Detekcija

1 Detekcija medijum Izvor (predajnik) prijemnik Jednozračno vjerovatnoća prostor Preslikavanje prelaska odlučivanje Opšti model detekcijeodlučivanja Izvor medijum prijemnik Model

- iz konačnog skupa mogućih porukadogađaja prostor događanja (event space) izvor emituje jednu u trejanju simbola

- jednoznačno se predstavlja signal (iz prostora signala) i emituje (može i nekoliko sukcesivnih poruka) predstaviti jednim signalom

bitna jednoznačnost Vjerovatnoće pojedinih poruka mogu biti poznate na prijemu ali ne moraju Medij kanal smetnja Signal je podvrgnutldquoprobablističkoj transformacijildquo

opisana - vjerovatnoćom

Prostor događanja

poruka

Prostor odlukaProstor opažanja mjerenja

Prostor signala

+

- gustinom vjerovatnoće odgovarajućih skupova signala Skup primljenih signala prostor opažanjamjerenja (Observation) prijemink na osnovu bdquopravila odlučivanjaldquo (decision rule) primljenom signalu pridružuje se jedan od mogućih simbola (poruka) u prostoru bdquoodlukaldquo problem detekcije singularan neka oblast za ostale koju je vjerovatnoća hipotza nenulta Oblast preklapanja Greške cijena greške Određuje kriterije (zavisno od podataka cilja)

Odluka o hipotezama i odabiranju hipoteza Na osnovu Na osnovu Jednog mjerenja višestrukog mjerenja Primjer I Gaussova raspodjela sa srednjom vrijednošću -mo

Uz iste varijanse -m1 jednostavna (hipoteza) Primjer II binarni prijenos u osnovnom opsegu

- u prisustvu n(t)Gaussov šum - neka je jedan nivo Bit 0 mo= 0 - neka je drugi nivo nepoznat m1

alternativna hipoteza Gaussova raspodjela nepoznate srednje vrijednosti Složena (Composite) hipoteza

Moguće je da obadvije hipoteze budu složene (nepoznato mo ili je nepoznata i snaga šuma) Zaključak kod složenih hipoteza mogu se znati tipovi gustine raspodjele i neki njihovi parametri (ali ne svi) ili se čak ne znaju ni tipovi raspodjele

Model optimalnog pravila odlučivanjadetekcije

- mogu da budu poznate apriorne vjerovatnoće hipoteze (binarne ili višestruke)

- gustine vjerovatnoće pod pojedinim hipotezama (jednostavne - poznati svi parametri)

- cijene grešaka (ili čak i cijene za slučaj kada nema grešaka pri odlučivanju)

implicira različite kriterijume odlučivanja 1 Jedno osmatranja

- Bajesov kriterijum - Kriterijum maksimalne aposteriorne vjerovatnoće i

maksimalne vjerodostojnosti (najboljeg izbora) - Minimaksni kriterijum - Nejman- Pirsonov kriterijum

2 Odlučivanje na osnovu višestrukog osmatranja koncept prostih hipoteza gdje se posmatraju binarne odluke 3 Višestrukealternativne hipoteze 4 Sekvencijalno odlučivanje 5 Složene hipoteze 2 ELEMENTI ESTIMACIJE Prethodno razmatranje odlučivanje (detekcija) Naredni korak procjena (estimacija) parametara signala u šumu

Model estimacije

Prostor parametara Prostor(opa anja)

mjerenjaž Prostor

procjene

Prbabilsticko preslikavanje Pravilo procijene

Izvorpredajnik

PrijemnikMedij

Prostor parametara - skalaran θ - višedimenzionalan θ (vektor)

θ slučajna promjenjiva

Šum

Q1

Q

X( )1Q

X( )Q

Prostor opažanjaProstor parametra

Vektor

Promijena

Pitanje koje pravilo estimacije (procjene) izabrati da bi se dobila bdquooptimalnaldquo procjena vrijednosti nepoznatog parametra Daje se procjena vrijednosti Q (X)

