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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

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  • 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 MAP receiver

  • (0/2)ln[pm] . , pm , pm .

    .

  • Correlation Receiver r(t)s1(t)-1/2(0/2)ln(p1) r(t)sm(t)-m/2(0/2)ln(pm)r(t)sM(t)-M/2(0/2)ln(pM)....

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

    Correlation Receiver

  • 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

  • ASK (=2):

    :

    :

    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

  • M=4: Quadrature PSK (QPSK)QPSK:

    I/Q:

  • QPSK cc- c- cy(t)x(t): :

  • MPSK MPSK M ASK.

  • 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