Quadratic phase integration using a Chebyshev expansion

An integration algorithm is described which is particularly effective in the numerical treatment of integrands having rapidly varying phase and slowly varying amplitude. The algorithm involves approximating the phase function by a quadratic polynomial and rewriting the integrand without approximation as a slowly varying function multiplied by this quadratic phase exponential. The slowly varying function is then approximated by Chebyshev expansion and the desired integral is thus expressed as a sum of constituent integrals with integrands containing a Chebyshev polynomial multiplied by the quadratic phase factor. These constituent integrals are computed by means of LU decomposition applied to a system of linear equations with a banded coefficients matrix. Example results are presented indicating that a substantial reduction in computation time may be realized by means of this approach.     

Simplified technique for evaluating the radiation integrals

A simplified numerical method is described for computing the radiation integral of a reflector antenna which is associated with either the aperture field method (AFM) or the induced current method (ICM). The method involves linear approximation of the phase function and absorption of the approximation error into amplitude function. Then the integrand is rewritten as a slowly varying function multiplied by the phase exponential. The slowly varying function is expanded in terms of a polynomial for the radial direction and trigonometric functions for the circumferential direction. Expansion coefficients are determined by matching at points on the aperture. The matching points are chosen in a manner similar to that of a two-dimensional numerical integration scheme.     

A new integration algorithm and its application to the analysis of symmetrical Cassegrain microwave antennas

An efficient numerical algorithm for physical optics (PO) analysis of reflectors is presented. Cylindrical symmetry of the reflector is not essential but is assumed for purposes of illustration, and this leads to the use of Fourier representation in the azimuthal variable. The algorithm is based on quadratic approximation of the phase and polynomial approximation of the remaining portion of the integrand. The necessary integrals are generated recursively and they involve the Fresnel integral. Additional efficiency is achieved by means of an interpolation scheme. Therein one interpolates all but the rapidly varying edge diffraction contribution to the integral and then adds the explicitly calculated edge contribution at each far-field point. A few illustrative examples are included.     

Prony and Polynomial Approximations for Evaluation of the Average Probability of Error Over Slow-Fading Channels

A novel and simple semianalytical method for evaluating the average probability of transmission error for digital communication systems that operate over slow-fading channels is presented. The proposed method applies a sum of exponentials fit known as the Prony approximation to the conditional probability of error. Hence, knowledge of the moment-generating function of the instantaneous signal-to-noise ratio (SNR) at the detector input can be used to obtain the average probability of error. Numerical results show that knowledge of the conditional probability of error at only a small number of points and the sum of only two exponentials are sufficient to achieve very high accuracy; the relative approximation error of the exact average probability of error is less than 6% in most of the cases considered. Furthermore, a piecewise polynomial approximation of the conditional probability of error is investigated as an alternative to the sum of exponentials fit. In this case, knowledge of the partial moments of the instantaneous SNR at the detector input can be used to obtain the average probability of error. Numerical results indicate that, to achieve good accuracy, the method based on the polynomial approximation requires that the product of the polynomial degree and the number of approximation subintervals be larger than 10.      

New approximations to J0 and J1 Besselfunctions

New approximate solutions to the 0th- and 1st order Bessel functions of the first kind are derived. The formulations are based upon using a new integral with no previously known solution. The new integral in the limiting case is identical to the 0th-order Bessel function integral. It is solved in closed form, and the solution is expressed as a simple even order polynomial with integer coefficients. The polynomial coefficients are all of integer value. The 1st-order Bessel function approximation can then be found through a simple derivative. Comparisons are made between the exact solution, classic solutions, and the new approximation. The new approximation proves to be much more accurate than the classic small argument approximation. It is also sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing Bessel functions     

Application of the integral equation-asymptotic phase method totwo-dimensional scattering

A hybrid-procedure called the integral equation-asymptotic phase (IE-AP) method is investigated for scattering from perfectly conducting cylinders of arbitrary cross-section shape. The IE-AP approach employs an asymptotic solution to predict the relatively rapid phase dependence of the unknown current distribution, to leave a slowly varying residual function that can be represented by a coarse density of unknowns. In the present investigation, the current density appearing within the combined-field integral equation is replaced by the product of a rapidly varying phase function obtained from the physical optics current and a residual function. The resulting equation is discretized by the method of moments, using subsectional quadratic polynomial basis functions defined on curved cells to represent the residual function. Results show that the required density of unknowns can often be as few as one per wavelength on average without a significant loss of accuracy in the computed current density, even for scatterers with corners     

Phase Local Approximation (PhaseLa) Technique for Phase Unwrap From Noisy Data

The local polynomial approximation (LPA) is a nonparametric regression technique with pointwise estimation in a sliding window. We apply the LPA of the argument of cos and sin in order to estimate the absolute phase from noisy wrapped phase data. Using the intersection of confidence interval (ICI) algorithm, the window size is selected as adaptive pointwise varying. This adaptation gives the phase estimate with the accuracy close to optimal in the mean squared sense. For calculations, we use a Gauss-Newton recursive procedure initiated by the phase estimates obtained for the neighboring points. It enables tracking properties of the algorithm and its ability to go beyond the principal interval (-pi,pi) and to reconstruct the absolute phase from wrapped phase observations even when the magnitude of the phase difference takes quite large values. The algorithm demonstrates a very good accuracy of the phase reconstruction which on many occasion overcomes the accuracy of the state-of-the-art algorithms developed for noisy phase unwrap. The theoretical analysis produced for the accuracy of the pointwise estimates is used for justification of the ICI adaptation algorithm.     

