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Σχεδιαση βελτιστου δεκτη Δεκτης συσχετισης Δεκτης προσαρμοσμενου φιλτρου Πιθανοτητα σφαλματος. ΗΥ 4 30 Ψηφιακες Επικοινωνιες. Συναρτησεις Βασης ενος συνολου σηματων. Εχουμε ενα συνολο σηματων (κυματομορφων) { s 1 (t), s 2 (t),…,s M (t)} πληθους Μ, και εκπεμπεται μια απο αυτες καθε φορα. - PowerPoint PPT Presentation

### Transcript of £‡µ´¹±ƒ·...

• 430

• () {s1(t), s2(t),,sM(t)} , . {f1(t), f2(t),fK(t)}, , : , :

:

• f1(t)f2(t)..fK(t)s1(t)s2(t)...sM(t) si(t) K- fj(t)

{f1(t), f2(t),fK(t)}{s1(t), s2(t),,sM(t)}

• log2M bits SM . : - . - :

• log2M bits address k(t) fk(t)am,I sm,iLUT= Look-up-table

• , - K x M ( ) . look-up tables (LUTs) log2M log2M bits LUTs. LUTs . .

• ()

():

:

• : , . .

• s(t) {s1(t), s2(t),,sM(t)}, s(t) t [0,T]. , sm(t) m- .

p1= Pr[s1(t)],, pM=Pr[sM(t)]

To o r(t) = s(t) + n(t)

r(t), (t) s(t), Ps=Pr[ (t) s(t)]

• To Gaussian (dditive White Gaussian Noise AWGN) n(t)

O n(t) 0, Rnn() = [N0/2]() Snn(f) = N0/2.

n(t) Gaussian .s(t)n(t)r(t)

• :

:

n'(t) ( ). slide n'(t) sm(t) m[0,,M-1]

• H n'(t) sm(t):

• :

rk = sm,k + nkf1(t)f2(t)[r1, r2]n'(t)

• O : s = [s1, s2,, sK] {s1,s2,,sM}

r = [r1, r2,,rK]= s+n, s n = [n1, n2,nK].

r s Ps=Pr[s]

r

• MAP (Maximum a posteriory Probability) {s1,s2,,sM} , {p1, p2,,pM} , r.

H sm : Pr[sm |r] Pr[si |r], mi ( receiver) (Bayes)

• ML (Maximum Likelihood) p1=p2==pm =1/M, ( ), MAP ML

H sm : p(r|sm) p(r|si), mi. (ML receiver)

• MAP ML p(r|sm).

r = sm + n, sm , p(n) = p (n1, n2,,nK) pdf ni.

n(t) Gaussian

Gaussian . p(n1, n2,,nK) pdf Gaussian

• pdf , p(n) Gaussian ni nk :

• pdf , p(n) (2) [nink]=0 ik, .

[nk2] =N0/2, (variance) 0/2. :

p(n)=

• pdf r, p(r|sm)

O o , nk=rk-sm,k :

• MAP:

• sm(t) r(t).

f1(t)f2(t)[r1, r2]n'(t) [s1,1, s1,2]

• ()

:

• (0/2)ln[pm] . , pm , pm .

.

• ML: (p1=p2==pM) pm . (1=2=...=) . :s(t)

• i)

r(t)f1(t)r(t)fK(t)r1rKr=[r1,r2,,rK] r(t) ( rk)

• ii) r=[r1,r2,,rK]S-1/2-/2(0/2)ln(p1)(0/2)ln(pM)

• Matched Filter Receiver fk(t) [0,], hk(t) = fk(T t) fk(t) = hk(T t)

r(t)hk(t)|t=T r(t) hk(t) t=T.

r(t) fk(t) r(t) hk(t) = fk(T t). To " - matched"

• correlator h1(t)hK(t)r(t)r(t)

r1rKt=Tt=Tr=[r1,r2,,rK] hk(t) = fk(T t)

• correlator (2) s1(-t)r(t)-1/2(0/2)ln(p1) r(t)sm(-t)-m/2(0/2)ln(pm)r(t)-M/2(0/2)ln(pM)....s1(-t)t=T hm(t)=sm(T-t)

• , =4:

s1(t) s2(t)

s3(t) s4(t)

1-112t112t112t112t211 2T=2, E1=E2=E3=E4=2

• (Correlation Rx) r(t)s1(t)-1/2(0/2)ln(p1) r(t)s4(t)-4/2(0/2)ln(p4)....

