Correlation function estimation by a polarity method using stochastic reference signals

Instead of estimating correlation functions by conventional means, a specific polarity scheme may be used on bounded processes. The method is based on a rather simple relationship between the correlation functions before and after infinite clipping, provided that stochastic reference signals of uniform distribution are added to the process. This correlation technique has been known for some time. Because of the apparent computational advantages, its application to the estimation of correlation functions from discrete or sampled data is being examined. A general derivation of the appropriate moment relationship is given and a complete mean-square error analysis of estimates is provided under the assumption of white-noise-type reference signals. It is shown that correlation function estimates obtained by this polarity method possess a mean-square error that differs from the error of conventional estimates only by a term proportional to1/N, whereNis the sample size. This term may be made arbitrarily small. Thus, only small degradations in the accuracy of estimates have to be expected when using the polarity approach.     

Transport of the mutual coherence functions and the intensity of a backscattered pulse in a random medium

Starting with a Helmholtz wave equation, the transport equations for the coherent and incoherent parts of the two-frequency two-position mutual coherence function in the transform domain can be derived. These equations have their respective analogs in the conventional radiative transfer theory. By solving the equations for the coherent part and the incoherent part, the incoherent scattered intensity is expressed as an infinite series, each term of which can be interpreted as an "nth order" scattered contribution. A procedure is given to compute the effective wavenumber and the total cross section, starting from a proper form of the mean Green's function. The intensity of a backscattered pulse is next investigated. A series expression is obtained. In order to be able to localize the position of pulse returns, "higher order" scattering terms must not be important when compared with the "single" scattered pulse. Numerical examples, applicable to the microwave scattering, show that the backscattered intensity is reduced when it is compared with the Booker-Gordon formula, even when the mean-square dielectric fluctuation is very small. This reduction is attributed to the multiple scattering effects, especially the multiple large-angle scatterings.     

Nonlinear transformations of random processes

This paper provides a general method of calculating the mean-square bandwidth (and other spectral moments) of an arbitrary zero-memory nonlinear transformation of a stationary random process. The method is valid when the original process is an arbitrary combination of other random processes. It can be used to determine the mean-square bandwidth (or the spectral moments) of the transformed process either before or after that process is passed through a bandpass filter. Five examples of the application of this method are provided simplifying and generalizing known results, as well as providing new results.     

输入光束光斑半径变化对BOE整形环的影响

在已设计的16阶用于将He-Ne基模高斯光束转换为厚环形光束二元光学元件基础上,模拟计算了输入光束光斑半径变化对所设计的二元光学元件整形环效果的影响,模拟结果显示,当光斑半径偏离设计值在±50μm范围内,环上均匀性的均方误差不超出设计值的5%.同时,给出了输入光斑半径为1.745mm时的实验结果及660nm半导体激光实验结果.     

Mean-square performance of a convex combination of two adaptive filters

Combination approaches provide an interesting way to improve adaptive filter performance. In this paper, we study the mean-square performance of a convex combination of two transversal filters. The individual filters are independently adapted using their own error signals, while the combination is adapted by means of a stochastic gradient algorithm in order to minimize the error of the overall structure. General expressions are derived that show that the method is universal with respect to the component filters, i.e., in steady-state, it performs at least as well as the best component filter. Furthermore, when the correlation between the a priori errors of the components is low enough, their combination is able to outperform both of them. Using energy conservation relations, we specialize the results to a combination of least mean-square filters operating both in stationary and in nonstationary scenarios. We also show how the universality of the scheme can be exploited to design filters with improved tracking performance.     

Estimation of Second-Order Cross-Moments of Generalized Almost-Cyclostationary Processes

In this paper, the problem of estimating second-order cross-moments of generalized almost-cyclostationary (GACS) processes is addressed. GACS processes have statistical functions that are almost-periodic functions of time whose (generalized) Fourier series expansions have both frequencies and coefficients that depend on the lag shifts of the processes. The class of such nonstationary processes includes the almost-cyclostationary (ACS) processes which are obtained as a special case when the frequencies do not depend on the lag shifts. ACS processes filtered by Doppler channels and communications signals with time-varying parameters are further examples. It is shown that the second-order cross-moment of two jointly GACS processes is completely characterized by the cyclic cross-correlation function. Moreover, it is proved that the cyclic cross-correlogram is an asymptotically normal, mean-square consistent, estimator of the cyclic cross-correlation function. Furthermore, it is shown that well-known consistency results for ACS processes can be obtained by specializing the results of this paper.     