Phase local approximation (PhaseLa) technique for phase unwrap from noisy data.

The local polynomial approximation (LPA) is a nonparametric regression technique with pointwise estimation in a sliding window. We apply the LPA of the argument of cos and sin in order to estimate the absolute phase from noisy wrapped phase data. Using the intersection of confidence interval (HCI) algorithm, the window size is selected as adaptive pointwise varying. This adaptation gives the phase estimate with the accuracy close to optimal in the mean squared sense. For calculations, we use a Gauss-Newton recursive procedure initiated by the phase estimates obtained for the neighboring points. It enables tracking properties of the algorithm and its ability to go beyond the principal interval [-pi, pi] and to reconstruct the absolute phase from wrapped phase observations even when the magnitude of the phase difference takes quite large values. The algorithm demonstrates a very good accuracy of the phase reconstruction which on many occasion overcomes the accuracy of the state-of-the-art algorithms developed for noisy phase unwrap. The theoretical analysis produced for the accuracy of the pointwise estimates is used for justification of the HCI adaptation algorithm.     

A new procedure for the design of high order minimum phase FIR digital or CCD filters

The approximation problem for high-order minimum phase FIR filter is solved without requiring any polynomial factorization. A modified Parks-McClellan program is used to compute the amplitude function; the minimum phase function is then derived by a method using the FFT algorithm. The procedure is illustrated by the design of various high order filters; short computation time with no numerical troubles is achieved.     

Successive order scattering transport approximation for laser lightpropagation in whole blood medium

An analytical solution method of the radiative transport equation, describing light scattering distribution in whole blood, is derived by applying successive order scattering approximation and transport approximation. By separating coherent components of scattered fluxes, the transport equation can be represented in terms of each order scattering flux, and the equations for each order scattering flux have a simplified integration term of scattering contribution that usually makes the solution complicated or even impossible. Also, actual phase function can be used for calculation of angular dependent scattering distribution that is approximated by the sum of the zeroth- and first-order Legendre polynomial in diffusion theory, or the sum of isotropic and coherent components in transport approximation. The method is then used to calculate reflectance from a half-space blood medium. It is found that first-order scattering flux alone produces a good agreement with experimental data and higher-order scattering fluxes are negligible in whole blood     

A Low-Pass Prototype Network for Microwave Linear Phase Filters

A new approximation theory is presented for a low-pass prototype filter which simultaneously optimizes both the passband amplitude and phase response of the scattering transfer coefficient over the same finite band. This closed form solution is expressed in terms of single polynomial, which is readily generated through a simple recurrence formula, and has been termed the equidistant linear phase polynomial since its phase deviation from linearity vanishes at equidistant points along the real frequency axis. A synthesis procedure is presented for the realization of this transfer function using a resistively terminated, symmetrical, lossless, two-port network where extensive use is made of the immittance inverter concept. The even-mode admittance, which defines the network, possesses a simple closed form representation in terms of the equidistant linear phase polynomial and its derivative, and consequently, the entire theory is derived in an analytic form. Typical performance characteristics are graphically presented for networks of up to 14th degree, illustrating the superiority of this new approach over any other known form of approximation theory for selective linear phase filters.     

Adaptive region growing technique using polynomial functions for image approximation

This paper presents an approximation algorithm for two-dimensional signals (e.g., images) using polynomial functions. The proposed algorithm is based on an adaptive segmentation of the original signal into adjacent regions and on the approximation of the signal in each region by a two-dimensional polynomial function. The segmentation is obtained by an adaptive region growing technique which allows perfect adaptation between the chosen approximation and the inner structure of the signal. Results of this technique are presented in the context of image coding applications.     

Estimation of slowly changing components of physiological signals

A method for the estimation of slowly changing components of physiological signals is presented in this communication. The method is based on a sequential approximation of slowly changing components by a low-order polynomial function. The polynomial coefficients are obtained by minimizing the distance between the expected zero crossing density (ZCD) value of the fast components of the physiological signal and the estimated ZCD value of these components. The method has been tested and preliminary results were satisfactory     

A fast time-domain algorithm for the simulation of switching powerconverters

An efficient algorithm for the simulation of switched-mode power converters is developed. A Chebyshev series expansion is used to effectively solve the differential equations describing the system in each topology. The power of the new simulation technique lies both in the simple, but accurate, polynomial approximation for the state transition matrices and in the ability to explicitly obtain the instants at which the switching of the circuit topology takes place. The simulation technique is illustrated with reference to a simple Buck converter operating at a constant frequency. The derivation of the new algorithm is presented and its performance is analyzed. The case of a rapidly varying input forcing function is analyzed. Examples illustrating the generality and the computational efficiency of the algorithm are presented     

