Eigenvalue Distributions of Sums and Products of Large Random Matrices Via Incremental Matrix Expansions

This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common assumptions, and matches the results obtained from using - and -transforms in free probability theory. We also give a direct derivation of the a.e.d. of the sum of certain random matrices which are not free. This is used to determine the asymptotic signal-to-interference-ratio of a multiuser code-division multiple-access (CDMA) system with a minimum mean-square error linear receiver.     

Transient analysis of data-normalized adaptive filters

This paper develops an approach to the transient analysis of adaptive filters with data normalization. Among other results, the derivation characterizes the transient behavior of such filters in terms of a linear time-invariant state-space model. The stability, of the model then translates into the mean-square stability of the adaptive filters. Likewise, the steady-state operation of the model provides information about the mean-square deviation and mean-square error performance of the filters. In addition to deriving earlier results in a unified manner, the approach leads to stability and performance results without restricting the regression data to being Gaussian or white. The framework is based on energy-conservation arguments and does not require an explicit recursion for the covariance matrix of the weight-error vector.     

Pel-recursive motion estimation algorithm

A new pel-recursive motion estimation algorithm for video coding applications is presented. The derivation of the algorithm is based on recursive least-squares estimation that minimises the mean-square prediction error. A comparison with the modified steepest-descent gradient estimation technique algorithm shows significant improvement in terms of mean-square prediction error performance     

On mutual information, likelihood ratios, and estimation error for the additive Gaussian channel

This paper considers the model of an arbitrarily distributed signal x observed through an added independent white Gaussian noise w, y=x+w. New relations between the minimal mean-square error of the noncausal estimator and the likelihood ratio between y and w are derived. This is followed by an extended version of a recently derived relation between the mutual information I(x;y) and the minimal mean-square error. These results are applied to derive infinite-dimensional versions of the Fisher information and the de Bruijn identity. A comparison between the causal and noncausal estimation errors yields a restricted form of the logarithmic Sobolev inequality. The derivation of the results is based on the Malliavin calculus     

Mean-Square Error Optimization of Quadrature Receivers for CPM with Modulation Index 1/2

Detection of continuous phase modulation (CPM) signals can be accomplished using quadrature coherent detectors which include a pair of linear and time-invariant postdetection filters. Performance of the quadrature detectors is highly sensitive to the postdetection filters response. This paper presents a rigorous derivation of an optimum postdetection filter response. The derivation is based on minimizing the mean-square interference subject, to the constraint that the filter noise bandwidth is held constant. The amount of computations involved is fairly small and increases linearly with the receiver observation interval. Performance analysis results for several modulation techniques are presented.     

The Mean-Square Error of SSBAM Class 4 Partial Response Data Signals Distorted by Parabolic Group Delay

An expression is derived for the mean-square error of an SSBAM multilevel class 4 partial response data signal when distorted by a channel with parabolic group delay with the minimum value centered in the signal spectrum. The optimum carrier phase and sampling times are also determined. Numerically evaluated results indicate that the approximation errors made in the derivation are not significant in the regions of practical interest.     

A Fast Karhunen-Loeve Transform for a Class of Random Processes

The Karhunen-Loeve transform for a class of signals is proven to be a set of periodic sine functions and this Karhunen-Loeve series expansion can be obtained via an FFT algorithm. This fast algorithm obtained could be useful in data compression and other mean-square signal processing applications.     

Expansion of likelihood ratio for signals in Gaussian noise

It is shown that the coefficients of the Taylor expansion of the likelihood ratio for random signals in Gaussian noise are proportional to the mean-square estimates of the successive powers of the signal evaluated at zero. A similar relationship holds for the expansion of the logarithm of the likelihood ratio.     

Exact expectation analysis of the LMS adaptive filter

In almost all analyses of the least mean square (LMS) adaptive filter, it is assumed that the filter coefficients are statistically independent of the input data currently in filter memory, an assumption that is incorrect for shift-input data. We present a method for deriving a set of linear update equations that can be used to predict the exact statistical behavior of a finite-impulse-response (FIR) LMS adaptive filter operating upon finite-time correlated input data. Using our method, we can derive exact bounds upon the LMS step size to guarantee mean and mean-square convergence. Our equation-deriving procedure is recursive and algorithmic, and we describe a program written in the MAPLE symbolic-manipulation software package that automates the derivation for arbitrarily-long adaptive filters operating on input data with stationary statistics. Using our analysis, we present a search algorithm that determines the exact step size mean-square stability bound for a given filter length and input correlation statistics. Extensive computer simulations indicate that the exact analysis is more accurate than previous analyses in predicting adaptation behavior. Our results also indicate that the exact step size bound is much more stringent than the bound predicted by the independence assumption analysis for correlated input data     

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.     

