One- and two-dimensional minimum and nonminimum phase retrieval bysolving linear systems of equations

The discrete phase retrieval problem is to reconstruct a discrete time signal whose support is known and compact from the magnitude of its discrete Fourier transform. We formulate the problem as a linear system of equations; our methods do not require polynomial rooting, tracking zero curves of algebraic functions, or any sort of iteration like previous methods. Our solutions obviate the stagnation problems associated with iterative algorithms, and our solutions are computationally simpler and more stable than alternative noniterative algorithms. Furthermore, our methods can explicitly accommodate noisy Fourier magnitude information through the use of total least squares type techniques. We assume either of the following two types of a priori knowledge of the signal: (1) a band of known values (which may be zeros) or (2) some known values of a subminimum phase signal (whose zeros lie inside a disk of radius greater than unity). We illustrate our methods with nonminimum-phase one-dimensional (1-D) and two-dimensional (2-D) signals     

Turbo equalization for GMSK signaling over multipath channels basedon the Gibbs sampler

We consider the problem of Bayesian data restoration for Gaussian minimum shift keying (GMSK) signals over unknown multipath channels. As an alternative to the linear approximation method employed in the conventional finite impulse response (FIR) model, we develop a nonlinear signal model for this system. A Bayesian equalizer based on the Gibbs sampler, a Markov chain Monte Carlo (MCMC) procedure, is developed for estimating the a posteriori symbol probability in the GMSK system without explicit channel estimation. The basic idea of this technique is to generate ergodic random samples from the joint posterior distribution of all unknowns, and then to average the appropriate samples to obtain the estimates of the unknown quantities. Being soft-input soft-output in nature, the proposed Bayesian equalization technique is well suited for iterative processing in a coded system, which allows the Bayesian equalizer to successively refine its processing based on the information from the decoding stage, and vice versa     

Fast image restoration without boundary artifacts.

Fast Fourier transform (FFT)-based restorations are fast, but at the expense of assuming that the blurring and deblurring are based on circular convolution. Unfortunately, when the opposite sides of the image do not match up well in intensity, this assumption can create significant artifacts across the image. If the pixels outside the measured image window are modeled as unknown values in the restored image, boundary artifacts are avoided. However, this approach destroys the structure that makes the use of the FFT directly applicable, since the unknown image is no longer the same size as the measured image. Thus, the restoration methods available for this problem no longer have the computational efficiency of the FFT. We propose a new restoration method for the unknown boundary approach that can be implemented in a fast and flexible manner. We decompose the restoration into a sum of two independent restorations. One restoration yields an image that comes directly from a modified FFT-based approach. The other restoration involves a set of unknowns whose number equals that of the unknown boundary values. By summing the two, the artifacts are canceled. Because the second restoration has a significantly reduced set of unknowns, it can be calculated very efficiently even though no circular convolution structure exists.     

Input Design for Model Discrimination: Application to Respiratory Control During Exercise

The existence of a set of alternative models, all competing to characterize respiratoxy controller structure, leads naturally to a model discrimination problem. In these models the structure and0 parameter values are fixed. This paper considers the problem of input design for enhancing model discrimination. Using a class of models given by linear differential equations with time delays, an input consisting of a sum of sinusoidal components under a power constraint is considered. A procedure for designating the power spectrum of the signal is developed. Furthermore, a procedure for obtaining a binary signal with a similar spectrum is illustrated. These concepts are applied to input design for discrimination among models that characterize respiratory control during exercise.     

Phase noise characterization of SAW oscillators based on a Newtonminimization procedure

An iterative minimization technique is used to optimize the values of circuit and device parameters which determine the phase noise response of a voltage-controlled SAW-stabilized oscillator (VCSO). An expression-developed by T.E. Parker (1985) is used to calculate the double-sideband phase noise to carrier ratio from circuit parameter values; good agreement between calculations and phase noise measurements is achieved by minimizing the squared error through the use of a steepest-descent/Newton-Raphson minimization scheme. Less accurately known circuit parameters are thus optimized in an iterative fashion. Exact expressions for the elements of the Hessian Matrix are used in the Newton-Raphson procedure, allowing for fast computations. The numerical findings suggest that useful results can be obtained in the determination of VCSO circuit parameters     

