made to investigate in a systematic way the changes inR^{y}when certain mathematical manipulations are performed on some given device whose correlation function is known. These manipulations are the "elementary combinations and transformations" used in the theory of Fourier integrals, such as addition, differentiation, integration, shifting, etc. To each of these, the corresponding law governingR^{y}is established. The same laws are shown to hold for the envelope of signal plus noise for narrow-band noise with spectrum symmetric about signal frequency. Throughout the text and in the Appendix it is shown how the results can be used to establish unknown correlation function quickly with main emphasis on power-law devicesy = x^{m}withmeither an integer or half integer. Some interesting recurrence formulas are given. A second-order differential equation is derived which serves as an alternative means for calculatingR^{y}.     

Stochastic processes in estimation theory

We describe the role of various stochastic processes, especially martingales and related concepts, in estimation theory. It is shown, in the simplest context, that in nonlinear estimation theory martingales play the same fundamental role as uncorrelation and white noise do in linear estimation.     

Interlacing properties of shift-register sequences with generator polynomials irreducible overGF(p)(Corresp.)

Interlacing properties of shift-register sequences with generator polynomials irreducible over GF(p)-herein called elementary sequences--are analyzed. The most important elementary sequences are maximal-length sequences (m-sequences). If the periodqof an elementary sequence is not prime, the sequence can always be constructed by interlacing shorter elementary sequences of periodq_{i}, providedq_{i}dividesq. It is proved that all interlaced elementary sequences are generated by one and the same irreducible polynomial. Some relationships between equal-length elementary sequences are derived, including some rather unexpected crosscorrelation properties. As an example of an application of the theory, a new time-division multiplex technique for generating high-speedm-sequences is presented.     

Optimal functional expansion for estimation from counting observations

A systematic method to develop approximate nonlinear estimators is presented, in the form of a functional series, for the signal that modulates the rate of a counting process. The estimators are optimal for the given structure and approach the minimum variance (MV) estimator when the approximation order increases. Two kinds of functional series, the iterated integral (II) series and the Fourier-Charlier (FC) series, are used. Product-to-sum formulas for the II and FC functionals are derived. By using the formulas, the MV estimate is projected onto the Hilbert subspaces of the II and the FC series driven by the counting observations with the given index set. The projection results in a Wiener-Hopf type equation for the II kernels and a system of linear algebraic equations for the FC coefficients. The FC series estimator consists of finitely many single Wiener integrals of the counting observations and a nonlinear postprocessor. The nonlinear postprocessor, however, is not memoryless.     

A general likelihood-ratio formula for random signals in Gaussian noise

It is shown that the likelihood ratio for the detection of a random, not necessarily Gaussian, signal in additive white Gaussian noise has the same form as that for a known signal in white Gaussian noise. The role of the known signal is played by the casual least-squares estimate of the signal from the observations. However, the "correlation" integral has to be interpreted in a special sense as an Itô stochastic integral. It will be shown that the formula includes all known explicit formulas for signals in white Gaussian noise. However, and more important, the formula suggests an "estimator-correlator" philosophy for engineering approximation of the optimum receiver. Some extensions of the above result are also discussed, e.g., additive finite-variance, not necessarily Gaussian, noise plus a white Gaussian noise component. Purely colored Gaussian noise can be treated if whitening filters can be specified. The analog implementation of Itô integrals is briefly discussed. The proofs of the formulas are based on the concept of an innovation process, which has been useful in certain related problems of linear and nonlinear least-squares estimation, and on the concept of covariance factorization.     

A new approach to the problem of wave fluctuations in localized smoothly varying turbulence

In the past, smoothly varying turbulence has been studied by changing the structure constant to the functionC_{n}^{2}(bar{r}). The purpose of this paper is to show that this approach is insufficient, and that a random process developed by Silverman can be used to describe the wave fluctuations in localized smoothly varying turbulence. The localized turbulence is characterized by a correlation function which is a product of a function of the average coordinate and a function of the difference coordinate. The corresponding spectrum is also given by a product of a function of the difference wavenumber and a function of the average wavenumber. They are related to each other through two Fourier transform pairs. Making use of the preceding representations, the fluctuations of a wave propagating through such a turbulence can be given either by the integrals with respect to the two wavenumbers or by a convolution integral of the structure constantC_{n}^{2}(bar{r}) and a function involving the outer scale of the turbulenceL_{0}. It is shown that for a plane wave case, if the distanceLis within (L_{0}^{2}/lambda), then the usual formula given by Tatarski is valid. But if the distance is betweenL_{0}^{2}/lambdaand(bL_{0})/lambdawherebis the total transverse size of the turbulence, the variance of the wave is nearly constant, and ifL gg (bL_{0})/lambda, the variance decays asL^{-2}. Similar conclusions are shown for a spherical wave case. Some examples are shown illustrating the effectiveness of this method.     

