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Efficient approximation of a family of noises for application in adaptive spatial processing for signal detection

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3 Author(s)

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.

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