مرکز منطقه ای اطلاع رساني علوم و فناوري - Efficient approximation of a family of noises for application in adaptive spatial processing for signal detection

Abstract :

Two solutions are presented to the problem of efficiently approximating a family of noises parameterized by a scalar $\\Upsilon , 0 \\leq \\Upsilon \\leq \\infty$ . The noises are represented in the form of vectors with $m$ random components, and their covariance matrices are such that the number of significant eigenvalues increases with $\\Upsilon$ . The noise sample vector is to be approximated, within a specified error $\\epsilon$ , by a linear combination of vectors taken from a fixed set of $m$ vectors that are independent of $\\Upsilon$ . Furthermore, for each $\\Upsilon$ the number of approximating vectors is to be mlnlmlzed while keeping the error below $\\epsilon$ . This number increases with $\\Upsilon$ as does the number of significant eigenvalues. The problem is to find a sequence of parameter values $\\Upsilon {1} < \\cdots < \\Upsilon _{m}$ ,andasetofvectors $u_{1}, \\cdots ,u_{m}$ such that, for each $j, \\Upsilon _{j}$ is the maximum value of $\\Upsilon$ for which the noise can be approximated within the error of $\\epsilon$ by using only $j$ vectors, and $u_{1}, \\cdots , u_{j}$ are the approximating $j$ vectors corresponding to $\\Upsilon _{j}$ The critical constraint is that the set of $m$ approximating vectors be independent of $\\Upsilon$ . In the first solution, the root-mean-square error is used for the error that is to remain below $\\epsilon$ . In the second, the sample error is used but the $\\epsilon$ -approximation is limited to only those noise samples which have nonnegligible average power. In both solutions a recursive scheme is given for obtaining $\\Upsilon _{1}, \\cdots , \\Upsilon _{m}$ and $u_{1} , \\cdots , u_{m}$ , the resultant $\\Upsilon$ -sequence and $u$ -set (orthonormal) are unique. The result is applied to adaptive spatial processing for signal detection in the case where the signal wave, though temporally incoherent, has a known wavefront, the dominant noise ls spatlally localized, and the processor must be nearly opthnum for a wide range of frequencies.