Lower bounds on estimator performance for energy-invariantparameters of multidimensional Poisson processes

Using rate distortion theory, lower bounds are developed for the mean-square error of estimates of a random parameter of an M-dimensional inhomogeneous Poisson process with respect to which the energy, i.e. the average number of points, is invariant. The bounds are derived without stringent assumptions on either the form of the intensity or the prior distribution of the parameter, and they can handle random nuisance parameters. The derivation makes use of a side-information averaging principle applied to the distortion-rate function and a maximum-entropy property of energy-constrained Poisson processes. Under the additional assumption of conditional entropy invariance of the point process with respect to the parameter of interest, an explicit bound is given which depends on the information discrimination between the inhomogeneous conditionally Poisson process and a nearly homogeneous Poisson process. The application of the explicit bound is illustrated through a treatment of the problems of time-shift estimation and relative time-shift estimation for Poisson streams     

Worst sources and robust codes for difference distortion measures

It has long been known that for a mean-square error distortion measure the Gaussian distribution requires the largest rate of all sources of a given variance. It has also been stated that a code designed for the Gaussian source and yielding distortiondwhen used with a Gaussian source will yield distortionleq dwhen used with any independent-letter source of the same variance. In this paper, we extend these results in two directions: a) instead of assuming that the source has a fixed variance, we fix an arbitrary moment; b) instead of mean-square error distortion measures, we consider nearly arbitrary continuous difference distortion measures. For each moment constraint, we show that there is a given distribution that has the largest rate for (nearly) any difference distortion measure and that a code designed for this source yielding distortiondyields distortionleq dfor any ergodic source satisfying the same moment constraint. Furthermore, digital encoding of the output of this encoder may yield a lower rate when this encoder is used with a source for which it was not designed. We also extend these results to the case of a random process or random field of known correlation function under a difference distortion measure.     

Generating rice processes with given spectral properties

The autocorrelation function (acf) of a stochastic Rician fading process is numerically evaluated, and useful approximations are given. The spectral properties of the fading are related to the component processes constituting the fading process. The result is that for Rayleigh fading and first-order Butterworth characteristic of the component processes the fading bandwidth is double the component bandwidth. The fading bandwidth decreases to the component bandwidth, however, when the Gaussian channel is approached or when high-order faltered components are assumed.     

Optimal lattices for sampling

The generalization of the sampling theorem to multidimensional signals is considered, with or without bandwidth constraints. The signal is modeled as a stationary random process and sampled on a lattice. Exact expressions for the mean-square error of the best linear interpolator are given in the frequency domain. Moreover, asymptotic expansions are derived for the average mean-square error when the sampling rate tends to zero and infinity, respectively. This makes it possible to determine the optimal lattices for sampling. In the low-rate sampling case, or equivalently for rough processes, the optimal lattice is the one which solves the packing problem, whereas in the high-rate sampling case, or equivalently for smooth processes, the optimal lattice is the one which solves the dual packing problem. In addition, the best linear interpolation is compared with ideal low-pass filtering (cardinal interpolation).     