A new approach for evaluating the performance of a symbol timingrecovery system employing a general type of nonlinearity

A new technique is presented for evaluating the jitter performance of a symbol timing recovery (STR) subsystem for digital data transmission systems. The STR system consists of any even-symmetric zero-memory nonlinear device followed by a narrowband filter tuned to the pulse repetition frequency. Exact analytical expressions are derived for the mean and the mean-square values of the timing wave, based on iterative computations of high-order moments of the input signal. Then, the root mean-square (RMS) jitter performance is determined as a function of various system parameters such as the power series expansion of the zero-memory nonlinear device, the rolloff factor of the input pulse shape, and the postfiltering. Finally, the numerical results obtained from some specific examples serve to illustrate several aspects of the timing recovery problem     

The probability distribution of gains in avalanche photodiodes

The probability density function for the gain of an avalanche photodiode is derived. The derivation is based on a Lagrange expansion of the implicit form of the moment generating function. The results confirm previously published conjectures.     

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     

Receiver Design for Single-Carrier Block Transmission Over Doubly Selective Channels

Single-carrier block transmission is an alternative scheme to orthogonal frequency-division multiplexing (OFDM) for wireless broadband communications. In this paper, a receiver is designed for single-carrier block transmission with cyclic prefix for mobile broadband communications. As the wireless transmission is over doubly selective channels, a basis expansion model is used to capture both the time- and frequency-selectivity of the channel and is parameterized for the receiver design. The receiver estimates the channel model coefficients in the time domain and uses these coefficients for equalization in the frequency domain. The channel estimation is assisted by time-domain pilot insertion. The structure of the frequency-domain channel matrix is exploited and a linear minimum mean-square error equalizer is used for the equalization. When the basis expansion model well matches the physical channel, simulation results show superior receiving performance of the proposed system compared with the OFDM system with a similar complexity.     

一种新颖的求解离散Gabor展开对偶窗的方法

本文给出了一种新颖的简捷的求解离散Gabor展开最佳对偶窗的方法．首先推导了离散Gabor展开的公式，给出了连续Gabor展开和离散Gabor展开间的关系．最后利用连续展开和离散展开间的关系给出了一求解离散Gabor展开最佳对偶窗的表达式及实例．     

Further Results on Robust Variance-Constrained Filtering for Uncertain Stochastic Systems with Missing Measurements

This paper revisits the problem of robust filtering for uncertain discrete-time stochastic systems with missing measurements. The measurements of the system may be unavailable at any sample time. Our aim is to design a new filter such that the error state of the filtering process is mean-square bounded. Furthermore, the steady-state variance of the estimation error of each state does not exceed the individual prescribed upper bound subject to all admissible uncertainties and all possible incomplete observations. It is shown that the design of a robust filter can be carried out by directly solving a set of linear matrix inequalities. The nonsingular assumption on the system matrix A and the inequality which is used to handle the uncertainties are not necessary in the derivation process of our results. Thus, it is expected that a less conservative condition can be obtained. The advantage of the new method is demonstrated via an illustrative example.     

On digital Wiener filters

The design of optimal digital fiters to minimize the mean-square error is well known and can be obtained by solving [Sud(z)/Su^-(z)]+/Su⁺(z). However, the decomposition of [.]+ by partial fraction expansion, unlike the analog case, may not yield the optimal filter. In this letter, this problem is discussed.     

On the Question of Computation of the Dyadic Green's Function at the Source Region in Waveguides and Cavities (Short Papers)

The derivation of the dyadic Green's function for rectangular waveguides and cavities is approached systematically by using the theory of distributions. It is shown that, in order to obtain a complete solution for the field distribution in the entire structure, one must add an additional term to the classical expansion which is valid only outside the source region.     

Array pattern synthesis with a controlled mean-square sidelobelevel

The author presents a technique for synthesizing an antenna pattern with a controlled mean-square sidelobe level and a smallest possible beamwidth. The basic idea is to minimize the mean-square error between the array response and the desired response over a mainlobe width subject to a mean-square sidelobe constraint. This formulation results in a quadratically constrained minimization problem. An efficient numerical technique to obtain the optimum weights is presented. Numerical results showed that, under high interference-to-white-noise ratio, the new design approach performs better, on the average, than the Chebyshev technique, in terms of interference rejection     

Theory of the circular harmonics expansion method formultiple-optical-fiber system

The circular harmonics expansion method that has been applied to solve the modal characteristics of multiple-core fiber structure is given a formal derivation. The validity of the field expansion expression in the method is rigorously proved by expanding the Green's function for the scalar wave equation in a surface integral equation into circular harmonics. It is found that for coupled round cores only the modified Bessel functions of the second kind are needed in the expansion, and that the modified Bessel functions of the first kind do not appear in the expansion