I'm not sure what you are getting at here.
Rotating the feed around an axis perpendicular to the beam is likely
just to increase noise and reduce signal, but still take the same signal
from the dish.
Translating it perpendicular to the beam will move you across the
focal plain and pick up different directions but won't reduce the gain
a lot for small movements. However it will mean that you are no longer
picking up the original direction so the integrated signal from that
direction will stop accumulating, but the noise will continue to do
so. (You should really tilt the feed to keep it pointing at the dish
centre, as well.)
Translating it just fast enough to compensate for the earth's rotation,
will actually keep the source in beam longer and allow longer integration
times. I believe this is what Arecibo does.
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