the hypothesis that the channel 39 s delay doppler response begins at u03c4 0 u03bd 0 in the delay doppler plane and extends over the range of its maximum delay doppler spread since the channel response h u03c4 u03bd is not known it is a nuisance variable that needs to be removed this may be accomplished by applying the same method used in the generalized likelihood ratio test glrt the first step is to estimate the nuisance variable h u03c4 u03bd assuming the hypothesis u03c4 0 u03bd 0 is correct from eq 6 this is given by for u03c4 0 u2266 u03c4 u2266 u03c4 0 u03c4 max and u03bd 0 u2266 u03bd u2266 u03bd 0 u03bd max the next step is to replace the true channel response with its estimate in the log likelihood function u03bb u03c4 0 u03bd 0 u2212 u222b r t u2212 u222b u03bd 0 u03bd 0 u03bd max u222b u03c4 0 u03c4 0 u03c4 max u0125 u03c4 u03bd s p t u2212 u03c4 e j2 u03c0 u03bdt d u03c4d u03bd 2 dt 9 after rearranging and removing irrelevant terms the log likelihood function for the hypothesis becomes in the generalized maximum likelihood sense the optimal likelihood measure for such hypothesis is thus given by equation 10 where i u03c4 u03bd is the delay doppler image observed by the hypothesized signal s p t the detection of the pilot pattern is then to evaluate and compare the values of this log likelihood function among possible hypotheses of s p t and u03c4 0 u03bd 0 or to compare them with a certain threshold direct evaluation of u03bb u03c4 0 u03bd.