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

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.