Antonia Oya

Signal detection using approximate Karhunen-Loeve expansions

A new approach to the signal detection problem in continuous time is presented on the basis of approximate Karhunen-Loeve (K-L) expansions. This methodology gives approximate solutions to the problem of detecting either deterministic or Gaussian signals in Gaussian noise. Furthermore, for this last problem an approximate estimator-correlator representation is provided which approaches the optimum detection statistic.

Widely Linear Simulation of Continuous-Time Complex-Valued Random Signals

An approach to the simulation of continuous-time complex-valued random signals under widely linear processing is investigated. The technique is based on a version of the Karhunen-Loève expansion for improper stochastic signals defined on any interval of the real line which takes into account the full statistical information of the signal, i.e., the correlation and complementary correlation functions. It is especially designed for simulating approximately a broad class of signals including complex Gaussian or transformation of complex Gaussian signals and it can be accommodated to simulate any widely linear operations of the signal of interest. Its feasibility and accuracy is examined numerically by considering an example in which the technique devised is compared with the classical approach that ignores the complementary correlation information.

A numerical solution for multichannel detection

A numerical approach to the vector signal detection problem is proposed by using an approximate series representation for vector complex-valued random processes. This technique allows us to provide computationally feasible suboptimum receivers for the problem of detecting an improper or proper complex signal in additive white noise. Its implementation requires the knowledge of the correlation matrix of the vector process involved. The performance of the technique devised is assessed by some communication examples.

Detection of continuous-time quaternion signals in additive noise

Different kinds of quaternion signal detection problems in continuous-time by using a widely linear processing are dealt with. The suggested solutions are based on an extension of the Karhunen-Loève expansion to the quaternion domain which provides uncorrelated scalar real-valued random coefficients. This expansion presents the notable advantage of transforming the original four-dimensional eigen problem to a one-dimensional problem. Firstly, we address the problem of detecting a quaternion deterministic signal in quaternion Gaussian noise and a version of Pitcher’s Theorem is given. Also the particular case of a general quaternion Wiener noise is studied and an extension of the Cameron-Martin formula is presented. Finally, the problem of detecting a quaternion random signal in quaternion white Gaussian noise is tackled. In such a case, it is shown that the detector depends on the quaternion widely linear estimator of the signal.

Numerical Evaluation of Reproducing Kernel Hilbert Space Inner Products

A new approach to numerically evaluate an inner product or a norm in an arbitrary reproducing kernel Hilbert space (RKHS) is considered. The proposed methodology enables us to approximate the RKHS inner product with the desired accuracy avoiding analytical expressions. Furthermore, its implementation is illustrated by means of some classic examples and compared with the standard iterative method provided by Weiner for this purpose. Finally, applications in both the problem of representing approximately second-order stochastic processes by means of series expansions and in the problem of signal detection are studied.

An approximate solution to the simultaneous diagonalization of two covariance kernels: applications to second-order stochastic processes

We define the approximate simultaneous orthogonal (ASO) expansions of two second-order stochastic processes from the Rayleigh-Ritz eigenfunctions and prove its convergence. We consider an example that illustrates the implementation of the proposed method and that allows us to assess the accuracy of the approximations achieved with such finite expansions. Finally, we give two specific applications: in the problem of estimating a Gaussian signal in noise and in the Gaussian signal detection problem.

An approach to RKHS inner products evaluation. Application to signal detection problem

The aim of this paper is to propose a new methodology based on Hilbert spaces with reproducing kernels (RKHS) obtained from the Rayleigh-Ritz method that allows us to evaluate a RKHS inner product approximately with the desired accuracy and obtain approximate series representations for second-order stochastic processes. Furthermore, we apply the introduced technique in the signal detection problem providing computationally feasible suboptimum detectors.

Approximate simultaneous orthogonal expansions. Applications to mean-square estimation and signal detection problems

Approximate simultaneous orthogonal expansions of two second-order stochastic processes are defined and their convergence is showed. The technique is based on the Rayleigh-Ritz method to solve the homogeneous equation involving both covariance kernels simultaneously. Finally, two specific applications of these finite expansions are given: in the Gaussian estimation and detection problems.