Efficient approximation of a family of noises for application in adaptive spatial processing for signal detection
Alfons J. Claus, Toshihito Kadota, D. Romain · IEEE Transactions on Information Theory · 1980
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 withmrandom 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 ofmvectors that are independent of\Upsilon. Furthermore, for each\Upsilonthe number of approximating vectors is to be mlnlmlzed while keeping the error below\epsilon. This number increases with\Upsilonas does the number of significant eigenvalues. The problem is to find a sequence of parameter values\Upsilon{1} < \cdots < \Upsilon_{m},andasetofvectorsu_{1}, \cdots ,u_{m}such that, for eachj, \Upsilon_{j}is the maximum value of\Upsilonfor which the noise can be approximated within the error of\epsilonby using onlyjvectors, andu_{1}, \cdots , u_{j}are the approximatingjvectors corresponding to\Upsilon_{j}The critical constraint is that the set ofmapproximating 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}andu_{1} , \cdots , u_{m}, the resultant\Upsilon-sequence andu-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.