Multiple starting points for phase determination are created by the magic integer method which is described in [39]. The values of phases, , may be expressed in the form of a set of equations by multiplying a variable by successive magic integers, in each equation,

for . Any set of phases with values between and may be approximately represented in this way as a set of values ranging from and with the appropriate choice of , where also takes values between 0 and 1. There will in general be a residual error, , in the representation of which depends on the choice of magic integers [79] [80]. The r.m.s. error in the phase representation is .

A starting set of phases is produced by choosing several values of at intervals which permute the phases such that the mean change in , as x is incremented, is approximately equal to . The advantage of magic integer permutations over quadrant permutations (where each phase takes one of the four values, , , or ) is the large reduction in the number of permutations required to efficiently sample the -phase space. Suitable magic integers have been found to be those of the Fibonacci series defined as (), with .

Fri Oct 7 15:42:16 MET 1994