Stationary and nonstationary learning characteristics of the LMS adaptive filter

This paper describes the performance characteristics of the LMS adaptive filter, a digital filter composed of a tapped delay line and adjustable weights, whose impulse response is controlled by an adaptive algorithm. For stationary stochastic inputs, the mean-square error, the difference between the filter output and an externally supplied input called the "desired response," is a quadratic function of the weights, a paraboloid with a single fixed minimum point that can be sought by gradient techniques. The gradient estimation process is shown to introduce noise into the weight vector that is proportional to the speed of adaptation and number of weights. The effect of this noise is expressed in terms of a dimensionless quantity "misadjustment" that is a measure of the deviation from optimal Wiener performance. Analysis of a simple nonstationary case, in which the minimum point of the error surface is moving according to an assumed first-order Markov process, shows that an additional contribution to misadjustment arises from "lag" of the adaptive process in tracking the moving minimum point. This contribution, which is additive, is proportional to the number of weights but inversely proportional to the speed of adaptation. The sum of the misadjustments can be minimized by choosing the speed of adaptation to make equal the two contributions. It is further shown, in Appendix A, that for stationary inputs the LMS adaptive algorithm, based on the method of steepest descent, approaches the theoretical limit of efficiency in terms of misadjustment and speed of adaptation when the eigenvalues of the input correlation matrix are equal or close in value. When the eigenvalues are highly disparate (λmax/λmin> 10), an algorithm similar to LMS but based on Newton's method would approach this theoretical limit very closely.     

Recursive short-data-record estimation of AV and MMSE/MVDR linear filters for DS-CDMA antenna array systems

The presence of the desired signal during estimation of the minimum mean-square error (MMSE)/minimum-variance distortionless-response (MVDR) and auxiliary-vector (AV) filters under limited data support leads to significant signal-to-interference-plus-noise ratio (SINR) performance degradation. We quantify this observation in the context of direct-sequence code-division multiple-access (DS-CDMA) communications by deriving close approximations for the mean-square filter estimation error, the probability density function of the output SINR, and the probability density function of the symbol-error rate (SER) of the sample matrix inversion (SMI) receiver evaluated using both a desired-signal-"present" and desired-signal-"absent" input covariance matrix. To avoid such performance degradation, we propose a DS-CDMA receiver that utilizes a simple pilot-assisted algorithm that estimates and then subtracts the desired signal component from the received signal prior to filter estimation. Then, to accommodate decision-directed operation, we develop two recursive algorithms for the on-line estimation of the AV and MMSE/MVDR filter and we study their convergence properties. Finally, simulation studies illustrate the SER performance of the overall receiver structures.     

On Divergence-Power Inequalities

Expressions for (entropy-power inequality (EPI) Shannon type) divergence-power inequalities (DPIs) in two cases (time-discrete and time-continuous) of stationary random processes are given. The new expressions connect the divergence rate of the sum of independent processes, the individual divergence rate of each process, and their power spectral densities. All divergences are between a process and a Gaussian process with same second-order statistics, and are assumed to be finite. A new proof of the Shannon EPI, based on the relationship between divergence and causal minimum mean-square error (CMMSE) in Gaussian channels with large signal-to-noise ratio, is also shown     

Shaping filter representation of nonstationary colored noise

The problem of determining a shaping filter for nonstationary colored noise is considered. The shaping filter transforms white noise into a possibly nonstationary random process (having no white noise component) with a specified covariance function. A set of conditions to be satisfied by the covariance function leads to the determination of a shaping filter. The shaping filter coefficients are simply related to the solution of a matrix Riccati equation. In order to formulate the Riccati equation, basic results concerning the mean-square differentiability of a random process are developed. If the Riccati equation can not be defined, an autonomous (zero-input) shaping filter may be easily determined.     

Spectrum Conservation and Characteristics of Single-Sideband Phase Modulation

Phase modulation with an analytic signal, which is a Gaussian random process, is examined in order to determine the amount of spectrum conservation that may be achieved by using single-sideband phase modulation (SSB-PM) rather than conventional phase modulation (PM). The autocorrelation function is derived and found to be an analytic signal in terms of the autocorrelation function of the actual modulating signal and its Hilbert transform. When the modulating signal strength is very low, the sideband spectral distribution is the same as that of the actual modulating signal or single-sideband amplitude modulation. As the modulating signal mean-square value is increased, the sideband spectrum broadens and approaches a Gaussian shape. The average power output of an SSB-PM system increases exponentially with input modulating signal strength, while the carrier power remains constant. For the same modulating signal mean-square value, a greater fraction of power is in the one sideband of an SSB-PM system than in the two sidebands of conventional PM. Single-sideband phase or frequency modulation always effectuates spectrum conservation in the continuum when it is compared with conventional phase or frequency modulation on the basis of equal relative sideband power. A Fourier transform computer program is used to generate SSB-PM spectral distributions with varying modulating signal mean-square values, when the modulating signal spectrum is a low-pass rectangular spectrum, a narrowband pass spectrum, and the shape of an average voice spectrum. These examples illustrate the power series formulation of the output spectrum as well as the theoretical analysis of bandwidth.     

