Almost Equiripple Low-Pass FIR Filters

An approximation of the linear phase almost equiripple low-pass finite impulse response filter is introduced. The frequency response of an almost equiripple low-pass finite impulse response filter closely approaches the frequency response of an optimal equiripple low-pass finite impulse response filter in the Chebyshev sense. The presented approximation is based on the generating polynomial. Despite that the generating polynomial has no iso-extremal behavior, it is related to the class of iso-extremal polynomials. The zero phase transfer function of an almost equiripple low-pass finite impulse response filter follows from the generating polynomial. The closed form solution for the direct algebraic computation of the impulse response of the filter has been developed on the basis of generalization of the differential equation suitable for the half-band specifications. No numerical procedures are involved. The practical design procedure based on the developed approximation is presented. For illustration of the design procedure one example of the design is included here.     

Error Rate Bounds for Digital PAM Transmission with Phase Jitter and Intersymbol or Cochannel Interference

A simple method is presented for computing upper and lower bounds for binary PAM signaling over channels with phase jitter. Other impairments to reliable communication are taken to be additive thermal noise and intersymbol or cochannel interference. The important cases of binary and quadrature phase shift keying are cases for which the analysis presented is valid. The method consists of finding upper and lower bounds conditioned on the phase jitter, φ, in a form in which the auguments of theQ-functions involved are linear incos phiandsin phi. When these functions are represented by a piecewise linear approximation, the average of these bounds can be given in closed form. In most cases 5 piecewise linear segments provides an error rate estimate that is accurate to at least 1 dB in system signal-to-noise ratio (SNR).     

Optimal Filtering for Polynomial Systems over Switching Delayed Observations

The problem of designing an optimal filter for a class of nonlinear systems confused with white Gaussian noise over linear time delayed observations is presented in this paper. Output observations may or may not experience sensor delays due to a random variable which is modeled as a Bernoulli distributed one that takes the quantities of zero and one. A closed form of this filter is obtained by expressing the conditional expectation of polynomial terms as a function of its expectation and covariance matrix. As a particular case, finite-dimensional filtering equations are obtained for a third degree polynomial function. Numerical simulations are performed on an induction motor in order to show the effectiveness of proposed filter.     

A New Leaky Waveguide for Millimeter Waves Using Nonradiative Dielectric (NRD) Waveguide-Part I: Accurate Theory

An almost-rigorous analysis is presented for a new leaky waveguide of simple configuration based on a recent nonradiative modification of H guide suitable for millimeter wavelengths, The analysis employs a transverse equivalent network that yields a dispersion relation in closed form, and contains some interesting and subtle features. Numerical values are presented for the phase and leakage constants in terms of the different geometric parameters and the dielectric constant comparisons with measurements are made in the companion paper, part II.     

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     

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.     

On the linear phase approximation

A simple technique for linear phase approximation is illustrated. The approximation polynomial used is of the form Pn(s) = M(s) + kN(s), where M(s) and N(s) are even and odd polynomials with interlaced, and equally spaced, imaginary zeros. The influence of constant k is investigated, and it is shown that a proper selection of that constant leads to good approximation.     

An interpolation technique in the design of pulse-forming networks

This paper presents a new method of synthesis of low-pass filters for pulse applications. A piecewise linear model is first introduced to define the phase characteristic of the idealized minimum phase network and then approximated by interpolation technique at equidistant points. The transient response to a unit step and the attenuation characteristic are controlled by two variable parameters which are determined by iterative procedure. The necessary computations are simple, since only linear equations are involved in computational process. The transient responses of the resulting filters are superior to other known systems and compare favourably even with the Schüssler filters with equi-ripple step response.     

BER performance of BPSK transmissions over multipath channels

A new closed form expression for the bit error rate (BER) performance of binary phase shift keying (BPSK) transmissions over frequency selective channels is presented. The expression is obtained through a novel approximation of the Gaussian Q(middot) function by a fixed series of sinusoids with exponentially decreasing amplitudes. Numerical results demonstrate the accuracy of the derived expression     

Complex polynomial phase integration

A previously published integration algorithm applicable to the numerical computation of integrals with rapidly oscillating integrands is generalized. The previous algorithm involved quadratic approximation of the phase function which was assumed to be real. The present generalization concerns approximation of a complex phase function by a polynomial of arbitrary degree. As before, the integrand is then written without approximation as a slowly varying function multiplied by the polynomial phase exponential and the slowly varying factor is approximated by a finite sum of Chebyshev polynomials. The integral is thus expressed as a sum of constituent integrals which are computed recursively via LU decomposition applied to a system of linear equations with a banded coefficients matrix. Examples are presented comparing various degree phase approximants.     

