Illustrative Problem 4.1 Output of correlator(Error free case) dr So(t) r(t)= S,(t) So(t) Od: 3/2 3/2 Odr r(t)= s (t) S1(t) Odr
Illustrative Problem 4.1 ◼ Output of correlator (Error free case) 0 ( ) t d 0 ( ) t d r(t)= S0(t) 0 1 ( ) ( ) s t s t Tb Tb /2 0 ( ) t d 0 ( ) t d r(t)= S1(t) 0 1 ( ) ( ) s t s t Tb Tb /2
Illustrative Problem 4.1 ■Noise components AWGN:E[n()]=0,E[n(t)"n(t)]= No 2 no,n are Gaussian process With zero mean n]=E可s,(r)m(r)dr=∫s,(e)Lrdr]=0 En]=E可s()nr)dr]=∫s(r)Erdr]=0 ■And variances o=En]=E可∫s)s(r)n0n(r)dida]=∫∫s()s,(a)E[nu)n(didr =六50x(e-h-0h=3i=l2
Illustrative Problem 4.1 ◼ Noise components ◼ AWGN: ◼ n0 , n1 are Gaussian process ◼ With zero mean ◼ And variances 0 [ ( )] 0, [ ( ) ( )] 2 T N E n t E n t n t = = 0 0 0 0 0 1 1 1 0 0 [ ] [ ( ) ( ) ] ( ) [ ( )] ] 0 [ ] [ ( ) ( ) ] ( ) [ ( )] ] 0 b b b b T T T T E n E s n d s E n d E n E s n d s E n d = = = = = = 2 2 0 0 0 0 0 0 0 2 0 0 [ ] [ ( ) ( ) ( ) ( ) ] ( ) ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) , 1, 2 2 2 2 b b b b b b T T T T i i i i i i T T i i i E n E s t s n t n dtd s t s E n t n dtd N N N s t s t dt s t dt i = = = = − = = =
lllustrative Problem 4.1 p(r lso(t)) pdf of ro and r p(n lso(t)) e(6-d/2o e1212o2 2πo ro(t) V2π 5o(I) r(t)= S,(t) So(t)+n(t) Od: r1() Odr p(s(t)) ro(t) p(rls(t)) So(t) r(t)= S1(t)+n(t) Odr ri(t) 3
Illustrative Problem 4.1 ◼ pdf of r0 and r1 2 2 0 0 0 ( ) / 2 ( | ( )) 1 2 r p r s t e − − = 0 ( ) t d 0 ( ) t d r(t)= S0(t)+n(t) 0 1 ( ) ( ) s t s t r 2 2 1 1 0 / 2 ( | ( )) 1 2 r p r s t e − = 0 ( ) t d 0 ( ) t d r(t)= S1(t)+n(t) 0 1 ( ) ( ) s t s t r 0 1 p r s t ( | ( )) 1 1 p r s t ( | ( )) r0(t) r1(t) r0(t) r1(t)