On the efficiency of on-line density estimators

An on-line density estimator may be defined to be one where each update, following the arrival of a new data value, may be accomplished after no more than a fixed number of calculations. This definition should also apply to any empirical bandwidth selection rule for such an estimator. Recursive estimators comprise only a special case of on-line estimators, but even there, on-line bandwidth formulas have not been developed. The authors introduce a class of on-line estimators, and discuss efficiency in this context. It is shown that some nonrecursive members of the class achieve greater efficiency than any recursive estimators, and that efficiency increases to 100% as the order of the estimated derivative increases. On-line bandwidth selection rules, enabling these high orders of efficiency to be achieved asymptotically, are introduced     

On the Cramer-Rao bound of finite-length cepstrum spectral estimators

Spectral estimators based on finite-length cepstrum modelling are useful in several applications. The Cramer-Rao lower bound of the asymptotic variance of their logarithmic spectral estimates can be obtained in explicit form. This does not depend on the specific underlying spectrum.<>     

An efficient Newton-type method for the computation of ML estimators in a uniform linear array

In the problem of estimating the angles of arrival to a uniform linear array, we present an efficient method to compute Maximum Likelihood (ML) estimations, based on the Modified Variable Projection (MVP) algorithm. In contrast to methods like Iterative Quadratical Maximum Likelihood (IQML) or the Iterative Method of Direction Estimation (IMODE), it is not based on a polynomial parameterization but on directly exploiting the Vandermonde structure through analytical tools like the Fast Fourier Transform (FFT), the geometric series summation formula, and Horner's synthetic division. The computational burden of the proposed method is significantly smaller than the burden of IMODE and of the Relaxation (RELAX) algorithm. Besides, it is shown that the computation of the ML estimation can be divided in a preliminary step in which a few FFTs are computed and an iterative step with a complexity that is independent of the array size.     

On the existence of cyclic Hadamard difference sets

The main conjecture of this article is the following: if a cyclic (v=4n-1, k=2n-1, λ=n-1) Hadamard difference set exists, the the value of v must be either a prime, or a product of “twin primes,” or one less than a power of 2. Six cases, v=399, 495, 627, 651, 783, and 975, which were once listed as the possible exceptions for v<1000, are now fully investigated, and all the cases of v<10000 are now verified relative to this conjecture, with at most 17 possible exceptions     

On the existence of the Mazo-limit on MIMO channels

Mazo, in 1975, showed that the signaling rate of a linear modulation can be significantly higher than the maximum rate for orthogonal signaling without any loss of minimum square Euclidean distance. In subsequent literature the highest such rate is referred to as the Mazo-limit. In this letter we ask whether there exists a Mazo-limit also on MIMO channels. The answer is yes, but it applies to the largest pairwise error probability rather than to minimum square Euclidean distance. Moreover, it occurs at exactly the same rate as in the AWGN case. As a special case results for single-input single-output fading channels are obtained.     

On the existence of optimum cyclic burst- correcting codes

It is shown that for each integerb geq 1infinitely many optimum cyclicb-burst-correcting codes exist, i.e., codes whose lengthn, redundancyr, and burst-correcting capabilityb, satisfyn = 2^{r-b+1} - 1. Some optimum codes forb = 3, 4, and5are also studied in detail.     

On the existence of universal nonlinearities for blind sourceseparation

Many density-based methods for blind signal separation employ one or more models for the unknown source distribution(s). This paper considers the issue of density model mismatch in maximum likelihood (ML)-type blind signal separation algorithms. We show that the score function nonlinearity, which was previously derived from the standpoint of statistical efficiency, is also the most robust in maintaining a separation solution for the ML algorithm class. We also consider the existence of a universally applicable nonlinearity for separating all signal types, deriving two results. First, among nonlinearities with a convergent Taylor series, a single fixed nonlinearity for universal separation using the natural gradient algorithm cannot exist. Second, among nonlinearities with a single adjustable parameter, a previously proposed threshold nonlinearity can separate all signals with symmetric amplitude distributions as long as the threshold parameter is properly chosen. The design of "difficult-to-separate" signal distributions is also discussed     

On the existence of group codes for the Gaussian channel

A number of theorems are given, which give insight into the algebraic structure of group codes for the Gaussian channel, as defined by Slepian. The problem of the existence of group codes for a given pair of defining parameters (i.e., number of codewordsM, and dimension of the signal spacen) is partially solved.     

On existence of good self-dual quasicyclic codes

For a long time, asymptotically good self-dual codes have been known to exist. Asymptotically good 2-quasicyclic codes of rate 1/2 have also been known to exist for a long time. Recently, it was proved that there are binary self-dual n/3-quasicyclic codes of length n asymptotically meeting the Gilbert-Varshamov bound. Unlike 2-quasicyclic codes, which are defined to have a cyclic group of order n/2 as a subgroup of their permutation group, the n/3-quasicyclic c codes are defined with a permutation group of fixed order of 3. So, from the decoding point of view, 2-quasicyclic c codes are preferable to n/3-quasicyclic c codes. In this correspondence, with the assumption that there are infinite primes p with respect to (w r t.) which 2 is primitive, we prove that there exist classes of self-dual 2p-quasicyclic c codes and Type II 8p-quasicyclic c codes of length respectively 2p/sup 2/ and 8p/sup 2/ which asymptotically meet the Gilbert-Varshamov bound. When compared with the order of the defining permutation groups, these classes of codes lie between the 2-quasicyclic c codes and the n/3-quasicyclic c codes of length n, considered in previous works.     

