7.2 Unknown phase Ignore any unknown frequency and time-of- arrival,and assume the attenuation is known y=ae+z,i=0,1j。=l,k, 0=1,.,D(7.4) The joint conditional pdf of all the measurements 寸={y1y61y12,Jk,2,y1…yk,y1DykD}（7.5) under hypothesis H,is Each pulse is independent plA-2an2a-a元eU-a元e） (7.6) UESTC Bp={B…,BD} 6
6 7.2 Unknown phase Ignore any unknown frequency and time-ofarrival, and assume the attenuation is known The joint conditional pdf of all the measurements under hypothesis is 6 UESTC , 0,1, 1, , , =1 (7.4) j j ij j y z = + = = e u i j k D ， ， 1 2 11 1 12 2 1 1 { , , , , , , , , , , , , , } (7.5) D = k k k D k D y y y y y y y y y Hi 2 2 1 1 1 ( | ) exp [ ] [ ] (7.6) (2 ) 2 D j j T i p i i k p y y u e y u e − = = − − − 1 { , , } p D = Each pulse is independent
Further assume all phases are uniformly distributed in0≤B,≤2π，From Eq.(6.28),then -r2 r晋2- (7.7) From Eq.(6.30): D m=n-字-2 (7.8) Do =h是交器2片=0=l 0(7.9 UESTC 7
7 Further assume all phases are uniformly distributed in , From Eq.( 6.28), then UESTC0 2 βl 2 2 2 1 1 2 2 0 2 2 1 1 1 1 ( ) exp | | (2 ) 2 exp | | , 0,1 (7.7) 2 D k i j k j k k j ij ij j j p y y I y u u i = = = = = − − = 1 0 1 1 0 0 1 1 ( ) ln ( ) ln ln (7.8) ( ) D D D B D p y L y U U p y = = = = − 2 2 0 2 2 1 1 ln | | , 0,1, 1, , (7.9) 2 k k i j ij ij j j U I y u u i D = = = − = = From Eq.( 6.30):
In the commonly assumed case of a binary communication system,the sum of all the bias terms is equal for both hypothesis,i.e., 2 (7.11) 0-1 0-1 where: ,2%尺、i-0Lt=1D i。= UESTC 8
8 UESTC 2 2 2 2 1 0 1 1 (7.11) D D = = = In the commonly assumed case of a binary communication system, the sum of all the bias terms is equal for both hypothesis, i.e., 2 1 1 = | | , 0,1, 1, , 2 k i ij j u i D = = = where:
2→o 36 a6() 9 aisu/o -9 In() 9 D D plaiD apeip/iD Pick largest 4wg-回-9- n6() 4o1 ajsolon wg回9 n6() Lop D aplaip a☑eop/oip For ML criterion Figure 7.3.Optimal noncoherent detector with diversity UESTC 9
9 UESTC For ML criterion 2 2 zi i →
7.2.1 Square-Law Processing If a value x is very small (low SNR),then nl,x)≈4 (7.12) therefore 2 ,i=0,1,0=1,D (7.13) Square-Law noncoherent detector. D For ML criterion 0=1 <0=1 Do 10 UESTC
10 7.2.1 Square-Law Processing If a value x is very small (low SNR), then therefore 10 UESTC 2 0 ln ( ) (7.12) 4 x I x 2 2 2 2 1 y , 0,1, 1, , (7.13) 2 k i j ij i j u i D = − = = U Square-Law noncoherent detector. 1 0 1 0 1 1 For ML criterion D D D D U U = =