In this design, the phase difference is reduced further by a factor e, e.g., A0 is only With A/ = 50 MHz and J Nds = 5 x 1017 electrons/m2, this difference is only about 0.04°. Even with A/ = 100 MHz, this difference is only 0.66°, which is still very small. In all these three designs the uplink frequencies are all on the lower side of the downlink frequency and they are all equally spaced. These are not the only choices. One can use all frequencies on the upper side of the downlink frequency and they need not be equally spaced either. As a simple example one may very well have a two-tone uplink with and </> = 3d>(f') — It will work just as well. So far we have used the ionosphere as an example to show how the dispersion of the transmission medium could introduce sizable biases between the uplink and downlink phases. For a two-tone uplink signal, this phase difference is about 8°. We suggested one way to suppress it is to use a three-tone uplink. However, it turns out that if the biases were purely due to the ionosphere, it is really not necessary to use a three-tone uplink. This is because even though the phase difference between and in our two-tone uplink design is large, its variation from subarray to subarray will be very small. To elaborate on this statement, from Eq. (6) and the definition of B. the difference between the uplink and downlink phases from the pilot to the Ath subarray for a two-tone uplink is where f Ndsk is the columnal electron density along the path from the pilot to the Ath subarray. Since the horizontal dimension of the region of the ionosphere that would be transversed by the pilot signal to any subarray is very small, typically of the order less than 100 m, the transverse variation of the total electron along a path of several hundred kilometers within a tube of diameter less than 100 m would be very small. Hence even though Ad>fc is estimated to be about 8°, its variation is at least several orders less. Then with the use of the phase from one of the subarrays as the reference phase, this difference A^*. can be subtracted out and the remainder be treated as a constant phase offset which has no effect on the retrodirective beam. In this sense, though the ionosphere is dispersive, it does not cause any problem, and a two-tone uplink is sufficient. On the other hand, with the use of “central phasing” we cannot avoid the extra path length of transmission lines for some subarrays. Since these lines are not dispersionless, they will introduce a sizable phase difference. This phase difference for a two-tone uplink could be very large but this number can be greatly reduced in a three-tone or four-tone uplink. The use of these multitone uplink signals can be an alternative method to suppress biases due to the dispersion of the transmission line and the medium.
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