M-ary Detection Problem. Lecture Notes 2: Detection Theory. Example 1: Additve White Gaussian Noise

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1 Hi ue Hi ue -ay Deecio Pole Coide he ole of decidig which of hyohei i ue aed o oevig a ado vaiale (veco). he efoace cieia we coide i he aveage eo oailiy. ha i he oailiy of decidig ayhig ece hyohei H whe hyohei H i ue. Lecue oe : Deecio heoy he udelyig odel i ha hee i a codiioal oailiy deiy (a) fucio of he oevaio give each hyohei H. Goal: Oiu Deecio i AWG Oiu Deecio wih uiace (Uwaed) Paaee Pe P i decide H P decide H i i Ri i i Ri i πid πid II- II- he deciio ule ha iiize aveage co aig o R i if i a πi Le e a aiay deiy fucio ha i ozeo eveywhee i i ozeo he a equivale deciio ule i o aig o R i if i a hu fo hyohee he deciio ule ha iiize aveage eo oailiy i o chooe i o ha i i. Le whee πi π, Λ i Chooe i if Λ i i π. he he oial deciio ule i: π fo all i We will uually aue i. (If o we hould do ouce ecodig o educe he eoy (ae)). Fo hi cae he oial deciio ule i Chooe i if Λ i i π. ale : Addive Whie Gauia oie Coide hee igal i addiive whie Gauia oie. Fo addiive whie Gauia oie K δ. Leϕi e ay colee ohooal e o. Coide he cae of 3 igal. Fid he deciio ule o iiize aveage eo oailiy. Fi ad he oie uig ohooal e of fucio ad ado vaiale. i ϕ i whee i ad Va i ad i i a ideede ideically diiued (i.i.d.) equece of ado vaiale wih Gauia deiy fucio. Le ϕ ϕ oe ha he eegy of each of he hee igal i he ae, i.e. ϕ ϕ ϕ ϕ i d i 5. he II-3 II-4

2 we have a hee hyohei eig ole. H : H : H : i i i i i i he deciio ule o iiize he aveage eo oailiy i give a follow Decide H i if i a π Fie u oalize each ide y he deiy fucio fo he oie aloe. he oie deiy fucio fo vaiale i π he he oial deciio ule i equivale o i Decide H i if a π i ϕi ϕi ϕi A uual aue. he π π i 4 5 i i i i i i i ow ice he aove doe deed o we ca le adhe eul i he ae, i.e. Siilaly li i II-5 II-6 φ Deciio Regio φ II-7 II-8

3 ale : Oiu Deecio of -ay ohogoal igal fo iiu i eo oailiy I hi ecio we coide he ole of deecio wih uwaed aaee. o illuae coide he ole of iiizig he i eo oailiy i a -ay ohogoal igal e. Le e ohogoal igal. he eceive coi of a a of ached file (coelao) ha geeae a ufficie aiic. If igal i aied he δ δ η η φ φ δ η φ Le e he equece of i deeiig which of he igal i aied. Aue he i ae ideede ad equally liely. Coide he deecio of daa i. ha i, we ae ieeed i iiizig he oailiy of eo fo daa i. Le H e he eve ha ad H e he eve ha. Le. he he oial eceive u coae he wo aoeioi oailiie H π H H H π II-9 II- o calculae H we oceed a follow. H Siilaly π π πσ πσ πσ πσ π πσ πσ σ σ σ σ σ σ l l l l δl δl l l lδl δl σ σ σ l δ l σ H π πσ π oice ha ay of he faco i aio fo i he log-lielihood aio i log H H π π hi ca e aoiaed y log H H π π H H π π H log a σ l π ad σ σ H σ σ π ae he ae. hu he lielihood σ σ log a σ σ II- II-

