Fluctuation analysis of a two-layer backpropagation algorithm usedfor modeling nonlinear memoryless channels

Neural networks have been used to model the nonlinear characteristics of memoryless nonlinear channels using the backpropagation learning (BP) with experimental training data. The mean transient and convergence behavior of a simplified two-layer neural network has been studied previously in order to better understand this neural network application. The network was trained with zero mean Gaussian data. This paper extends these results to include the effects of the weight fluctuations on the mean square error (MSE). A new methodology is presented that can be extended to other nonlinear learning problems. The new mathematical model is able to predict the MSE learning behavior as a function of the algorithm step size μ. The performance analysis is based on the derivation of linear recursions for the variance and covariance of the weights that depend nonlinearly on the mean weights. These linear recursions can be used to predict the local mean-square stability of the weights. As in linear gradient search problems (LMS, etc.), it is shown that there exists an optimum p (minimizing the MSE), which is the result of the tradeoff between fast learning and small weight fluctuations. Monte Carlo simulations display excellent agreement between the actual behavior and the predictions of the theoretical model over a wide range of μ values     

Statistical analysis of neural network modeling and identificationof nonlinear systems with memory

The paper presents a statistical analysis of neural network modeling and identification of nonlinear systems with memory. The nonlinear system model is comprised of a discrete-time linear filter H followed by a zero-memory nonlinear function g(.). The system is corrupted by input and output independent Gaussian noise. The neural network is used to identify and model the unknown linear filter H and the unknown nonlinearity g(.). The network architecture is composed of a linear adaptive filter and a two-layer nonlinear neural network (with an arbitrary number of neurons). The network is trained using the backpropagation algorithm. The paper studies the MSE surface and the stationary points of the adaptive system. Recursions are derived for the mean transient behavior of the adaptive filter coefficients and neural network weights for slow learning. It is shown that the Wiener solution for the adaptive filter is a scaled version of the unknown filter H. Computer simulations show good agreement between theory and Monte Carlo estimations     

A note on the Arimoto- Blahut algorithm for computing the capacity of discrete memoryless channels (Corresp.)

In this work a termination condition for the Arimoto-Blahut algorithm for computing the capacity of a discrete memoryless channel will be discussed which turns out to decrease the number of iterations needed in many cases.     

Statistical analysis of two nonlinear least-squares estimators of sine-wave parameters in the colored-noise case

This paper establishes the large-sample accuracy properties of two nonlinear least-squares estimators (NLSE) of sine-wave parameters: the basic NLSE, which ignores the possible correlation of the noise, and the optimal NLSE, which, besides the sine-wave parameters, also estimates the noise correlation (appropriately parametrized). It is shown that these two NLS estimators have thesame accuracy in large samples. This result provides complete justification for preferring the computationally less expensive basic NLSE over the optimal NLSE. Both estimators are shown to achieve the Cramér-Rao Bound (CRB) as the sample size increases. A simple explicit expression for the CRB matrix is provided, which should be useful in studying the performance of sine-wave parameter estimators designed to work in the colored-noise case.The work of A. Nehorai was supported by the Air Force Office of Scientific Research, under Grant No. AFOSR-88-0080.     

Analysis of the dynamics of a memoryless nonlinear gradient IIR adaptive notch filter

A nonlinear dynamical model of a memoryless nonlinear gradient IIR adaptive notch filter for estimating the frequency of a noisy sinusoid is derived. The model is verified through simulations, where simulated responses of the estimated frequency are compared with the responses obtained from the model with good agreement. Convergence properties of the filter are studied using the model, and maximum step sizes and initial frequency ranges for convergence are determined. The performance of the adaptive filter in tracking a time-varying signal frequency is also examined.     

Statistical modeling of W-CDMA signals for use over frequency-selective multipath channels

This letter presents a mathematical model for the statistics of wideband code-division multiple-access signals which experience multipath fading. The statistics are obtained from numerically generated signals, and are then modeled with an offset noncentral chi distribution with three degrees of freedom. The mathematical model is tested against the numerically generated data by the use of a chi-square goodness-of-fit test.     

Statistical modeling and simulation of the RMS delay spread ofindoor radio propagation channels

Using a large empirical data base of 12000 impulse response estimates collected in two office buildings the root mean square (rms) delay spread for the individual impulse responses have been calculated and analyzed. Statistical modeling of τ_rms includes investigation of the local and global distributions, correlations in space, dependence on transmitter-receiver antenna separation, correlations with large scale path losses, and dependence on the dynamic range of measurements. Using the results of the statistical analysis, a simulation model capable of generating a set of τ_rms values for spatially-adjacent points has been developed     

On the capacity of a cascade of identical discrete memoryless nonsingular channels (Corresp.)