Izbor pravila

- dostupni podatci - bdquoželjeldquo posmatrača

1 Diskretni slučajni proces koji se posmatra ima poznatu gustinu vjerovatnoće i M nepoznatih parametara

1 2 ( )MQ Q Q Q 2 Poznato je n uzoraka slučajnog procesa u prostoru opažanja (predpostavlja se da se osobine procesa ne mjenjaju u toku intervala posmatranja) 3 Definisan je prostor parametara dimenzije M 4 Poznata je gustina vjerovatnoće W(Q ) definisana u prostoru parametara 5 za svaku sekvencu mjerenja (vektor X ) vektor parametara (Q ) i procjenu parametara ( )Q X poznata je skalarna funkcija

cijene ( )C Q Q X⎡ ⎤⎢ ⎥⎣ ⎦

U zavisnosti od dostupnih podataka mogu se primjenjivati različiti kriterijumi

situacije

I Bajesov kriterijum ( minimalna srednja cijena) Ukoliko se ne raspolaže sa svih pet nabrojanih tipova podataka II Ako nedostaje gustina vjerovatnoće može se primjeniti minimaksni kriterijum III Ako se ne poznaju niti cijene ( dostupna samo prva tri tipa podataka) Može se primjeniti kriterijum maksimalne vjerodostojnosti (najboljeg izbora) parametarska procjena parametra suprotno neparametarska procjena parametra (ne poznaje se ni tip gustine vjerovatnoće slučajnog procesa) IV Poznaje se ipak originalna gustina vjerovatnoće i zna se da se bdquosamo maloldquo modifikovana (kontaminirana ) šumom Robusna procjena izmjenjeni bdquorepovi raspodjeleldquo Zaključak o teoriji detekcije i estimacije Osnova Midlton An introdution to statistical communication theory 1958 god dobre osobine slabosti teorije detekcije i estimacije I 1 Pristup veoma generalan -koristi sve raspoložive informacije

-uključuje bdquoposmatračevoldquo neznanje (kroz gustine vjerovatnoće parametara ili apriorne vjerovatnoće parametara ili apriorne vjerovatnoće hipoteze ) -dovodi do optimalne strukture sistema - daje procjenu performansi optimalnog sistema i kvantitativnu metodu za poređenje stvarnih sistema sa teorijskim optimalnim - suboptimalni Prijemnici -realni Konkretno riješavaju probleme II- izvjesna proizvoljnost u pripisivanju cijena odlukama

- obično su i apriorne informacije nedovoljne i izbor kriterija može biti proizvoljan (proizvoljnost cijena) ndash nije toliko kritično u telekomunikacijama

konačno teorija omogućava unificiran ali pomalo i ograničen pristup kvantitativnoj teoriji optimalnosti procjene i poređenja sistema metoda zavisi od posmatrača 3 OPTIMALNI PRIJEMNIK Cilj - prikaz direktne primjene rezultata statističke teorije detekcije u telekomunikacijama

- sinteza na osnovu izložene teorije optimalni digitalnih telekomunikacionih sistema

31 odlučivanje na osnovu višestrukog osmatranja Model 1 donošenje odluke na osnovu višestrukih osmatranja Posmatraju se slučajne promjenjive 1 2 nξ ξ ξ koje mogu da predstavljaju sukcesivna mjerenja

-iste veličine -istovremeno mjerenje različitih veličina ili -kombinacija ove dvije mogućnosti 2dobijeni rezultat mjerenja Vektor 1 2( )nX x x x 3 pod hipotezama posmatraju se združene gustine vjerovatnoće

0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=

Ako ove gustine ne sadrže nikakve nepoznate parametre tada se opet radi o prostim hipotezama (jednostavnim) 4 i dalje će biti posmatrane binarne odluke Može se proširiti teorija na slučaj višestrukih hipoteza Za ovakve gustine vjerovatnoće može se nači odnos vjerodostojnosti (najboljeg izbora)

1

0

( )( )( )

