  We present an improved system for orbit and betatron phase measurement
utilizing nonlinear models of BPM pickup response. We first describe the
calculation of the BPM pickup signals as nonlinear functions of beam position
using Green's reciprocity theorem with a two-dimensional formalism. We then
describe the incorporation of these calculations in our beam position
measurements by inverting the nonlinear functions, giving us beam position as a
function of the pickup signals, and how this is also used to improve our
measurement of the betatron phase advance. Measurements are presented comparing
this system with the linearized pickup response used historically at CESR.