Error probability of 2DPSK with phase noise

A simple tight upper bound on the BEP of 2DPSK over the AWGN channel with phase noise in the received signal is obtained. The phase is modeled as a Gaussian random process which is slowly varying compared to the bit rate so that a piecewise-constant approximation can be made. The bound is verified by computer simulations, and it provides good estimates of the error probability. It shows that for high SNR the error probability decreases as the reciprocal of the square-root of the SNR. The results are applicable in particular to heterodyne optical communications     

一种新的机载SAR回波高阶多项式相位误差提取方法

当载机在SAR回波方位子孔径时间内运动较复杂时,二次相位误差模型不能准确描述载机运动造成的相位误差。针对此情况,该文借鉴PACE算法的思想,提出了一种提取SAR回波中时域高阶多项式相位误差的TPACE算法。TPACE算法将图像对比度函数作为目标函数,以时域高阶多项式相位误差模型系数作为自变量,通过最优化方法提取时域误差系数。文中详细推导了对比度函数关于误差模型系数的梯度表达式,分析了TPACE与以往提取时域高阶多项式相位误差的算法计算量之差别。实际超宽带SAR回波数据处理结果表明,TPACE能有效提取时域高阶多项式误差,是一种计算量相对较小的SAR自聚焦算法。     

Polynomial-Approximation-Based Control for Nonlinear Systems

This paper is concerned with the stabilization problem for nonlinear systems. A new polynomial-approximation-based approach for modeling nonlinear systems is first proposed. The nonlinearity is approximated by polynomials, and the approximation errors are treated as modeling uncertainties. The original nonlinear systems are converted into polynomial systems with modeling uncertainties. In order to highlight the approximation accuracy, the piecewise polynomial approximation functions are utilized. A novel polynomial state-feedback controller is designed to solve the stabilization problem. Furthermore, switched polynomial state-feedback controllers are designed to improve the performance. The stabilization conditions are presented in terms of sum of squares, which can be numerically solved via SOSTOOLS. Finally, simulation examples are provided to demonstrate the feasibility of the proposed method and show its advantage over the polynomial-fuzzy-model-based approach.     

AM/FM signal estimation with micro-segmentation and polynomial fit

Amplitude and phase estimation of AM/FM signals with parametric polynomial representation require the polynomial orders for phase and amplitude to be known. But in reality, they are not known and have to be estimated. A well-known method for estimation is the higher-order ambiguity function (HAF) or its variants. But the HAF method has several reported drawbacks such as error propagation and slowly varying or even constant amplitude assumption. Especially for the long duration time-varying signals like AM/FM signals, which require high orders for the phase and amplitude, computational load is very heavy due to nonlinear optimization involving many variables. This paper utilizes a micro-segmentation approach where the length of segment is selected such that the amplitude and instantaneous frequency (IF) is constant over the segment. With this selection first, the amplitude and phase estimates for each micro-segment are obtained optimally in the LS sense, and then, these estimates are concatenated to obtain the overall amplitude and phase estimates. The initial estimates are not optimal but sufficiently close to the optimal solution for subsequent processing. Therefore, by using the initial estimates, the overall polynomial orders for the amplitude and phase are estimated. Using estimated orders, the initial amplitude and phase functions are fitted to the polynomials to obtain the final signal. The method does not use any multivariable nonlinear optimization and is efficient in the sense that the MSE performance is close enough to the Cramer–Rao bound. Simulation examples are presented.     

基于CORDIC的精确快速幅相解算方法

针对传统CORDIC算法进行高精度幅度相位解算时迭代次数过多、时延较长、相位收敛较慢等局限,提出了一种基于最佳一致逼近方法的幅度与相位补偿算法,即利用传统CORDIC算法迭代一定次数后得到的向量信息,采用最佳一致逼近方法对幅度和相位分区间进行一阶多项式补偿,有效提高了计算精度.仿真及实测结果表明,对传统CORDIC算法4次迭代后的结果进行补偿,幅度相对误差可达到10^-5量级、相位绝对误差可达到10^-5度量级,最大输出时延不大于100ns.在使用部分专用乘法器的条件下,寄存器消耗降低了42.5%,查找表消耗降低了15.5%.采用该补偿算法,每多一次CORDIC迭代其相位精度可提高约一个数量级.因此,本文提出的补偿CORDIC算法在迭代次数、计算精度等方面优于传统CORDIC算法,适合于高精度计算的场合.     

On the numerical analysis of the subreflector of an offset Cassegrain microwave antenna

The physical optics analysis of the scattering from the subreflector of an offset Cassegrain microwave antenna is formulated in a novel way using the geometrical properties of the conic sections. The resulting integral is computed by means of an algorithm involving quadratic approximation of the phase and polynomial approximation of the remainder of the integrand. The formulation allows for displacement of the phase center of the feed pattern representation. Examples are presented which provide a comparison with previously reported results obtained using the geometrical theory of diffraction rather than physical optics.