4 ale 3: Oiu Deecio of iay igal i fadig chael Coide a ye wih L aea. Aue ha he eceive ow eacly he faded aliude o each aea. he deciio aiic ae he give y l z l η l l whee l ae Rayleigh, η l i Gauia ad eee he daa i aied which i eihe + o -. he ado vaiale l eee he fadig fo he aie o he l deiy l σ e σ L h aea ad ha We aue he fadig o each aea i ideede. he oial ehod o coie he deodulao ouu ca e deived a follow. Le z z L L e he codiioal deiy fucio of z z L give he aied i i + ad he fadig aliude i L. he ucodiioal deiy i z z L L z z L he codiioal deiy of z give ad, i Gauia wih ea l. he oi diiuio of z z L i he oduc of he agial deiy fucio. he oial coiig ule i deived fo he aio Λ z z z z z z z L z L z L z L z L z L L l z l 4 L l L l L L L L L L L l zl l z l l L L L ad vaiace II-3 II-4 he oiu deciio ule i o coae Λ wih o ae a deciio. hu he oial ule i L l l z l oe ha we do o eed o ow he deiy of he aliude fo hi deciio ule. hi deciio ule i called aiu aio coiig (RC). I he ecial cae whee hee i u oe aea he oiu eceive educe o z hu he oiu eceive fo u oe aea (ad BPSK) doe o eed he ifoaio aou he eceived aliude o ae a (had) deciio. Howeve, he efoace deed ciically o he diiuio of he fadig aliude. Fo he Rayleigh faded cae he eo oailiy ecoe P e Ē Ē z Lielihood Raio fo Real Sigal i AG Aue wo igal i Gauia oie. H : H : Goal: Fid deciio ule o iiize he aveage eo oailiy. Le have covaiace K eigefucio ϕi wih ad eigevalue. We aue ha i a zeo ea Gauia ado oce. he eige fucio ϕ i ae ohooal fucio ad eal ue uch ha (ee Aedi) K d ϕi λiϕi II-5 II-6

5 d d By Kahue-Loeve aio Kahue-Loeve aio i i ϕ i whee i i Gauia ea i vaiace. i i i π πλi i i i i whee i ae Gauia ado vaiale wih ea vaiace ad i i ideede i eal). Sice ϕ i ae a colee ohooal e ad we aue ha fiie eegy we have hu Defie H : Λ i iϕ i. Λ i i i i i i i li ϕi i Λ i Le Λ l l li πλi πλi i i i ϕ i i i i l i i i i iϕ i i i l i i i i i l i II-7 II-8 he hu Λ l li Λ l i i l ϕ l i l i i ϕ i d l iϕ i ϕ l d ϕ i ql ϕl d ql So K d i ϕ i K K d If he oie i whie, he he oie owe i each diecio i coa (ay λ) ad hu he oial eceive he ecoe o equivalely Λ l λ i λ ϕ i λ oe: i oluio of he iegal equaio Λ l λ II-9 II-

6 Lielihood Raio fo Cole Sigal Fo equal eegy igal hi aou o icig he igal wih he lage coelaio wih he eceived igal. he oial eceive i owhie Gauia oie ca e ileeed i a iila fahio a how elow. hu Λ l K K K K K K K K K K K I i clea he ha hi iu he oial file fo igal K whe eceived i addiive whie Gauia oie. hi aoach i called whieig ecaue K will e a whie Gauia oie oce. I hi ecio we edeive he lielihood aio fo cole igal eceived i cole oie. We aue ha he igal ae he lowa eeeaio of ada igal ad he oie i he lowa eeeaio of a aowad ado oce. Le H : H : whee ha covaiace K, wih eigefucio ϕ i, eigevalue. Uig Kahue-Loeve aio we have i πλ e l e π H i : l l l i l i l l λl λl i l il ϕ λl II- II- Le i ϕ i he l l l l l i l l Re Re i l l a l l ϕ l λl l l l i l λl a l l l ϕ l λ l ϕ ϕ d d i l Re l i l So Λ i li H Λ i i i oe: Sice we ae dealig wih oie ha i deived fo a aowad ado oce we ca o ue he eul deived fo eal ado ocee we u ue he lielihood aio fo cole ado oce give aove. Fo eal ado oce he lielihood aio i Λ i Fo addiive whie Gauia oie (eal) q i i ϕ λ i Re i i ϕ q II-3 II-4

7 So he lielihood aio (fo eal igal) ecoe Λ lli l H H l ale: ohogoal igal i addiive whie Gauia oie I hi ecio we coide he oiu eceive fo -ay ohogoal igal ad he aociaed eo oailiy. Aue he igal ae equieegy igal ad equioale. he deciio ule deived eviouly fo AWG i Decide H i if i i Aue π. he α. A equivale deciio ule he i H l H H l H he oiu deciio ule fo addiive whie Gauia oie i he o chooe i if i i ow ice he igal ae ohogoal ad equieegy we ca wie hi a he fi e aove i coa fo each a i he la e. hu fidig he iiu i equivale o fidig he aiu of hu he eceive hould coue he ie oduc ewee he diffee igal ad fid he lage uch coelaio. If he igal ae all of duaio, i.e. zeo ouide he ieval he hi i alo equivale o fileig he eceived igal wih a file wih ilue eoe, alig he ouu of he file a ie ad chooig he lage a how elow. II-5 II-6