The equivalence of two test statistics for an optimum differential phase-shift keying receiver

Two recent papers [by J.J. Bussgang, and M. Leiter, "Error rate approximations for differential phase-shift keying," ibid., vol CS-12, Mar 1964, pp 18-27 and J.G. Proakis, P.R. Drouilhet, Jr., and R. Price, "Performance of coherent detection systems using decision-directed channel measurement," ibid., vol CS-12, Mar 1964, pp 54-63] presented test statistics for an optimum Differential Phase-Shift Keying (DPSK) receiver. In this correspondence the authors show that these test statistics are equivalent.