A Minimum Variance Method for Problems in Radio Antenna Placement M.V. Panduranga Rao Amrit Ahuja, V. Srinivasan, Kavita Iyer, Ranu Khade, Sachin Lodha, Dinesh Mehta, Balasz Nagy TCS Public
Outline • The Setting • Some Problems in Radio Antenna Placement • The Minimum Variance Method • Application to the Problems. • Questions for Future
The Setting • N antennas on the XY-plane (Earth Centered Coordinate system). • Each pair of antennas forms a baseline and generates a “UV-” point (a point on the sampling plane). • Therefore, we have NC2 UV-points; each generating a spatial correlation coefficient. • Image= FFT-1[spatial correlation vector]T when the UV-plane is perpendicular to the line joining the source.
XY to UV planes UV Plane XY Plane
Some Previous Work • Treloar (1988): Experimental examination of various configurations (Non-Random arrays and Random Arrays). An interesting empirical observation: A random antenna placement yields a UV distribution that tapers radially. • Keto (1997): Slightly perturbed curves of constant width like the Rayleigh triangle give the “most complete” UV coverage. • Lonsdale and Cappallo (1999): Advocates large N Arrays. Tries out (a) Single Arm Log Spiral and (b) Log Spiral in the Centre and pseudo random star for the outer regions. Report excellent UV coverage in the inner regions and economical in cable length. (~πD for the first).
Previous Work (contd) • Boone (2001): Compute pressure forces related to the discrepancy between the desirable distribution and the actual distribution. Move antennas so as to minimize the forces. • Cohanim, Hewitt and Weck (2004): Genetic Algorithm to solve the multi objective optimization problem of maximizing imaging performance and minimizing wire length costs. • Su, Nan, Peng (2004): Drop those antennas that contribute to the denser regions. Assign to a UV point a weight that is equal to its distance from the nearest UV point. An antenna has a weight equal to the sum of all relevant UV points. Remove the lightest weight antenna.
Previous Work (contd) • Karastergiou, Neri and Gurwell (2006): Several antenna pads, fewer antennas. Find all configurations and keep the best. • deVilliers (2007): UV Samples are projected to on to a one dimensional vector. Compared with projections of ideal distributions. Correction terms yielded, which are then translated to new antenna positions. Done in all directions until close match occurs.
Parameters of the Minimum Variance method: • Given: (1) A number M of potential antennas sites. (2) N antennas. • Output: N of the M sites on which to place antennas such that some criterion for UV distribution is met.
The Algorithm, Intuitively: • Removes M-N antennas from the M antennas iteratively. • Divide the UV plane into regions. • Removal of an antenna results in the removal of several UV points. Rir = the number of points that should be ideally removed from the ith region in the rth iteration (“ideally” decided by any logic). wij = the number of points actually removed from the ith region because of the removal of antenna j. • Remove the antenna that minimizes the discrepancy between the expected and the actual.
The Algorithm, Pictorially UV Plane XY Plane
Many points removed from the same region: Bad! UV Plane XY Plane
Remove a different antenna: XY Plane UV Plane
More Uniform Removal: Better! XY Plane UV Plane
The algorithm, Formally: Initially, there are M antennas. Draw p Regions while there are more than N antennas remaining do for each remaining antenna j Calculate Var(j)= ∑i=1p(wij-Rir)2 Remove the antenna that has least Var(j) end while • Solution O((M-N)M3) time.
For dense UV plots, a Gaussian along the radius and a uniform distribution across the azimuth is good for most experiments. • A random distribution of antennas on the XY plane yields a tapering distribution: • 1. Inherently approximate; • 2. Some antenna locations might be infeasible, and hence have to be discarded • Can we decrease the discrepancy between the desired Gaussian and the distribution generated by randomly placing the points?
M antennas to be constructed in all • However construction is staggered (only N sanctioned in the beginning): • Financial reasons: staggered funding • Logistical reasons: sheer scale of the telescope • Wish List- • Telescope should start functioning in the interim • Image quality should be as close to the final as possible • Upgrading should be graceful as new antennas are added
Not Working 2 1 3 4 5
Five antennas demanded at specific locations. • One down, only four available. • Now what? • Minimum Variance to the rescue! • Pick the four sites that best approximate the UV distribution generated by the five demanded. • If one of the four sites carries the dysfunctional antenna, move the good antenna to that site.
Minimum Variance: These four are almost as good.
Not Working 2 1 3 4 5
Future Directions: Wire-length Minimization Problem • Estimated communication cost for SKA = 330 Million Euros including fibre and trenches. • Minimizing cable length without sacrificing scientific merit. • Approach 1: Step 1: Compute antenna placement purely on scientific merit. Step 2: Optimize wire-length for the placement thus obtained. • Approach 2: Simultaneously optimize for wire-length and scientific merit.
Wire-length Minimization Problem (contd) • Multi-objective optimization (minimizing both trench and fibre costs). • Minimizing total trench-length precisely the Euclidean Steiner Tree problem. • EST is NP-hard. • Approximation algorithms?