APPENDIX A DOPPLER EFFECTS IN ACTIVE RETRODIRECTIVE ARRAYS We wish to calculate the phase error in the retrodirected signal produced by the radial component of the motion of the individual elements of the ARA. Accordingly, we denote the distance from the pilot source to the kth element of the ARA by [] where t is time. We assume that the array is centrally phased. Let k = 0 denote the reference element, and [] the length of the transmission line between the kth element and its PCC located at the reference element. We also assume, to simplify the algebra of our analysis, that unlike [], is fixed. We show later that this assumption does not greatly affect our result. The plan of our analysis is to calculate the phase of the conjugated signal transmitted by the kth element as seen by a receiver at the pilot source. To do this we calculate the phase of the signal at each stage of its round trip: from the pilot source to the kth element, down the transmission line to the kth PCC, and, after conjugation, back via the same path to the receiver at the pilot source. We then derive an expression for [] the phase difference between the conjugated signals arriving at the pilot source from two different elements, the jth and kth. Finally, we calculate upper bounds for 16, | as a function of the radial velocities of the elements. Since a perfectly retrodirective array means [] and [] is a measure of the phase error due to element motion. The phase of the pilot signal at time t at the pilot source (r = 0) is cot. Therefore, the phase of the pilot signal received by the kth element at time t is
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