Phaseless bi-polar planar near-field measurements and diagnosticsof array antennas

Antenna near-field measurements typically require very accurate measurement of the near-field phase. There are applications where an accurate phase measurement may not be practically achievable. Phaseless measurements are beginning to emerge as an alternative microwave antenna measurements technique when phase cannot be directly measured. There are many important aspects for successful implementation of a phaseless measurement algorithm. This paper presents appropriate phaseless measurement requirements and a phase retrieval algorithm tailored for the bi-polar planar near-field antenna measurement technique. Two amplitude measurements and a squared amplitude optimal sampling interpolation method are integrated with an iterative Fourier procedure to first retrieve the phase information and then construct both the far-field pattern and diagnostic characteristics of the antenna under test. In order to critically examine the methodologies developed in this paper, phaseless measurement results for two different array antennas are presented and compared to results obtained when the near-field amplitude and phase are directly measured     

Signal reconstruction from two close fractional Fourier power spectra

Based on the definition of the instantaneous frequency (signal phase derivative) as a local moment of the Wigner distribution, we derive the relationship between the instantaneous frequency and the derivative of the squared modulus of the fractional Fourier transform (fractional Fourier transform power spectrum) with respect to the angle parameter. We show that the angular derivative of the fractional power spectrum can be found from the knowledge of two close fractional power spectra. It permits us to find the instantaneous frequency and to solve the phase retrieval problem up to a constant phase term, if only two close fractional power spectra are known. The proposed technique is noniterative and noninterferometric. The efficiency of the method is demonstrated on several examples including monocomponent, multicomponent, and noisy signals. It is shown that the proposed method works well for signal-to-noise ratios (SNRs) higher than about 3 dB. The appropriate angular difference of the fractional power spectra used for phase retrieval depends on the complexity of the signal and can usually reach several degrees. Other applications of the angular derivative of the fractional power spectra for signal analysis are discussed. The proposed technique can be applied for phase retrieval in optics, where only the fractional power spectra associated with intensity distributions can be easily measured.     

Maximum likelihood estimation, analysis, and applications ofexponential polynomial signals

We model complex signals by approximating the phase and the logarithm of the time-varying amplitude of the signal as a finite order polynomial. We refer to a signal that has this form as an exponential polynomial signal (EPS). We derive an iterative maximum-likelihood (ML) estimation algorithm to estimate the unknown parameters of the EPS model. The initialization of the ML algorithm can be performed by using the result of a related paper. A statistical analysis of the ML algorithm is performed using a finite-order Taylor expansion of the mean squared error (MSE) of the estimate about the variance of the additive noise. This perturbation analysis gives a method of predicting the MSE of the estimate for any choice of the signal parameters. The MSE from the perturbation analysis is compared with the MSE from a Monte Carlo simulation and the Cramer-Rao Bound (CRB). The CRB for this model is also derived     

The Effect of Carrier Phase and Timing on a Single-Sideband Data Signal

Considering the recovery of carrier phase and timing for optimum detection of a single-sideband data signal, a relation is derived for simultaneous optimum setting of these parameters. To find the optimum settings, it is shown that the mean-square error (MSE) of the demodulated signal, defined as the sum of the squared deviations of the received signal from the nearest possible level of an undistorted signal, is a useful measure of carrier phase and timing errors only near the optimum settings, so that a rough estimate of both carrier phase and timing must be available at the receiver.     

Simulation and Design of Microwave Class-C Amplifiers through Harmonic Analysis

A method for analyzing microwave class-C amplifiers is proposed which satisfies the requirements of a wide application field, and, at the same time, operates with a fast runing time and without convergence problems. It is based on the partitioning of the circuit into linear and nonlinear subnetworks for which, respectively, frequency-domain and time-domain equations are written. Then, taking into account that the time-domain and frequency-domain representations are related by the Fourier series, the circuit behavior is described by means of a system of nonlinear equations whose unknowns are the harmonic components of the incident waves at all the connections. To overcome the numerical problems arising in the search for the solution of this system when strong nonlinearities are involved, a special step-by-step procedure is adopted. The problem is transformed into the search for the solution of a sequence of well-conditioned systems of equations corresponding to a sequence of well-chosen circuits obtained from the original one through progressive changes of the input signal starting from 0 up to the nominal value. The program which implements the method is also described and the results of the analysis relative to a class-C amplifier are compared with measured values.     