A simplified procedure for calculating the channel capacity of a cascade ofNidentical discrete memoryless nonsingular channels is presented. The result depends only upon theMeigenvalues andMeigenvectors of any one of the subchannel transition matrices. Thus, for smallMand largeN(the usual case of interest) the result represents a considerable saving in computation relative to the standard technique of finding the overall channel transition matrix. The procedure is illustrated for anNcascade of binary symmetric channels.     

Steady-state tracking analysis of the RLS algorithm for time-varying channels: a general state-space approach

We present a general technique for analyzing the steady-state tracking performance of the recursive least squares (RLS) algorithm for estimating Rayleigh fading channels in terms of the mean squared identification error (MSIE) or mean squared deviation (MSD). The technique may be used with a general-order vector-autoregressive (VAR) model. Numerical results obtained using the technique agree closely with experimental results.     

An algorithm for optimal prefix parsing of a noiseless and memoryless channel

We discuss the prefix encoding of aQ-ary source(Q geq 2)into anL-symbol channel alphabet withL geq Q. We present an optimal encoding scheme that minimizes the expected cost per symbol in the case of equally probable source symbols and arbitrary channel symbol costs.     

Statistical analysis of the main parameters involved in the designof a genetic algorithm

Most genetic algorithm (GA) users adjust the main parameters of the design of a GA (crossover and mutation probability, population size, number of generations, crossover, mutation, and selection operators) manually. Nevertheless, when GA applications are being developed it is very important to know which parameters have the greatest influence on the behavior and performance of a GA. The purpose of this study was to analyze the dynamics of GAs when confronted with modifications to the principal parameters that define them, taking into account the two main characteristics of GAs; their capacity for exploration and exploitation. Therefore, the dynamics of GAs have been analyzed from two viewpoints. The first is to study the best solution found by the system, i.e., to observe its capacity to obtain a local or global optimum. The second viewpoint is the diversity within the population of GAs; to examine this, the average fitness was calculated. The relevancy and relative importance of the parameters involved in GA design are investigated by using a powerful statistical tool, the analysis of the variance (ANOVA)     

基于单神经元PSD算法控制串级倒立摆系统

倒立摆是具有非线性,多变量,强耦合性的综合系统模型,传统的控制方法在解决其复杂多变的控制问题上的能力是有限的.本文在基于串级PID倒立摆的基础上,利用单神经元的自学习和自适应能力设计其外环控制器.在单神经元PID控制中,增益K的选取与实际系统的控制对象特性息息相关,且K值的给定对系统有很大影响.针对单神经元中固定增益K...     

Statistical performance analysis of the algebraic constant modulus algorithm

This paper presents a large sample analysis of the covariance of the beamformers computed by the analytical constant modulus algorithm (ACMA) method for blindly separating constant modulus sources. This can be used to predict the signal-to-interference plus noise ratio (SINR) performance of these beamformers, as well as their deviation from the (nonblind) Wiener receivers to which they asymptotically converge. The analysis is based on viewing ACMA as a subspace fitting optimization, where the subspace is spanned by the eigenvectors of a fourth-order covariance matrix. The theoretical performance is illustrated by numerical simulations and shows a good match.     

A novel approach to modeling of OQPSK-type digital transmissionover nonlinear radio channels

Offset quaternary PSK-type (OQPSK) digital modulation schemes are often recommended for radio transmission systems employing grossly nonlinear amplifiers, and have found widespread use in land mobile and/or satellite communications. Most of the recommended OQPSK-type schemes-also called MSK-type schemes-jointly exhibit the following features: a constant envelope or, at least, a low envelope fluctuation; a compact spectrum after a power-efficient, saturated amplifier, and a high detection efficiency obtained with simple, low-cost receivers. We introduce a novel approach to computer-aided modeling of OQPSK-type digital transmission over nonlinear radio channels. By taking into account the specific nature of OQPSK-type modulations, when using the Volterra approach as proposed by Benedetto et al. (1987), we conclude that two signal representations (“parallel” and “serial”) can be achieved which are similar to the conventional Laurent (1986) representation of binary CPM signals with h=1/2. With the proposed approach, the OQPSK-type signals have an invariant structure along the transmission path (which is supposed to include, at least, one nonlinearity) in the sense that they are all made of several linear components of the same type. Some examples are given, including signal representation and power spectrum results. The proposed unified signal representations can be very useful for design and performance evaluation of radio communication systems (mobile and/or satellite applications). They combine flexibility, accuracy, and simplicity, and allow high computational efficiencies     