W XXW X

Δ =

Ovaj količnik za svaki skup izvršenih mjerenja je nenegativni skalar Numerička vrijednost koja se dobije kada se u izraz zamijene konkretno izmjerene vrijednosti (za 0 ( ) 0W X = singularni slučaj) Na osnovu poznatih veličina i usvojenih kriterija mogu se izarčunati odgovarajući pragovi vjerodostojnosti Poređenjem sa tim pragovima

1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt

Biraju odgovarajuće hipoteze Parg 0Δ može biti određen po Bajesovom minimaksnom Nplman - Pirsonovom kriteriju ili na drugi način Umjesto primjera za ( )XΔ gdje se vrši direktno poređenje vjerodostojnosti sa pragom može se koristiti i bilo koja druga monotona funkcija ove veličine ( )G G X⎡ ⎤= Δ⎣ ⎦ Najčešće se koristi logoritamska funkcija logoritamski količnik vjerodostojnosti (logarithmic likelihood ratio)

( ) ln ( )L X X= Δ Primjer posmatra se binarni prenos u aditivnom bijelom Gassovom šumu 2( ) 0 1n t σ= = Bit 0 tri sukcesivna impulsa amplitude -1 (-1-1-1) Bit 1 +1 +1+1 tri impulsa

P(0) =P(1)= 12

Na svaki impuls se superponiraju nezavisne vrijednosti šuma n1 n2 i n3 Prema tome primljenje vrijednosti x1 x2 x3 predstavljaju superpoziciju -1 ili 1 sa ni

Odgovarajuće vjerovatnoće pod hipotenuzama su

( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod

Pretpodstavlja se cijene grešaka međusobno jednake i pošto su hipoteze podjednako vjerovatne primjenjeno pravilo maximalne vjerodostojnosti (najboljeg izbora)

( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo

zbog znaka (-1 u exponatu

g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum

Radi se o euklidskim rastojanjima primljenog vektora (tačke) od tačaka (-1-1-1) (111)

2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum

Novi aspekt gledanja ravan x1 + x2 + x3 = 0 daje oblast dijeljena na dvije oblasti odlučivanja (upravna na duž koja spaja tačke (-1-1-1) i (111) Na sličan način tretira se i slučaj Kada se vrši mjerenje u većem broju tačaka kada se jedan bit prenosi sa više sukcesivnih impulsa Ovo se može shavtiti i kao zaštitno kodiranje trostrukim ponavljanjem vjerovatnoće greške pri ovakvom odlučivanju su

( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m

P P H H P H H P H H P H H

P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=

⎡ ⎤= minus minus minus = minus = + + =⎣ ⎦

int

sabiraju se srednje nezavisni procesi vrijednosti (sabiraju se 2

iσ ) Pitanje šta bi bilo da je korišteno odlučivanje koje se često primjenjivalo pri FEC-u ponavljanjem ( napušten princip) O svakom impulsu bi se posebno odlučivalo i odluka donosila majoritetnom logikom Greška bi bila ako bi od tri primljena impulsa za dva donijeta pogrešna odluka Ako je vjerovatnoća greške po jednom impulsu P

2( 1)

0

1 015872

xP e dxπ

infinminus += =int

Tada se vjerovatnoća greške za bit kodovan trostrukim ponavljanjem

3 23 (1 ) 00675Pe P P p= + minus = Posljedica bdquosubtimalneldquo strategije Tvrdo ( hard) odlučivanje Predhodni slučaj Pe=0042 Mehko (soft) odlučivanje Za n=2 (mjere se samo dvije vrijednosti) 32 Primjer optimalne sinteze Objašnjeno odlučivanje na osnovu višestrukog osmatranja primjer 1 (111) i 0 (-1-1-1) Po tri impulsa Problematski se posmatra sljedeći model

- koriste se dva elementarna signala S0(t) i S1(t) poznatog oblika i jednakog trajanja T

- na ulazu u prijemnik djeluje aditivni Gaussov šum n(t)

