Question

1. Let us consider a digital binary communication system, in which the fol- lowing signal s1(t) is transmitted for '0' and signal s2(t) is transmitted for '1' and these two signals are equal probability P('O' is transmitted) P(1' is transmitted). For these two signals and their correspond- ing basis functions, answer the following questions [40 points -2 0t<0.5; 0<t<0.5 -1 0.5 <t< 1 2 s2(t) -1 0.5<t1. s1(t) otherwise 0 otherwise 0 0<t<0.5; -1 0.5 <t1. otherwise 1 10t<1 0 otherwise t) 0 (a) Find the location of the signals in the , 2 coordinate by calcu- lating the projection elements s11, 812, 821, and s22 Note: s1(t)S11¢(t)+ 812%2(t), s2(t)21(t)8222(t) (b) Find the decision rule of the optimum receiver for an AWGN chan nel (c) Draw the decision line in the o, 2 coordinate and show the de cision regions

Answer 1

ANSWER ::

Given that

p('0' is transmitted)= p('1' is transmitted)=1/2

  
The binasy Vawes a be Jwo Signais bepbesen jed as si and Sz(t). They have a Common basis nameiy Sunchong we need Calwiate he oahog os hege to signavs Soncions ma? basis he a) The basis Sunchons ase o%-thogonai we an ind he eiemen b Pessomingn036ahon on a Ptojection ahe basis Sonctong unit me SCale SeJAINAL àt + -1\dt 2 0-5 Su2t o5 2 (05-0) -1-05) SuFO5 simiebi Sitt)e )dt S12 O 2dt 1 - dt o.5 S12

2.o.5-0) l1-0.5) S12.5 dt G.S S21 - dt -2 d S21 2t = -2 (G.S-o) - ¥ (\-0.5) =2 (o-5)- 14 (0.5) 0.5 Selt)2t)dt S22 -1 -1 dt -2 dt 0.S -10.5 -o-5

S22 5 P6ejecion hus, he elements abe S O5 S,2- 15, S2) 5, S22 5 and the ePsession os he signais age Soou S)0-5 (t) + 15 g (t Selt) 5 (A) - 0.5 øe lA) b) -he decision uie o the oPtimum eceives So6 AWGN channei i exPsessed ag Rise13)j (3ls) Rls>= P(3]se )f(S2j whebe Yhe 6eceived Signa and R(Si)R(S2)=0.S The above aeisjon Goie Sates -hat he Po Stegiog 96eates en s2 (l) hen Pobabiiy So Sit) Silt) eise s2 lt). he eceived signa i8 S60m The Psobabiity i8 eipgessed ag a disence fonction ieihood P6oach. 15 he P6ajecion osing maxium oS ase 3, ana 2 So% he basig eiements foncsion then me aisdence Sbom he signeis Can be Comfoted as fOnows

The obabii- 18 exp6essed as S s, -Su) 2- S12) aistances with the Reqiacing he Pobabiitties above Computed S (2 S) 91-s% +(2-S22) Dニ Removngme swate boot and Subsihting me P6ajechon eiements, we 3et Sot me Vaiues -1S 3 Y+0.25+ 92-33q +2.25 > = 9, +33i +2.25+ Stom bom -me sides 2-25, o.25 Remove we Set -y1 332> = 391 +92 Dこ S1 heswise S2 exptession ig the decision ule of he AWGN Chamnel whese ,2 The above oPmum 6eceive Sob an

eiemens oS ahe basis Sonciong abe he Pajecion ime nhe decision The SySem 1 shown below Coodinate Silo5,1.5) Decision ine Blt) O5 2(-1-5,-0.5) The decision uie Comuted in (b) 18 the boundasy two signals ag shown o decision ine befween figuse apove oovnda6y ies on the NOTE The decision ine sA and se (A), "The segio n between sigvial 2 wHtn ignt biue OSSA \g matked dissetensiate he géaph.