Generalized Interdigital Linear Phase Filter

The design theory is presented for narrow-band generalized interdigital linear phase filters which consist of a pair of identical cross-coupled interdigital lines. The procedure for the determination of the characteristic admittances of the elements which describe the structure based upon the element values of the low-pass linear phase prototype network is given, from which the physical dimensions of the filter may readily be obtained. The measured performance characteristics of two filters are presented. The first is a 2.5 percent bandwidth 14-element filter based upon the maximally flat prototype operating in L-band, and the second is a 1 percent bandwidth 18-element version based upon the finite band prototype in S-band. Both filters are shown to be in excellent agreement with theory, with the latter exhibiting transfer characteristics considerably superior to those obtainable from any form of conventionally equalized filter of similar overall degree.     

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.     

A NEW LEAKY WAVE ANTENNA BASED ON CHANNEL GUIDE FILLED WITH LEFT-HAND MATERIAL FOR MILLIMETER-WAVE APPLICATIONS

A new leaky wave antenna based on channel guide filled with left-handed material (LHM), operating in evanescent surface mode of TM polarization, is presented for millimeter wave applications. A transverse equivalent network that yields a dispersion relation in closed form is employed to analyze the leakage characteristics of the antenna. Numerical results are presented for the phase and leakage constants of the antenna in terms of different geometric parameters and dielectric distributions. Besides, the frequency dependences of the leaky characteristics for the present antenna are given, from which the backward radiation is clearly demonstrated.     

滤波器特性的非等波纹逼近方法

本文提出滤波器特性的非等波纹逼近方法。利用切比雪夫多项式构成有理函数，并与频率相关的系数组成特征函数，从而获得接近于同阶数椭圆函数滤波器的衰减特性，明显优于切比雪夫和逆切雪夫滤波器。理论明晰，过程简单，便于计算机编程。非等波纹逼近方法是最平坦逼近和等波纹逼近的一种中间形式，有其实用价值。     

Splines for electromagnetics

The theory of splines for approximation is developed. The theory is presented in a general form applicable to other than just piecewise polynomial approximation. The relationship of spline approximation and finite elements is shown with an example given to demonstrate the procedure as applied to the solution of Poisson's equation. The use of spline functions in the method of moments is also discussed. The results of this use is then presented for the solution of a center-fed wire antenna.     

Optimal Filtering for Incompletely Measured Polynomial Systems with Multiplicative Noise

In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations with an arbitrary, not necessarily invertible, observation matrix is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. Thus, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. A transformation of the observation equation is introduced to reduce the original problem to the previously solved one with an invertible observation matrix. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of linear and bilinear state equations. In an example, the performance of the designed optimal filter is verified against those of the optimal filter for a quadratic state with a state-independent noise and a conventional extended Kalman–Bucy filter. The authors thank the Mexican National Science and Technology Council (CONACyT) for financial support under Grants No. 55584 and 52953.     

Digital Filter Design Techniques in the Frequency Domain

The frequency domain of wide-band linear sampleddata filters is considered. The sampled-data filter is termed ?wideband? when the frequency range of useful approximation to its continuous counterpart approaches half the sampling frequency. Sampled-data filter representations for continuous filters can be obtained using several different design procedures.[1] A particular design method utilizing the bilinear transformation is developed. The method is especially useful in designing wide-band sampleddata filters which exhibit relatively flat frequency-magnitude characteristics in successive pass- and stop-bands. Filters of this type are widely used in network simulation and data processing problems.[2] The design method possesses two chief advantages over the standard z transform.[3] The first is that the transformation used is purely algebraic in form. This means it can be applied easily to a continuous filter having a rational transfer characteristic expressed in either polynomial or factored form. The second advantage is the elimination of aliasing[4] errors inherent in the standard z transform. Thus, the sampled-data filter obtained by this design method exhibits the same frequency response characteristics as the continuous filter, except for a nonlinear warping of the frequency scale. Compenation for this warping can be made by a suitable frequency scale modification. Some of the more common filter networks to which the design method can be applied effectively are the Butterworth, Bessel, Chebyshev, and elliptic-filter structures. The design method consists first of obtaining a rational transfer characteristic for a continuous filter that satisfies the design specifications.     

Chebyshev phase approximation

A method to determine a network transfer function with minimum phase shift, the phase of which approximates a given phase in the Chebyshev sense, is described. The approximation method is exclusively based on the solution of the linear equation system.     

一种新型径向基函数神经网络的非线性系统逼近

讨论了一种新的、正弦型径向基函数(SRBF)神经网络，并用来逼近n堆连续函数。该SRBF所采用的n堆正弦型的基函数是光滑的,并且是致密的。该SRBF网络的权因子是输入的低阶多项式函数。本文给出的一种简单计算程序，显著地降低了网络训练和计算时间。并且由于SRBF的基函数可以非均匀的量化格点为中心。因而降低了网络所需存储的样本数，网络的输出及其一阶导数都是连续的。对于非线性系统。该SRBF网络在系统定义城内的逼近是精确的。并且在存储参数的个数上是最优的。通过实例仿真，证明该方法步骤简单,训练速度快，精度也很理想。     

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.