Spatially and temporally correlated MIMO channels: modeling and capacity analysis

Wireless communication systems employing multiple antennas at both the transmitter and receiver have been shown to offer significant gains over single-antenna systems. Recent studies on the capacity of multiple-input-multiple-output (MIMO) channels have focused on the effect of spatial correlation. The joint effect of spatial and temporal correlation has not been well studied. In this paper, a geometric MIMO channel model is presented, which considers motion of the receiver and nonisotropic scattering at both ends of the radio link. A joint space-time cross-correlation function is derived from this model and variates with this joint correlation are generated by using the vector autoregressive stochastic model. The outage capacity of this channel is considered where the effects of antenna spacing, antenna array angle, degree of nonisotropic scattering, and receiver motion are investigated. When n transmit and n receive antennas are employed, it is shown that the outage capacity still increases linearly with respect to n, despite the presence of spatial and temporal correlation. Furthermore, analytical expressions are derived for the ergodic capacity of a MIMO channel for the cases of spatial correlation at one end and at both ends of the radio link. The latter case does not lend itself to numerical evaluation, but the former case is shown to be accurate by comparison with simulation results. The proposed analysis is very general, as it is based on the transmit and receive antenna correlations matrices.     

Convergence analysis of a continuous-time nonlinear LMS-type algorithm using stochastic calculus

In several applications least mean square (LMS) has served as a good tool for estimating the parameters of linear models but the success of continuous-time in nonlinear models has not reached its height. In this paper, we have developed a nonlinear continuous-time LMS type algorithm that estimates parameters of nonlinear systems considering the noisy input–output relationship. The nonlinear system has been assumed to be memoryless and an additive Gaussian noise component to the system has been assumed. The mean squared error between the true system output and the estimated output, when the estimated output is modeled using the same form of the nonlinear function as the original system but with the parameters unknown, is minimized using the gradient scheme with the expectation removed. The result is a least mean square algorithm for nonlinear systems. In particular, we have performed a convergence analysis of the continuous-time nonlinear LMS algorithm applied to nonlinear systems when the time step goes to zero. The resulting algorithm then behaves as a stochastic differential equation, and the standard methods of Itô calculus and Fokker–Planck theory are applied to obtain statistical properties of the mean and covariance evolution of the parameter estimates. Computer simulations corroborate the theoretical results.     

A nonlinear filter for estimating a sinusoidal signal and itsparameters in white noise: on the case of a single sinusoid

A nonlinear filter is proposed for estimating a complex sinusoidal signal and its parameters (frequency, amplitude, and phase) from measurements corrupted by white noise. This filter is derived by applying an extended complex Kalman filter (ECKF) to a nonlinear stochastic system whose state variables are a function of its frequency and a sample of an original signal, and then, proof of the stability is given in the case of a single complex sinusoid. Simulations demonstrate that the proposed nonlinear filter is effective as a method for estimating a single complex sinusoid and its frequency under a low signal-to-noise ratio (SNR). In addition, the effect of the initial condition in the filter on frequency estimation is also discussed     

Statistical modeling of the LMS channel

In this paper, a statistical model for the land mobile satellite (LMS) channel is presented. The model is capable of describing both narrow- and wide-band conditions. The other relevant characteristic is that it can be used to study links with geostationary as well as nongeostationary satellites. The model is of the generative type, i.e., it is capable of producing time series of a large number of signal features: amplitudes, phases, instantaneous power-delay profiles, Doppler spectra, etc. Model parameters extracted from a comprehensive experimental data bank are also provided for a number of environments and elevation angles at L-, S-, and Ka-bands     

一种非线性动态网络的离散分析算法设计及实现

在非线性动态网络响应分析中,采用Volterra级数法可以导出与线性系统传递函数相似的非线性传递函数,能使非线性系统用线性化和系统化方法达到精确分析.本文利用方波函数的积分变换具有将时域内的微分、积分运算变换成方波域内的矩阵代数运算的性质,将一类非线性动态网络响应求解的Volterra级数解的连续时域递推算法转化为离散算式,解决了连续算式中广义卷积积分的迭加计算麻烦的问题.文中给出了该算法,仿真结果证明了它的有效性.