Chapter 7 Multiple pulse detection with random parameters UESTC 1
1 UESTC Chapter 7 Multiple pulse detection with random parameters
Hypothesis Ho Hypothesis H Do Pulses D Pulses or channels or channels Each channel output Each channel output y(()=aoce/mo-ouo (-to)+z). ye(()=avehe-u (-t+), e=1,..,D0 €=1.，D1 Detector Figure 7.1.Conceptual depiction of multiple pulses arriving at detector Model the complex continuous measurements under hypothesis H,and during pulse number e as y,(0）=ae2+8'4,(t-te)+z(),i=0,l=l,,D,(7.1) UESTC 2
2 UESTC (2 ) ( ) ( ) ( ), 0,1, =1 (7.1) i i j t i i i i i t e u t t i D + = − + = f y t z ， ， Model the complex continuous measurements under hypothesis and during pulse number as Hi
7.1 Chapter highlights Pulse number is D Hypothesis Ho D=D1=Do Detector Pulses or channels Hypothesis H Figure 7.2.Conceptual depiction of multiple pulses arriving at detector(Do =D) Consider the situation in Figure 7.2,we have ye()=a,e2t+'4,(t-t)+z(),i=0,l,(=l,,D (7.2) Sampling ye0.1.j=1.. =1.,D(7.3) UESTC where uy (t)=u,(tit) 3
3 7.1 Chapter highlights Consider the situation in Figure 7.2, we have UESTC (2 ) ( ) ( ) ( ), 0,1, =1 (7.2) j t i t e u t t i D + = − + = f y t z ， ， (2 ) ( ) , 0,1, 1, , , =1 (7.3) j j t j ij j e u t i j k D + = + = = f y z ， ， ( ) ( ) ij i j where u t u t t = − Sampling Pulse number is D
In Radar Case:Multiple pulse accumulation In communication case: Different channel,Diversity Noncoherent accumulation: Each pulse contains a different and unknown phase. UESTC 4
4 • In Radar Case: Multiple pulse accumulation • In communication case: Different channel, Diversity • Noncoherent accumulation: Each pulse contains a different and unknown phase. 4 UESTC
·Unknown phase Unknown phase and amplitude Diversity approaches and performances UESTC 5
5 • Unknown phase • Unknown phase and amplitude • Diversity approaches and performances 5 UESTC
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 = =