Some integrals involving theQ_Mfunction (Corresp.)

Some integrals are presented that can be expressed in terms of theQ_Mfunction, which is defined as begin{equation} Q_M(a,b) = int_b^{infty} dx x(x/a)^{M-1} exp (- frac{x^2 + a^2}{2}) I_{M-1}(ax), end{equation} whereI_{M-1}is the modified Bessel function of orderM-1. Some integrals of theQ_Mfunction are also evaluated.     

Fundamental quantum 1/f noise in semiconductor devices

The quantum 1/f noise theory has been developed in the last two decades and has been applied to 1/f noise suppression in various electronic devices. This theory derives fundamental quantum fluctuations present in the elementary processes of physics at the level of the quantum mechanical cross sections and process rates. This paper demonstrates the basic simplicity of the theory with an elementary physical derivation followed by a short derivation of the conventional quantum 1/f effect in second quantization, for an arbitrary number of particles N defining the scattered current in the final state. A new derivation of the coherent quantum 1/f effect is also included. No adjustable parameters are present in the quantum 1/f theory. Practical applications to semiconductor materials, p-n junctions, SQUID's and quartz resonators are presented. Optimal design principles based on the quantum 1/f theory are described and explained     

F-近似下颗粒的多重散射计算公式的改进

利用多重散射基础理论和修正的Fraunhofer近似及其相位函数,我们对过去F一近似下得到的颗粒的多重散射计算公式进行了改进.与理论值的相对误差的计算表明,我们的公式的准确性较原公式有了较大程度的提高.此外,我们也发现,随着光学厚度增加,多重散射光强的角分布的震荡减弱.     

Electromagnetic theorems for complex anisotropic media

Complex anisotropic media can generally be described by a6 times 6macroscopic constitutive tensorhat{C}. Usinghat{C}properties, a reaction integral formula is derived from which an anisotropic reaction theorem (modified reciprocity theorem) is developed. Reduction of thehat{C}medium into a reciprocal medium is discussed including tensor symmetry attributes and limiting cases. The anisotropic reaction theorem is utilized to derive a zero reaction theorem, and then treated in relation to the moment method. Mutual and self-impedance elements of a network are also derived in terms of reaction integrals, symmetry covered using the anisotropic reaction theorem, and impedance elements related to moment calculations. Use of spectral domain analysis is also covered.     

直线阵列数字波束形成技术

基于直线阵列天线体制,介绍了数字波束形成(Digital Beam Forming,DBF)技术的原理、系统组成和实现方法,给出了详细的公式推导,对数字波束形成的正交校正、通道一致性校正和波束形成等关键技术进行了描述,并说明了关键技术的有效解决方法。结合工程实践,提出用现场可编程门阵列(FPGA)和数字信号处理器(DSP)相结合的架构来完成数字波束形成板的实现方式,并对数字波束形成板的工作模式和工作流程做出了详细说明。所给出的理论和方法都是通过实践检验的,具有一定的实用性。     

On the truncation error of the cardinal sampling expansion

Modern communication theory and practice are heavily dependent on the representation of continuous parameter signals by linear combinations, involving a denumerable set of random variables. Among the best known and most useful is the cardinal seriesf_{n} (t) = sum^{+n}_{-n} f(k) frac{sin pi (t - k)}{ pi ( t - k )}for deterministic functions and wide-sense stationary stochastic processes bandlimited to(-pi, pi). When, as invariably occurs in applications, samplesf(k)are available only over a finite period, the resulting finite approximation is subject to a truncation error. For functions which areL_{1}Fourier transforms supported on[-pi + delta, + pi - delta], uniform trunction error bounds of the formO(n^{-1})are known. We prove that analogousO(n^{-1})bounds remain valid without the guard banddeltaand for Fourier-Stieltjes transforms; we require only a bounded variation condition in the vicinity of the endpoints- piand+ piof the basic interval. Our methods depend on a Dirichlet kernel representation forf_{n}(t)and on properties of functions of bounded variation; this contrasts with earlier approaches involving series or complex variable theory. Other integral kernels (such as the Fejer kernel) yield certain weighted truncated cardinal series whose errors can also be bounded. A mean-square trunction error bound is obtained for bandlimited wide-sense stationary stochastic processes. This error estimate requires a guard band, and leads to a uniformO(n^{-2})bound. The approach again employs the Dirichlet kernel and draws heavily on the arguments applied to deterministic functions.     

Efficient computation for the general solution of a slot loaded by a hemispherical dielectric and/or backing cavity

The problem of a slot loaded by a hemispherical dielectric and/or a hemispherical backing cavity is solved using the method of moments (MoM). A novel recurrence formula is presented for fast calculations of the MoM admittance integrals associated with the spherical homogeneous solutions. The method enables the admittance integrals to be calculated without the need for any numerical integration. Measurements were carried out and good agreement between theory and experiment is obtained.     

