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An extension of Green's condition to cross-ambiguity functions

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The clear region in which a cross-ambiguity function is identically zero is investigated. The following theorem is proved, which limits the size of the permitted clear region. If the origin in the \tau , \phi plane is at the center of a centrally symmetrical convex region, which is bisected along one of its diameters, and if some cross-ambiguity function \chi _{u \upsilon }(\tau , \phi) is non-zero at the origin, and identically zero elsewhere in one of the halves of this region, then the area of the whole region cannot exceed 4 . The theorem is proved by considering the cross-ambiguity function to be part of a communication channel. The clear area cannot be larger than indicated; if it were, this channel could be made to decode more information than its channel capacity permits.

Published in:

IEEE Transactions on Information Theory  (Volume:13 ,  Issue: 1 )