( ) 0n t = i snage 2σ Cilj na kraju intervala T (osmatranje započinje u t = 0) prijemnik treba da se opredjeli za jednu od dvije hipoteze Vezane za signal X(t) X(t)=S0(t)+n(t) (H0) X(t)=S1(t)+n(t) (H1) Predpostavka - ulazni filter idealan i ne izobličava primljeni korisni signal - prijemnik bdquoima pravoldquo da primjeni bilo kakav način osmatranja i odlučivanja - na osnovu izabranog kriterija određuje se optimalna vrijednost praga 0Λ i količnik vjerodostojnosti poredi se sa pragom da bi se donijela odgovarajuća odluka -neka je u prijemniku primjenjeno višestruko osmatranje Na intervalu T prijemnik uzima n uzoraka ulaznog signala u ti ( i=12n) Kao rezultat dobije se slučajni vektor X (x1x2xn) sa komponentama

0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩

ni slučajne promjenjive s0i s1i uzroci determinističkih signala

xi Gassove slučajne promjenjive varijanse 2σ sa srednjim vriednostima s0i i s1i Predpostavka 1ulazni filtar idealan u opsegu [-fg fg]

2 spektralna gustina snage 0

2N zbog uslova nekorelisanosti

uzimaju se uzorci šuma u intervalima

1 uslov po teoremu o odmjeravanju2 g

Tn tt f

= Δ = rarrΔ

Teorema o odabiranju uslov nekoreliranosti Posljedica nezavisne slučajne promjenive imaju gaussovu raspodjelu sa srednjim vrijednostima s0i i s1i i istim varijansama 2σ

( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum

Količnik vjerodostojnosti je

( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii

N XX x s x s s sN X σ =

⎡ ⎤Λ = = minus minus + minus⎢ ⎥⎣ ⎦

sum

Predpostavka neka je izabran prag vjerodostojnosti 0Λ H1 usvojena 0( )XΛ ge Λ

2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt

x s x s s sXσ σ σ

σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum

Formalna blok šema prijemnika

X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i

Zaključak Realizirati model za vrijeme posmatranja T Cilj realizovati optimalni prijemnik za cijelokupni (kontinuirani) vremenski oblik signala Pretpostavka granica filtra gf rarrinfin

1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t

x s t x s t s sN N N

x t s t dt x t s t dt s t s t dtNo N No

s t dt s t s

o

t

Nt

σ

Δ rarr= = =rarrinfin

⎡ ⎤Δ minus Δ ge Λ minus minus⎢ ⎥

⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥

⎡sdot minus sdot ge

sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt

t dt bsdot minus sdot

⎤⎦

ge

⎣int

int int Korelaconi prijemnik Za notaciju x(t) equiv r(t) Ei equiv E0

01 0 0

0 0

( ) ( ) ( ) ( ) ln2

T T Nr t s t dt r t s t dtsdot minus sdot ge Λint int

( ) ( ) ( ) ( ) ( ) ( )ir t t u t T n t s t n tμ= sdot minus + = +

Implicira 2 slučaja

1 μ equiv const φ slučajan ( ) 12

pII

φ = )

kanal sa constantnim parametrima 2 p(μ ) i ( )p φ apriori nepoznate kanal sa slučajnim parametrima

Posljedica 1) Fluktuacija φ spora pa je moguće estimirati ndash koherentni prijemnik 2 složena estimacija φ nekoherentni prijemnik Zaključak potrebno je znati oblike (sliku) polaznih signala -potrebno je znati i vrijeme dolaska

X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX

Problem 0i iS uμ μ= sdot gt srtuktura zavisi od μ u prkasi se μ drzi u opsegu vriednosto ARP Optimalni prijemnik na bazi prilagođenog filtera

Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T

y t x t s t dt= sdotint

( ) ( ) ( )0

T

y t h x t dτ τ τ= sdot minusint

( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T

y T h x T d s T x T d

y T s t x t dt

T t

τ τ τ τ τ τ

τ

=

= sdot minus = minus sdot

=

= sdot

minus

minus

int int

int

Zaključak za t=T isti odziv imaju i korelator i prilagođeni filter Varijante optimalnog koherentnog prijemnika

