Joined: 13 Feb 99
While thinking about eliminating redundant multiplets across pixels (see the last blog entry) I realized that the way we find multiplets is "pixel-centric": it's more sensitive to ET signals whose sky position is near the center of a pixel. Details are below. Bottom line: I couldn't think of an easy way to address this, and it may not be worth addressing.
The Arecibo beam has a half-power width (diameter) W of .05 deg.
The "reported position" P(S) of a signal S is the center of the beam at the time of S. Let Q(S) denote the "true position" of S (assuming that S is of celestial origin).
Nebula assumes that for most signals, angle(P(S), Q(S)) < W. In other words, the true position is usually in the disc centered at the reported position, with twice the diameter of the half-power disc. This is true for signals whose power is close to the noise floor.
This assumption means that signals S1 and S2 may have the same true position if and only if angle(P(S1), P(S2)) < 2W.
Nebula's multiplet-finding algorithm work on pixel-by-pixel basis. For each pixel we form a "signal disc" with radius W centered at the pixel center. This has the desirable property that if S1 and S2 are in the signal disc, angle(P(S1), P(S2)) < 2W, so S1 and S2 may have the same true position. Hence, for any multiplet we form from the signal disc, it's possible that all its signals have the same true position.
Because of this, the multiplet-finding algorithm doesn't need to take position into account when selecting signals to add to a multiplet; it doesn't give preference to closely-spaced signals. Position is a factor in multiplet scoring (variance in position reduces the score) but not in multiplet formation.
The problem with current scheme
The current scheme can't find some potential multiplets, because they're not contained in any signal disc.
Suppose the true position of an ET signal is the center of a pixel. Then, for any signal S that's the result of the ET signal, the reported position of S will lie in the pixel's signal disc. If the multiplet-finding algorithm works like it's supposed to, we'll group most or all of these signals into a multiplet.
However, suppose the true position is not at the center of a pixel; for example, suppose that it's at the corner of a pixel. The disc of radius W centered at this point will overlap 4 signal discs, but it won't be contained in any of them. The signals resulting from the ET signal may not be contained
in any of the 4 discs, so we won't find the best possible multiplet; at best we'll find 4 lower-scoring multiplets.
In theory, we could do multiplet-finding separately for every signal S: form the signal disc of radius W centered at S, and look for multiplets in that. This isn't feasible; it would involve 10 billion signal discs instead of 16 million.
We could continue to use signal discs centered at pixel centers, but increase the radius enough to include everything within W of the pixel. This radius would be W plus the maximum half-diagonal of a pixel, which is about .025 degrees. There are two problems with this:
Joined: 20 Apr 00
I'd always assumed the pixel was the unitary unit, but now it seems WUs are compared according to their position within a pixel.
Is this perhaps an argument to have a smaller pixel size and then not worry about within-pixel differences?
Joined: 24 Jan 00
Maybe a pixel should be broken down to a 100th or 1000th grid just to get more accuracy into the equation.
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