Two solutions are presented to the problem of efficiently approximating a family of noises parameterized by a scalarUpsilon, 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 withUpsilon. The noise sample vector is to be approximated, within a specified errorepsilon, by a linear combination of vectors taken from a fixed set ofmvectors that are independent ofUpsilon. Furthermore, for eachUpsilonthe number of approximating vectors is to be mlnlmlzed while keeping the error belowepsilon. This number increases withUpsilonas does the number of significant eigenvalues. The problem is to find a sequence of parameter valuesUpsilon{1} < cdots < Upsilon_{m},andasetofvectorsu_{1}, cdots ,u_{m}such that, for eachj, Upsilon_{j}is the maximum value ofUpsilonfor which the noise can be approximated within the error ofepsilonby using onlyjvectors, andu_{1}, cdots , u_{j}are the approximatingjvectors corresponding toUpsilon_{j}The critical constraint is that the set ofmapproximating vectors be independent ofUpsilon. In the first solution, the root-mean-square error is used for the error that is to remain belowepsilon. In the second, the sample error is used but theepsilon-approximation is limited to only those noise samples which have nonnegligible average power. In both solutions a recursive scheme is given for obtainingUpsilon_{1}, cdots , Upsilon_{m}andu_{1} , cdots , u_{m}, the resultantUpsilon-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.

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Information Theory, IEEE Transactions on  (Volume:26 ,  Issue: 5 )

Sep 1980