X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1

PrilFilterS (t)2X ( )

0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT

Prilag FiltarS1(t)-S2(t)( )

0

T

int

Prednost prilagođenog filtera asinhrona detekcija usamljenog impulsa (RADAR)

B Vjerovatnoća greške Pretpostavka

- vjerovatnoća signala na ulazu u prijemnik P(s1) = P(so) = 12

- cijene 00 11C C =0equiv 10 01C C =1equiv

Posljedica 0Λ =1=gt 0ln 0Λ = Tada se odlučivanje o hipotezama vrši na osnovu odnosa

[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H

G x t S t S t dt S t S t dtgt

⎡ ⎤= minus + minus⎣ ⎦ ltint int

Uz ovaj uslov Pe minPe

2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o

NG x t s t dt x t s t dt S S dt⎡ ⎤

⎡ ⎤= minus ge Λ + minus⎢ ⎥⎣ ⎦⎣ ⎦int int int

0 0

0 0

1 ln 0 ne utiče na prag2

1 ln 0 prag zavisi od šuma2

No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne

Cilj odrediti vjerovatnoću greške treba odrediti raspodjelu signala x(t)(s1(t)-s0(t)) pod hipotezama s(t) konstanta u statističkom smislu s1(t)-s0(t)

x(t)[s1(t)-s0(t)] implicira normalnu raspodjelu x(t) Gaussov proces Za procese x(t)(s1(t)-s0(t)) +12 () Treba znati srednju vrijednost i varijansu I srednja vrijednost G pod hipotezom H0 je

[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T

E G H E s t n t s t s t dt

s t s t dt

E n t

E G H s t s t dt s t dt

s t s t dt

s t s t dt

φ

⎡ ⎤= + sdot minus⎢ ⎥

⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int

II srednja vrijednost G pod hipotezom H1 je

[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T

s t n t s t s t dtE G H E

s t s t dt

⎡ ⎤+ sdot minus⎢ ⎥

⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int

[ ] [ ]1 0E G H E G H= minus simetrija III izračunavanje varijanse G pod H0

[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H

G H E G H s t n t s t s t dt

s t s t dt s t s t dt

σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus

⎡ ⎤ ⎡ ⎤minus minus + minus⎣ ⎦ ⎣ ⎦

int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T

G H E G H n t s t s t dtminus = sdot minusint

Varijansa će biti [ ]0

2

21 0

0

( ) ( ) ( )T

G H E n t s t s tσ⎡ ⎤

= minus⎢ ⎥⎣ ⎦int

[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H

E n t n u s t s t s u s u dtdn

NE n t n u n t

No n t s t s t s u s u dtdu

δ

σ δ

⎡ ⎤= sdot sdot minus sdot minus⎣ ⎦

sdot = minus

= minus sdot minus sdot minus

int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T

G H G HNo s t s t dtσ σ= minus =int

Za slučaj da su energija signala nejednake

2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int

Može se pisati uslov za hipotezu pod GH0 i GH1 kao odnos vjerodostojnosti 0Λ 1Λ pri čemu če greška nastupiti kada se 1Λ lt 0Λ greška

