5 Using vector notation for the k samples,we can write the column vector as =i,exp(J阝，)+z Assume the noise is zero mean and white,then P,(|B)= o2=o/2 dap5-ameT-4 UESTC 16
16 UESTC Using vector notation for the k samples, we can write the column vector as exp( ) i i y u J z = + y Assume the noise is zero mean and white, then * * 2 2 ( | ) 1 1 exp (2 ) 2 i i i i i T J J k i p y y u e y u e − = − − − 2 2 = 2 z
P,(|B,)= 2dara5a[g-a 」取点a 2) esn台交gom-iean-号2 i=1 UESTC 17
17 UESTC 2 2 2 1 2 2 * 2 2 1 1 1 1 ( | ) exp( ) (2 ) 2 exp[ cos( )] exp( ) 2 k i i j k j k k j ij i i ij j j p y y y u u = = = = − − − * * 1 i k T J i j ij j y u y u e = = * * 2 2 ( | ) 1 1 exp (2 ) 2 i i i i i T J J k i p y y u e y u e − = − − −
Assume the random phase is uniformly distributed with p(B,)= 0≤B,≤2π Acording t4（-2edA we have Modified Bessel function of order zero p)= w,以g2 i=0,18 UESTC
Assume the random phase is uniformly distributed with According to , we have 18 UESTC 2 * 2 2 2 0 1 1 2 2 2 1 1 1 ( ) exp( ) ( ) (2 ) 2 exp( ), 0,1 2 k k i j j ij k j j k ij j p y y I y u u i = = = = − − = 2 cos( ) 0 0 1 ( ) 2 i i x i I x e d − = 1 ( ) , 0 2 2 i i p = Modified Bessel function of order zero
From these density functions,the log-likelihood ratio is bw=-hg器交d D 空w小品-u-u where Do 山-n片空4小安。 i=0,1 UESTC 19
19 UESTC From these density functions, the log-likelihood ratio is 1 0 2 2 * 0 1 1 2 2 1 1 2 2 * 0 0 0 0 2 2 1 1 ( ) ln ( ) ln ( ) 2 ln ( ) ln 2 k k j j j j j D k k j j j 1 B j j D y L y I y u u I y u u u u = = = = = = − − + = − where 2 2 * 0 2 2 1 1 ln ( ) , 0,1 2 k k i j ij ij j j u I y u u i = = = − = 2 2 i
The optimal non-coherent receiver can be shown as: MAC In I() 4 a/o2 MAC In lo() &o2 a21o2 站 UESTC 20
20 UESTC The optimal non-coherent receiver can be shown as: