Walsh Function Choices for 64 Antennas Abstract
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ALMA Memo # 586
Walsh Function Choices for 64 Antennas
Darrel Emerson
2009-03-12
ALMA Memo #537 studied the loss of orthogonality between pairs of Walsh functions, when one function had undergone a small time shift with respect to the other. It also showed that 5461 of the 8128 possible cross-product pairs from a set of N=128 Walsh functions remain perfectly orthogonal in the presence of a time shift. This memo shows that in general a fraction (N2)/3 of all possible cross-products of a set of N Walsh functions remain orthogonal in the presence of a relative time slip. This memo investigates the optimum choice of a subset of M functions, corresponding to M antennas, from a complete set of N, with the aim of minimizing crosstalk between antennas in the presence of electronic timing errors. ALMA has already adopted N=128, and it is found here that there would be relatively little gain for additional effort required to implement N=256 or greater. This Memo concentrates in particular on arrays of M=64 antennas, but the same approach may be used for any number of antennas. Different optimization strategies are examined, aimed at maximizing the number of zero cross-products, at minimizing the cumulative crosstalk level, and at minimizing the loss of sensitivity. Minimum crosstalk is not obtained by maximizing the number of zero cross-products, nor does the set of 64 functions chosen from 128 having lowest cumulative crosstalk include the highest number of zero cross-product pairs. Specific results and recommendations are given; the favored subset of functions for 64 antennas is WAL 0-31, 47-63 & 113-127.
View a pdf version of ALMA Memo #586.