2Λ equiv GH1 2Λ equivGH0

Zaključak 1 raspodjele pod hipotezama su Gaussove i to

[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int

2 neka je emitovan x(t) = s1(t) + n(t) Pojava greške je ako je

2Λ lt 1Λ 1Λ s1 2Λ s0 tada je

[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T

s t n t s t s t dt E E

n t s t s t dt s t s t dt

+ sdot minus lt minus

minus lt minus minus

int

int int

Gaussov proces

05 ekEξ lt minus Pa je

1 122 2

22

1( ) exp22

ekvEekv E

Pe P d dξξ ξ ξσσ π

minus minus

minusinfin minusinfin

⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int

I slučaj s1(t)=s0(t) antipodni

21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0

4 2 odnos signal šum2

B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int

Postoje različite klase ovih signala

1 0 1 0FSK2

n kf f f fT T

sdot = minus =

- gustinom vjerovatnoće odgovarajućih skupova signala Skup primljenih signala prostor opažanjamjerenja (Observation) prijemink na osnovu bdquopravila odlučivanjaldquo (decision rule) primljenom signalu pridružuje se jedan od mogućih simbola (poruka) u prostoru bdquoodlukaldquo problem detekcije singularan neka oblast za ostale koju je vjerovatnoća hipotza nenulta Oblast preklapanja Greške cijena greške Određuje kriterije (zavisno od podataka cilja)














Model estimacije


mjerenjaž Prostor

procjene


Izvorpredajnik

PrijemnikMedij



Šum

Q1

Q

X( )1Q

X( )Q


Vektor

Promijena


Izbor pravila




cijene ( )C Q Q X⎡ ⎤⎢ ⎥⎣ ⎦


situacije








0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=


1

0

( )( )( )

W XXW X

Δ =


1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =














Model estimacije


mjerenjaž Prostor

procjene


Izvorpredajnik

PrijemnikMedij



Šum

Q1

Q

X( )1Q

X( )Q


Vektor

Promijena


Izbor pravila




cijene ( )C Q Q X⎡ ⎤⎢ ⎥⎣ ⎦


situacije








0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=


1

0

( )( )( )

W XXW X

Δ =


1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =









Model estimacije


mjerenjaž Prostor

procjene


Izvorpredajnik

PrijemnikMedij



Šum

Q1

Q

X( )1Q

X( )Q


Vektor

Promijena


Izbor pravila




cijene ( )C Q Q X⎡ ⎤⎢ ⎥⎣ ⎦


situacije








0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=


1

0

( )( )( )

W XXW X

Δ =


1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

Model estimacije


mjerenjaž Prostor

procjene


Izvorpredajnik

PrijemnikMedij



Šum

Q1

Q

X( )1Q

X( )Q


Vektor

Promijena


Izbor pravila




cijene ( )C Q Q X⎡ ⎤⎢ ⎥⎣ ⎦


situacije








0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=


1

0

( )( )( )

W XXW X

Δ =


1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

Izbor pravila




cijene ( )C Q Q X⎡ ⎤⎢ ⎥⎣ ⎦


situacije








0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=


1

0

( )( )( )

W XXW X

Δ =


1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =








0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=


1

0

( )( )( )

W XXW X

Δ =


1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =







0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=


1

0

( )( )( )

W XXW X

Δ =


1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =


0 0 1 2

1 1 1 2

( ) ( )

( ) ( )n

n

W X W x x x i

W X W x x x

=

=


1

0

( )( )( )

W XXW X

Δ =


1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

1 2 0

1( )

0

n

H

x x x

H

gtΔ Δ

lt



P(0) =P(1)= 12



( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =


( )( )

( )( )

2

2

132

0 1 2 31 2

132

1 2 31

1 2 11 2

i

i

x

i

x

ni

W x x x e

W x x x e

π σ

π

+minus

=

minusminus

=

⎫⎪=⎪ =⎬⎪

= ⎪⎭

prod

prod


( ) ( )

( ) ( )

1 1 2 3 1 2 3

0

32 2

1

0

1

1 lo


g

01 1

1

)i ii

HWo x x x

HH

W

H

Hx

x x x

xH=

rArr

gtminus +

gt⎡⎢ lt⎣

gtlt

ltsum sum


2 2

2

( 1) ( 1)i i

i

x x

x

gtminus +lt

2 1ixminus + 2ixgt

lt 2 1ix+ +

13

3

0

21

1( 00 )ii

H

T x x

H

x x=

⎡

+

⎤⎢ ⎥ rArr⎢ ⎥⎢ ⎥⎣ ⎦

gt=

lt+ =sum


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =


( ) ( ) ( ) ( )

( )2323

0 1 1 0 1 0 0 1

0

2

20 3

23 3

1 12 2

( 0 )