Blind deconvolution of symmetric noncausal impulse responses usingtwo-sided linear prediction

Considers the problem of estimating the time-symmetric, noncausal impulse response of a linear time-invariant system from measurements of the response of the system to an unknown input signal, which is assumed to be a realization of a white random process. The symmetric impulse response is modeled by a two-sided AR or ARMA system model. The two-sided AR coefficients are estimated using a two-step procedure. First, an estimate of an unconstrained parameter vector is computed by solving a close-to-Toeplitz-plus-Hankel system of equations using previously developed fast algorithms. Then, the polynomial square root of the result is obtained by solving a constrained least-squares problem which has a simple solution. Unlike previous methods, this approach requires no iterative procedure. However, it may lead to an unstable model in some extreme cases. Simulation results illustrate the performance of the proposed methods     

Least-square-exponential approximation

A method is presented for solving the problem of fitting decay-type experimental data by sums of exponentials. Both the exponents and the coefficients of the exponentials are considered as unknowns. The novel idea presented here is to consider the sum of exponentials as a solution of an integral equation. A step-by-step procedure is given for the solution of the problem in the least-squares sense. An advantage of the proposed method is that the data need not be equidistant.     

The rate-distortion function for the quadratic Gaussian CEO problem

A new multiterminal source coding problem called the CEO problem was presented and investigated by Berger, Zhang, and Viswanathan. Recently, Viswanathan and Berger have investigated an extension of the CEO problem to Gaussian sources and call it the quadratic Gaussian CEO problem. They considered this problem from a statistical viewpoint, deriving some interesting results. In this paper, we consider the quadratic Gaussian CEO problem from a standpoint of multiterminal rate-distortion theory. We regard the CEO problem as a certain multiterminal remote source coding problem with infinitely many separate encoders whose observations are conditionally independent if the remote source is given. This viewpoint leads us to a complete solution to the problem. We determine the tradeoff between the total amount of rate and squared distortion, deriving an explicit formula of the rate-distortion function. The derived function has the form of a sum of two nonnegative functions. One is a classical rate-distortion function for single Gaussian source and the other is a new rate-distortion function which dominates the performance of the system for a relatively small distortion. It follows immediately from our result that the conjecture of Viswanathan and Berger on the asymptotic behavior of minimum squared distortion for large rates is true     

利用Duffing方程组检测随机相位的正弦信号

讨论用16个能产生混沌现象的方程来检测带有未知相位的微弱正弦信号。16个相位已经确定的方程的阈值事先求解出来。当未知相位的信号通过系统时,逐个验证方程,看是否有混沌向大尺度周期运动转变的方程,如果有,则此方程可以作为检测信号的方程,若没有则继续实验,最后必然会出现一个甚至多个相轨迹转变的方程。用能够出现相轨迹转变的方程...     

正负交替反转法解析电子衍射相位问题

与X射线晶体学中存在的相位问题类似,在电子衍射中也存在相位问题:在电子衍射实验中只能收集到衍射强度而丢失了相位.最近,衍射重构成像方法(diffractive imaging),即直接从衍射重构出晶体结构的方法,从理论和实验都有了重大进展.从理论上,人们提出和发展许多有效的相位解析方法.从实验上,高强度的X射线源,场发射电子枪以及高灵敏度的记录媒介的发展都对此有贡献.直接从衍射重构出晶体结构有许多的优点:首先在重构像中,物镜球差的影响很小.这是由于物镜传递函数对衍射强度的影响远远小于对相位的影响;其次,从同一晶体收集的电子衍射有更多的高阶衍射斑,使得衍射重构能得到较高的分辨率(小于0.1 nm);同时,在同样辐射条件下晶体的电子衍射比其高分辨像具有更高的信噪比.这对于用电镜解析对辐射损伤敏感的有机物和生物蛋白晶体是有用的.本文叙述了一个解决电子衍射相位的新方法.在本文的程序中,同时使用了Oszlányi和Süto提出的正负交替反转法(charge-flipping algorithm)和Fienup的重构方法(hybrid input-output algorithm).作者用模拟数据来验证该方法的有效性.在程序中输入计算的运动学电子衍射强度,模拟晶体的二维静电势场分布能被重构出来.使用归一化结构因子可以提高正空间的重构像衬度;这对解决相位问题是有利的.使用Fienup的重构方法可以有效地解决由局域最小值而引起程序停滞问题.在正负交替反转法中通常会有停滞问题而不能找到全局最小值.正负交替反转法会逐步地在正空间中产生较大的零值电势区域,从而减小了正空间中未知数的数目.当未知数数目小于或等于从傅立叶变换建立起来的等式数目时,晶体的相位就可以解决了.     