• (Matched Filter Rx) h1(t)h4(t)r(t)r(t)

t=2t=2hk(t) = sk(2 t)(0/2)ln(p1)(0/2)ln(p4)

• 4 : f1(t) f2(t)

s1(t) = 1f1(t) +1f2(t), s2(t) = 1f1(t) - 1f2(t) s3(t) = -1f1(t) +1f2(t), s4(t) = -1f1(t) -1f2(t)

1 2-111 2-11

• f1(t)f2(t)r(t)r(t)r=[r1 r2] r1r2h1(t)r(t)h2(t)r(t)r1r2r=[r1 r2]hk(t) = fk(2 t) 1 2h1(t) 1 2h2(t)t=2

• r=[r1,r2]S

s1s3s4s2f1f2

• ( )r(t)f(t)=(1/T) 0tT = 0

trigger at t=kT ()

• r(t)h(t)=f(T-t)trigger at t=kT h(t)= 1/T 0tT = 0 ()

• MAP sm(t) r(t) .

f1(t)f2(t)[r1, r2]n'(t) [s1,1, s1,2]

• (coherent) AWGN :

. . . .

• -. d.

• :

jm

Rm m- " " (= r m- )

• .

, .

, .

• Ps(e) = Pr[s] , :

Pr[si|s=si ]=P(E|si) si si. : P(E|si)=

Ri pdf

• , sm. m=1,,M . : s1 P(E|s1). s2 P(E|s2). : P(E) = P(E|s1 )Pr{ s1}+ P(E|s2 )Pr{ s2}4. Pr{ s1}=Pr{ s2}=1/2 P(E) =(1/2)P(E|s1) + (1/2)P(E|s2)= (1/2)[P(E|s1) + P(E|s2)}

• BPSK =2: ( = )

:

:

Pr[s1] = Pr[s2] = 0.5 ( ) s2s1-EbEb

• BPSK s1 :p(r|s1)Pr(s1) p(r|s2)Pr(s2)

s2= - Ebs1 = Eb rR2R10

• BPSK :

• BPSK slide :

:

BPSK, . Ps(e)= Q(di,j / 2N0) di,j i j

• BPSK

• (coherent) FSK (=2):

f1-f2=k/2T, k = . (??). :

:

• FSK f2(t)f1(t)R1R2s2s1 :s2'= - Eb/2 s1'= Eb/2R2R1

• (coherent) FSK , , .

BFSK b Eb/2 :

• BER (bit error rate) BPSK FSKTo FSK 3db BPSK ( bit )3db

:

:

FSK : s2=0 Eb/2 s1=2EbR2 R1XXR2 R1s2=-Eb/2 0 s1=Eb/2 =>

• m(t) : m(t) {0,1}M-ary : m(t) {0, 1,,M-1}To =2kk= bits/symbol.: (M-ary PSK) (M-ary ASK) (Quadrature Amplitude Modulation QAM)

• : :b bitTs Rb = 1/Tb bitsRs = 1/Ts bits

• M-ary PSK (MPSK) :

m(t) {0, 1,,M-1}To Ac . : =2 BPSK

I/Q:

• QPSK cc- c- cy(t)x(t): :

• QPSK 4 QPSK :

:

:

s = PT=

• QPSK s1s2s3s4f1f2

• To QPSK 450 s1s2s3s4f1f2s2=[-Es/2, Es/2 ]s3=[-Es/2, -Es/2 ]s4=[Es/2, -Es/2 ]

• QPSK

• QPSK (2)

• QPSKH - 4 , :

QPSK BPSK:

BER (Bit Error Rate) BPSK QPSK BER.

• R QPSK BPSK

• - . , ,

Union Bound