1 00422

1 1 1 3 1 1 1 3

e

t m


P T H

e dt

m

σ

πσ

σ

+infin minus

= + = =

= gt =

=


int



2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

2( 1)

0

1 015872

xP e dxπ

infinminus += =int






0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =




0 0

1 1

( )( )

( )i i

i ii i

s n Hx t x

s n H+⎧

= ⎨ +⎩







Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =






Tn tt f

= Δ = rarrΔ


( )( )

( )( )

2

01

0 22

2

11

1 22

1( ) exp22

1( ) exp22

n

i ii

n

n

i ii

n

x sW X

x sW X

σπσ

σπσ

=

=

⎡ ⎤minus⎢ ⎥

⎢ ⎥= minus⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

minus⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

sum

sum


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =


( )2 210 1 1 02

10

( ) 1( ) exp 2 22( )

n

i i i i i ii



sum


2 20 1 0 12 2 2

1 1 1

2 00

20

0

log ( )2

2

ln2

2 2g g

n n ni i i i i i

i i i

NNof f Nt

Nt


σ

σ

= = =

minusΛ = minus + +

ge Λ

⎡ ⎤= sdot sdot = =⎢ ⎥Δ⎢ ⎥⎢ ⎥=⎢ ⎥Δ⎣ ⎦

sum sum sum

( )

2 221 0 1

02 2 21 1 1

2 2 21 0 0 1 0

1

( )ln 2

1ln2

n n ni i i i i oi

i i in

i i i i i ii

x s x s s s

x s x s s s

σσ σ σ

σ

= = =

=

minusminus ge Λ +

minus ge Λ + minus

sum sum sum

sum sum sum


X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

X

+XiH0

H1

Odluka

( )1

n

i =sum

( )1

n

i =sum

S1i

S0i


1 02

1g

tf

nt

Δ = rarr

= rarrinfinΔ

Odnosno

0limt

n t TΔ rarr

sdotΔ =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

( )

( ) ( )( )

2 21 0 0 0 10 1 1 10 0 0

2 21 0 0 1 0

00

2

2 21 0 0 1

0

00 0

0

2 2 1lim ln

2 2 1( ) ( ) ( )

1( ) ( ) ( ) ( ) ln ( ) (

ln

2

( )

)2 2

n n n

i i i i i it

T

i i in

T T T

T Nox t s t dt x t



s t dt s t s

o

t

Nt

σ



⎣ ⎦

sdot minus sdot

⎡ ⎤= ⎢ ⎥


sdotΔ⎣ ⎦

Λ

+

ge minus minus

Λint int

sum sum sum

int int int

1 00

0

0

( ) ( ) ( ) ( )T

o

T

T

x t s t dt x t s

dt


⎤⎦

ge

⎣int


01 0 0

0 0

( ) ( ) ( ) ( ) ln2





pII

φ = )



X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =


X

+r(t)H0

H1

Odlukat=T

Kopija So(t)

Kopija S1(t)

( )0

T

int

( )0

T

intX


Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

Xx(t) y(t)

s(t)x(t) y(t)

( )0

T

int

( ) ( ) ( )0

T


( ) ( ) ( )0

T


( ) ( )

0

0

( ) ( ) (

( ) ( ) (

)

)T T

T

O

t T


y T s t x t dt

T t

τ τ τ τ τ τ

τ

=


=

= sdot

minus

minus

int int

int


X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

X

+r(t)

nT

Kopija S (t)0

Kopija S (t)1

+

nT

PrilFilterS (t)1


0

T

int

( )0

T

intr(t)

XnT

Kopija S1-S2(t)

r(t) r(t) nT


0

T

int






[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =





[ ]1

0

2 21 0 0 1

0 0

1( ) ( ) ( ) ( ) ( ) 02

HT T

H




2 201 0 0 1 0

0 0

1( ) ( ) ( ) ( ) ln2 2

T T T

o



0 0

0 0



No

No

Λ = rArr Λ rarr

Λ ne rArr Λ ne



[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =


[ ] [ ]