On the efficient prediction of fractal signals

A novel prediction scheme for self-affine fractal signals is presented. The signal is modeled by self-affine linear mappings, whose contraction factors are assumed to follow an auto-regressive (AR) process. In this way, the highly nonlinear time evolution of the fractal signal is captured by the linear AR process of the contraction factors, thereby exploiting the simplicity and ease of computation inherent in the AR model. An adaptive version of the proposed scheme is applied in simulations using the Weierstrass-Mandelbrot cosine fractal, as well as, in practice, using real radar sea clutter data     

On the existence of spurious modes in Battle-Lemarie based MRTD

A distinction between the effects of inaccuracy of the Battle-Lemarie wavelet based multiresolution time domain (W-MRTD) scheme for high spatial frequencies and spurious modes is drawn and explained. Investigating the performance of the scheme under various excitation methods, it is concluded that what was described earlier in the literature as a spurious mode effect actually corresponds to alias frequencies stemming from the excitation of higher order modes by the wavelets, given that the latter are spectrally supported at higher wavenumbers     

On the errors of two estimators of sub-pixel fractional cover whenmixing is linear

The authors consider the problem of estimating ground cover at sub-pixel scales from remotely sensed imagery. In particular, they examine two strategies that make use of a set of reference pixels, or training pixels, for which fractional ground cover is already known. These strategies are the so-called classical and inverse methods. The former proceeds by assuming that signals received at a sensor are area-weighted averages of characteristic signals for each ground cover component, estimates those characteristic signals from the training pixels, and predicts fractions for a general pixel to be those that give the best match of the modeled and observed image signals. The latter approach proceeds by direct multivariate regression of ground cover proportions on pixel spectral values. The authors show that when ground cover types are spectrally well separated, and mixing is indeed linear, the difference between the two estimators is much smaller than the prediction error associated with either. This means that it is perfectly acceptable to use standard methods of multivariate regression to perform spectral unmixing. They also show that the inverse estimator can be regarded as a regularized form of the classical estimator and that the supposed optimality of the inverse method may be compromised if the training dataset is not a random subset of the complete set of image pixels     

On the equivalence of three reduced rank linear estimators with applications to DS-CDMA

This correspondence shows the equivalence of three previously proposed reduced-rank detection schemes for direct-sequence code-division multiple-access (DS-CDMA) communication systems. The auxiliary vector filtering (AVF) algorithm is simplified through a key observation on the construction of the auxiliary vectors. After simplification, it is shown that the AVF algorithm is equivalent to the multistage Wiener filtering (MWF) algorithm of Honig and Goldstein (2002). Furthermore, these schemes can be shown to be equivalent to the multistage linear receiver scheme based on the Cayley-Hamilton (CH) theorem when the minimum mean-square error (MMSE) criterion is applied to the reduced dimensional space of the received signal.     

On the performance of cyclic moments-based parameter estimators ofamplitude modulated polynomial phase signals

Parameter estimation of amplitude-modulated polynomial phase signals embedded in additive white Gaussian noise is considered. The amplitude modulation is modeled as the sum of a real-valued deterministic function and a zero mean correlated stationary random process. It is shown that cyclic moments-based estimators, previously proposed for parameter estimation of polynomial phase signals modulated by stationary random processes, can be adapted to the more general signal model considered here. The covariance matrices of the cyclic moments-based amplitude and phase parameter estimators are derived for large sample lengths. Using this result, it is shown how the lags can be chosen to minimize the large-sample variances of the cyclic moments-based phase parameter estimators. Comparisons with the Cramer-Rao bounds are performed under the assumption of a Gaussian modulating process. The theoretical derivations are confirmed by simulation results     

On the existence of the solution for one-dimensional discrete phase retrieval problem

We consider the discrete form of the one-dimensional phase retrieval (1-D DPhR) problem from the point of view of input magnitude data. The direct method can provide a solution to the 1-D DPhR problem if certain conditions are satisfied by the input magnitude data, namely the corresponding trigonometric polynomial must be nonnegative. To test positivity of a trigonometric polynomial a novel DFT-based criterion is proposed. We use this DFT criterion for different sets of input magnitude data to evaluate whether the direct method applied to the 1-D DPhR problem leads to a solution in all explored cases.     

On the existence of self-excited oscillations in systems with discontinuous elements

Conditions are established which ensure the existence (or non-existence) of limit cycles in feedback systems containing discontinuous elements or elements with hysteresis. The results are applied to a specific example.     

On the (non)existence of undesirable equilibria of Godard blindequalizers

Existing results in the literature have proved that particular blind equalization algorithms, including Godard algorithms, are globally convergent in an ideal and nonimplementable setting where a doubly infinite dimensional equalizer is available for adaptation. Contrary to popular conjectures, it is shown that implementable finite dimensional equalizers which attempt to approximate the ideal setting generally fail to have global convergence to acceptable equalizer parameter settings without the use of special remedial measures. A theory based on the channel convolution matrix nullspace is proposed to explain the failure of Godard algorithms for such practical blind equalization situations. This nullspace theory is supported by a simple example showing ill convergence of the Godard algorithm     

On the beamforming design for efficient interference alignment

An efficient interference alignment (IA) scheme is developed for k-user single-input single-output frequency selective fading interference channels. The main idea is to steer the transmit beamforming matrices such that at each receiver the subspace dimensions occupied by interference-free desired streams are asymptotically the same as those occupied by all interferences. Our proposed scheme achieves a higher multiplexing gain at any given number of channel realizations in comparison with the original IA scheme, which is known to achieve the optimal multiplexing gain asymptotically.