Experimental performance of point process estimation procedures for log-normal fading (Corresp.)

The experimentally measured performance of equations which govern the evolution of both causal minimum mean-square error (MMSE) and approximate (suboptimal) MMSE estimates of the intensity function of a doubly stochastic Poisson process for the case of log normally faded optical field intensities is reported.     

Mixing of Two Renewal Processes and Its Applications to Reliability Theory

The probabilistic behavior of repairable two-state systems is studied. It is assumed that the reliability of the system can be measured by the amount of time the system has been operative. The operating and repair states are a pair of renewal processes, a particular mixture of which describes the statistical behavior of the system. The object of this contribution is to extend the results of Muth who has earlier obtained the average value of the downtime and its variance, when one of the constituent renewal processes has its interval lengths distributed exponentially. This paper, by the repeated use of the method of regeneration points, obtains the mean and mean-square values of the uptime distribution. In addition the correlation of the uptime for different times has been derived and a proof of Takacs' theorem is provided. Since the criteria for the reliability of the system include the associated cost, it is worthwhile investigating the operating cost over any period of time for arbitrary distribution of the two states. In particular a demonstration of the calculation of the first two moments of the total cost is given.     

A nonparametric-Bayes reliability-growth model

The authors propose a nonparametric reliability-growth model based on Bayes analysis techniques. By using the unique properties of the assumed prior distributions, the moments of the posterior distribution of the failure rate at various stages during a development test can be found. The proposed model is compared with the US Army Material Systems Analysis Activity (AMSAA) model based on relative and mean-square prediction errors. In all but one circumstance, the proposed model performed better than either the AMSAA or nonparametric models. The one exception appears to be when no information about the failure rate is available at the start of test and the actual failure process is nonhomogeneous Poisson, with power-law intensity function, as assumed by the AMSAA model     

Steady-state analysis of quantized distributed incremental LMS algorithm without Gaussian restriction

In this paper, we analyze the steady-state performance of the distributed incremental least mean-square (DILMS) algorithm when it is implemented in finite-precision arithmetic. Our analysis in this paper does not consider any distribution of input data. We first formulate the update equation for quantized DILMS algorithm, and then we use a spatial-temporal energy conservation argument to derive theoretical expressions that evaluate the steady-state performance of individual nodes in the network. We consider mean-square error, excess mean-square error, and mean-square deviation as the performance criteria. Simulation results are generated by using two types of signals, Gaussian and non-Gaussian distributed signals. As the simulation results show, there is a good match between the theory and simulation.     

On the structure of optimal entropy-constrained scalar quantizers

The nearest neighbor condition implies that when searching for a mean-square optimal fixed-rate quantizer it is enough to consider the class of regular quantizers, i.e., quantizers having convex cells and codepoints which lie inside the associated cells. In contrast, quantizer regularity can preclude optimality in entropy-constrained quantization. This can be seen by exhibiting a simple discrete scalar source for which the mean-square optimal entropy-constrained scalar quantizer (ECSQ) has disconnected (and hence nonconvex) cells at certain rates. In this work, new results concerning the structure and existence of optimal ECSQs are presented. One main result shows that for continuous sources and distortion measures of the form d(x,y)=ρ(|x-y|), where ρ is a nondecreasing convex function, any finite-level ECSQ can be "regularized" so that the resulting regular quantizer has the same entropy and equal or less distortion. Regarding the existence of optimal ECSQs, we prove that under rather general conditions there exists an "almost regular" optimal ECSQ for any entropy constraint. For the squared error distortion measure and sources with piecewise-monotone and continuous densities, the existence of a regular optimal ECSQ is shown