[ ]

[ ]

[ ]

0 0 1 0

2 20 1

0

20 0 1 0

0

2 20 1

0

20 1

0

( ( ) ( ) ( ) ( )

1 ( ( ) ( )2

( )

1 ( ) ( ) ( )2

( ) ( )

1 ( ( ) ( )2

T

o

T

T T

oT

T


s t s t dt

E n t


s t s t dt

s t s t dt

φ


⎣ ⎦

+ +

=

= sdot minus +

⎡ ⎤minus⎣ ⎦

= minus minus

int

int

int int

int

int


[ ][ ] [ ]1 1 0

01

2 20 1

0

( ) ( ) ( ) ( )

1 ( ) ( )2

T

T


s t s t dt


⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥+ minus⎣ ⎦⎢ ⎥⎣ ⎦

int

int


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =


[ ][ ]

[ ] [ ] [ ]

22 0 0

21 1

0 0 0 00

2 2 2 20 1 1 0

0 0

simetrija

( ) ( ) 1( ) ( )

1 1( ) ( ) ( ) ( )2 2

G Ho

T

T T

E G H E G H

E G H E G H



σ ⎡ ⎤= minus =⎣ ⎦

⎡ ⎤= minus⎣ ⎦

minus = + minus


int

int int

i

[ ] [ ]0 0 1 00

( ) ( ) ( )T



2

21 0

0

( ) ( ) ( )T



[ ] [ ]

[ ] ( )

[ ] [ ]0

1 0 1 00 0

0

21 0 1 0

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )2

( ) ( ) ( ) ( ) ( )2

T T

T T

G H


NE n t n u n t


δ

σ δ


sdot = minus


int int

int int

[ ]0 1

22 21 0

0

( ) ( )2

T



2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =


2 21 0

0 0

( )

1

T T

s dt s t dt

E Eo

ne

ne

int int




[ ]

0 0

0 1

2 2 2

2

0 1 00

1 0 1 00

0 1

1 ( ) ( )2

2 ( ) ( )

G H G H

T

ekv ek

T

G H G H

N E E s t s t dt

E E s t s t dt

n m m m

σ σ σ= = =

= sdot = minus

= + minus sdot sdot

= = minus = minus

int

int



[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

[ ] [ ]

[ ] [ ]

1 1 0 1 00

21 0 1 0

0 0

1( ) ( ) ( ) ( ) ( )2

1( ) ( ) ( ) ( ) ( )2

T

T T





int

int int

Gaussov proces


1 122 2

22

1( ) exp22

ekvEekv E


minus minus


⎛ ⎞= = minus⎜ ⎟

⎝ ⎠int int

2EekvPe Q

No⎛ ⎞

= ⎜ ⎟⎝ ⎠

2

2

0

1( ) exp22

2( )

xx

u

uQ x du

erf x e du

π

π

infin

minus

⎛ ⎞= minus⎜ ⎟⎝ ⎠

=

int

int


21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

21 2

1 2

4

1

ekvE E

E E E E

E E E

μ

μμ

μ

= sdot

= = sdot =

== =

( )2

2

1 ( )1( )2 2

( ) 2 ( 2 )

u

cx

c

c

erf x e du

erf xxQ x erf

erf x Q x

π

minusinfin

=

= minus

⎛ ⎞= hArr⎜ ⎟⎝ ⎠

hArr =

int

( )2 2

0

1 0

0 0


B-PSK ( ) cos ( )( ) cos( )

E EPe Q QNo N

s t w ts t w t

ρ ρ

π

⎛ ⎞= = sdot⎜ ⎟

⎝ ⎠

== +

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

II slučaj

1 0

1 20

2

( ) ( )

( ) ( ) 0

2

T

s t s t

s t s t dt

EPe Q QNo

ρ

perp

sdot =

⎛ ⎞= =⎜ ⎟

⎝ ⎠

int


1 0 1 0FSK2

n kf f f fT T

sdot = minus =

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