Zero-mean Process Research Articles

A New Method for Calculating the Position for a Sound Source by Passive Sensor Systems

Passive sensor systems (passive range meter, sonar, etc.) are often used for location of a sound source. The sound wave emitted by the sound source will be picked up by passive sensors at different times. These times as well as environmental parameters are known within a given measurement error. A new method for calculating the position the sound source is presented. The method is based on discrete stochastic linear-quadratic control theory. The noise source are treated as Gaussian white zero-mean processes. Some simulation results are presented.

Optimum detection of signals in non-Gaussian noise

The purpose of this paper is to derive optimum processing structures for use in the detection of signals in additive non-Gaussian noise. The cases chosen for analysis are of particular interest in sonar detection problems since it has been reported that ambient ocean noise may, under sone conditions, deviate from the Gaussian model. The processing structures are considered to be models of the likelihood ratio which is an optimum test no matter what the signal and noise statistics, and optimum single channel and array processors are derived for the small signal-to-noise ratio cases where (1) the signal is completely known, and (2) the signal is a noiselike, not necessarily Gaussian, zero-mean process. Expressions are derived which compare the performance of processors optimized for non-Gaussian noise with those optimized for Gaussian noise for each of the two cases, with transfer functions for the required optimum nonlinear filters and the expected improvements in performance determined using some ’’typical’’ non-Gaussian probability density functions. Justification of these particular density functions is beyond the scope of this paper except to note that very good agreement is obtained with some published experimental data. New results include the derivation of optimum array processors for the detection of plane wave signals when the array is steered at the signal arrival angle, in a non-Gaussian noise field and the development of expressions to predict their performance for the case where the signal is a zero-mean, noiselike process.

Estimating the Solution of a Stochastic Difference Equation

Best linear unbiased estimates are discussed assuming data $y_k $, $k > T$, $y_k = a_1 f_1 ( k ) + \cdots + a_J f_J ( k ) + n_k $, where $f_1 , \cdots ,f_J $ are the J solutions of a linear homogeneous difference equation of order J, the coefficients $a_1 , \cdots a_J $ are unknown, and $\{ {n_k } \}$ is a sequence of zero-mean random variables with variances, $\operatorname{var} ( {y - z} )$. Estimates are considered for future values of $\{ {y_k } \}$ and for a process $\{ {x_k } \}$ which differs from $\{ {y_k } \}$ by a zero-mean process of known covariance. The method is based on discussion of the zero-mean process $\{ {s_k } \}$ obtained by applying the difference operator to $\{ {y_k } \}$. In the case that $\{ {s_k } \}$ is weakly stationary, the results provide for the prediction of trend processes using the theory of weakly stationary processes.

Integral Equation for Simultaneous Diagonalization of Two Covariance Kernels

Let K 1 (s,t) and K 2 (s,t), − T s, T, be real, symmetric, continuous and strictly positive-definite kernels, and denote by K 1 and K 2 the corresponding integral operators. Let x(t) be a sample function of either of two zero-mean processes with covariances K 1 (s,t) and K 2 (s,t). We prove a generalized version of the following: If the integral equation $(K_{2}\psi_{i})(t)=\lambda_{i}(K_{1}\psi_{i}(t),\qquad -T \leqq t leqq T$,$ has formal solutions λ i and ψ i (t) which may contain δ-functions, and if {K 1 ψ i } forms a complete set in L 2 [-T,T], then (i) the two kernels have the following simultaneous diagonalization: $\eqalignno{& K_{1}(s,t) \Sigma_{i}(K_{1} \psi_{i})(s)(K_{1}\psi_{i})(t)\cr & K_{2}(s,t) \Sigma_{i}\lambda_{i}(K_{1} \psi_{i})(s)(K_{1}\psi_{i})(t)}$ uniformly on [-T,T] × [-T, T], and (ii) the sample function has an expansion $x(t) \Sigma_{i}(x,\psi_{i})(K_{i} \psi_{i})(t)$ in the stochastic mean, uniformly in t, and the coefficients are simultaneously orthogonal, i.e., $E_{1}\{(x, \psi_{i})(x,\psi_{i})\} = \delta_{ij}, \qquad E_{2}\{(x,\psi_{i})(x, \psi_{i}\}=\lambda_{i} \delta_{ij}$, where (x,ψ i ) is obtained by formally integrating ψ i (t) against x(t).