# notes9 - ECE 562 Fall 2011 Coherent Detection of...

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ECE 562 Fall 2011 Coherent Detection of Orthogonally Modulated Signals The signal set is described as: s m ( t ) = E g m ( t ) , m = 1 , 2 , . . . , M, 0 t T s Here we consider the case where { g m } are orthonormal signals, i.e., ρ k = 0, for k = (both in real and imaginary parts). For the case where only Re [ ρ k ] = 0, for k = , see HW 4. The signal s m ( t ) = E g m ( t ) belongs to span( g 1 ( t ) , . . . , g M ( t )), and hence R k = r ( t ) , g k ( t ) , k = 1 , . . . , M , form sufficient statistics. If symbol m is sent, then the vector of sufficient statistics is given by R = [ W 1 · · · E + W m · · · W M ] where { W k } are easily seen to be i.i.d. CN (0 , N 0 ) random variables. The likelihood function is given by: p m ( r ) = 1 πN 0 exp - ( r m,I - E ) 2 + r 2 m,Q N 0 k = m 1 πN 0 exp - r 2 k,I + r 2 k,Q N 0 We showed in class that for equal priors, ˆ m MPE ( r ) = ˆ m ML ( r ) = arg max m p m ( r ) = arg max m r m,I Note that only the real part of r contains information about the signal and hence the test only uses the real part.

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