A SPECTRAL ESTIMATION ALGORITHM USING THE HOUSEHOLDER TRANSFORM

Householder transform is used to triangularize the data matrix, which is basedon the near prediction error equation. It is proved that the sum of squared residuals for eachAR order can be obtained by the main diagonal elements of upper triangular matrix, so thecolumn by column procedure can be used to develop a recursive algorithm for AR modeling andspectral estimation. In most cases, the present algorithm yields the same results as the covariancemethod or modified covariance method does. But in some special cases where the numerical ill-conditioned problems are so serious that the covariance method and modified covariance methodfail to estimate AR spectrum, the presented algorithm still tends to keep good performance. Thetypical computational results are presented finally.     

用Householder变换的递推AR模型化与谱估计算法

Housecholder变换用于上三角化是基于线性预测误差方程的数据阵。可以证明,由上三角阵的主对角元素便可得到各阶AR模型的残差平方和。因此用逐列处理的方法可以构成 AR 模型化与谱估计的递推算法。在大多数情况下,本文的算法不仅给出与协方差算法或修正协方差算法相同的计算结果;而且当计算中存在严重的数值病态问题时,协方差法和修正协方差法无法获得好的AR谱估计,而本文的算法则仍然可以获得好的估计。文中给出了典型的计算例子。     

Blind turbo equalization in Gaussian and impulsive noise

We consider the problem of simultaneous parameter estimation and restoration of finite-alphabet symbols that are blurred by an unknown linear intersymbol interference (ISI) channel and contaminated by additive Gaussian or non-Gaussian white noise with unknown parameters. Non-Gaussian noise is found in many wireless channels due to the impulsive phenomena of radio-frequency interference. Bayesian inference of all unknown quantities is made from the blurred and noisy observations. The Gibbs sampler, a Markov chain Monte Carlo procedure, is employed to calculate the Bayesian estimates. The basic idea is to generate ergodic random samples from the joint posterior distribution of all unknowns and then to average the appropriate samples to obtain the estimates of the unknown quantities. Blind Bayesian equalizers based on the Gibbs sampler are derived for both Gaussian ISI channel and impulsive ISI channel. A salient feature of the proposed blind Bayesian equalizers is that they can incorporate the a priori symbol probabilities, and they produce as output the a posteriori symbol probabilities. (That is, they are “soft-input soft-output” algorithms.) Hence, these methods are well suited for iterative processing in a coded system, which allows the blind Bayesian equalizer to refine its processing based on the information from the decoding stage and vice versa-a receiver structure termed as blind turbo equalizer     

An efficient block adaptive decision feedback equalizer implemented in the frequency domain

In this paper, a new block adaptive decision feedback equalizer (DFE) implemented in the frequency domain is derived. The new algorithm is suitable for applications requiring long adaptive equalizers, as is the case in several high-speed wireless communication systems. The inherent "causality" problem appearing in the block adaptive formulation of the DFE equations is overcome by using tentative decisions in place of the unknown ones within each block. These tentative decisions are subsequently improved by using an efficient iterative procedure, which finally converges to the optimum decisions in a few iterations. This procedure is properly initialized by applying a minimization criterion that utilizes all the available information. The whole algorithm, including the iterative procedure, is implemented in the frequency domain and exhibits a considerable reduction in computational complexity, as compared with the conventional DFE, offering, at the same time, a noticeable increase in convergence speed. Additionally, the level of the steady-state MSE, which is achieved by the new algorithm, is practically insensitive to the block length.