First and second passage times of Rayleigh processes (Corresp.)

The first and second passage times of a stationary Rayleigh processR(t,a)are discussed.R(t,a)represents the envelope of a stationary random process consisting of a sinusoidal signal of amplitude and frequencyf_{0}plus stationary Gaussian noise of unit variance having a narrow-band power spectral density which is symmetrical aboutf_{0}. Approximate integral equations are developed whose solutions yield approximate probability densities concerning the first and second passage times ofR(t,a). The resulting probability functions are presented in graphs for the case when the power spectral density of the noise is Gaussian. Related results concerning the approximate distribution function of the absolute minimum or absolute maximum ofR(t,a)in the closed interval[0,tau]are also presented. The exact probability densities are expressed in the form of an infinite series of multiple integrals.     

基于多方法集成的复杂工艺优化

介绍了基于模式识别、神经网络、遗传算法、非线性回归等多种智能技术集成的复杂工艺过程优化系统的设计思想、体系结构、关键技术和实现方法,主要解决多因子、高噪声、非线性、非高斯分布和非均匀的复杂工艺系统难题。采用Agent技术设计系统的体系结构,用偏最小二乘法和Chelyshev多项式建立领域模型,通过演化计算进行最优问题求解,并用正交实验取得模型学习的样本数。实际应用证明,利用这些方法可以在很少的实验情况下,使建立的模型能在较大误差范围内指导生产实践。     

Error probability for incoherent diversity reception

The problem of computing error probabilities for multi-channel communications using an incoherently terminated receiver is analyzed. The signaling alphabet is composed of two equal-energy, equiprobable, correlated waveforms and the multichannel model is presumed to be of the slowly fading "Rician" type, i.e., each subchannel is presumed to be composed of a fixed or specular component and a scatter-like or Rayleigh fading component. The main result of this paper is a generalization (21) of an earlier result derived by Helstrom. Novel by-products of this generalization include, as special cases, results derived by Turin, Pierce, Price, and Lindsey. Also closed-form solutions to very general integrals (22) (heretofore seemingly unknown) involving Bessel function products are presented as part of the main result. These integrals are known to arise in the analysis of multichannel adaptive communication systems. Numerical computations for the error probabilities are given for special values of the signal cross-correlation coefficientlambdaand multi-channel order. These results graphically illustrate that the optimum set of equal-energy binary signals which minimize the error rate for the Rayleigh fading multichannel are orthogonal. Specifically, to maintain the same error probability in two systems, one employing nonorthogonal signals having correlation coefficientlambda, the other employing orthogonal signals, the transmitter power must be increased in the former. In fact, for large SNR's, the graphical data indicate that the required increase in transmitter power is approximately10 log_{10}(1 - lambda^{2})^{-1}dB.     

DESIGN OF 2-D QUASI-OPTICAL BEAM SPLITTERS USING ANALYTICAL GRADIENTS

Quasi-optical systems comprised of diffractive phase elements (DPEs) are designed as beam splitters. The design tasks are reformulated as optimization problems in which target functionals are defined. Analytical gradients are derived to those functionals, which can be passed to gradient methods to efficiently determine the DPEs. Rayleigh-Sommerfeld diffraction integrals are applied to compute the wave propagation in free-space between parallel planes in the quasi-optical systems. Numerical results for several beam splitters are depicted. Furthermore a procedure is proposed, how the phase functions of the DPEs could be smoothed out.     

Performance of anM-ary orthogonal communication system using stationary stochastic signals

This paper considers the performance of a communication system which transmits forTseconds the real part of a sample function of one ofMstationary complex Gaussian processes whose spectral densities are all frequency translations of the functionS_{xi (f). At the receiver white Gaussian noise of one-sided densityN_{0}is added. The center frequencies of the processes are assumed to be sufficiently separated that theMcovariance functions are orthogonal overT. Exponently tight bounds are obtained for the error probability of the maximum likelihood receiver. It is shown that the error probability approaches zero exponentially withTfor all ratesR = (ln M)/Tup toC= int_{-infty}^{infty} [S_{xi (f)/N_{0}] df - int_{- infty}^{infty} ln [1 + S_{xi}(f)/N_{0}] dfwhich is shown to be the channel capacity. Similar results are obtained for the case of stochastic signals with specular components.     

Simple formula for current on a cylindrical receiving antenna

An approximate formula is presented for the induced current on a loaded (or unloaded) thin cylindrical receiving antenna which is illuminated by an incident plane wave from an arbitrary direction. This formula involves only elementary functions and therefore can be calculated easily. Its derivation is based on the multiple reflection approach originated by Weinstein, Shen, and others. Currents calculated by the present formula are found to be in good agreement with experimental data.