We assume that the ARA is so far from the pilot source that angles a and a’ are infinitesimal. That being so, we may write We assume [], hence []. Therefore, the peak of the dowlink beam, i.e., the direction, if it exists, for which [] for all j, does not point towards 0, the pilot source. Let’s assume that = 0 at some other point, [], which is also at distance [] from the reference element but displaced from 0 by the small angle []. The distance from the kth element to [] is Let [] be the phase of the retrodirected signal from the kth element as observed at []. By our assumption, Since 0 and 0’ are equidistant from the reference element, []. Therefore, So, by (A15),
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