The signal vectors s1 and s2 are used to represent binary symbols 1 and 0, respectively, in a binary FSK system using coherent detection. The receiver decides in favor of symbol1 when
where xTsi is the inner product of the observation vector x and the signal vector si , i = 1, 2.
Show that this decision rule is equivalent to the condition x1 > x2, where x1 and x2 are the two elements of the observation vector x.
Assume that the signal vectors s1 and s2 have equal energy.