Σχεδιαση βελτιστου δεκτη Δεκτης συσχετισης Δεκτης προσαρμοσμενου φιλτρου Πιθανοτητα σφαλματος - PowerPoint PPT Presentation

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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.

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Σχεδιαση βελτιστου δεκτη Δεκτης συσχετισης Δεκτης προσαρμοσμενου φιλτρου Πιθανοτητα σφαλματος

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4609813

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  • () {s1(t), s2(t),,sM(t)}

    , .

  • {f1(t), f2(t),fK(t)}, , :

    • , :

    • :


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{s1(t), s2(t),,sM(t)}

s1(t)

s2(t)

.

.

.

sM(t)

si(t)

K-

fj(t)

{f1(t), f2(t),fK(t)}

f1(t)

f2(t)

.

.

fK(t)


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  • log2M bits SM . :

    • - .

  • - :


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k(t) fk(t)

am,I sm,i

LUT= Look-up-table

log2M bits address


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  • , - K x M ( ) .

  • look-up tables (LUTs) log2M

  • log2Mbits LUTs.

  • LUTs .

  • .


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  • ()

  • ():

  • :


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  • :

    • ,

    • .

    • .


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  • 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)]


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To

  • Gaussian

    (dditive White Gaussian Noise AWGN)n(t)

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

  • n(t) Gaussian .

r(t)

s(t)

n(t)


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  • :

  • :

  • n'(t) ( ).

  • slide n'(t) sm(t) m[0,,M-1]


H n t s m t

H n'(t) sm(t)

  • :


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  • :

  • rk = sm,k + nk

n'(t)

f2(t)

[r1, r2]

f1(t)


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O

r

s

n

  • : 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

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

ML (Maximum Likelihood)

  • p1=p2==pm =1/M, ( ), MAP ML

  • H sm :

    p(r|sm) p(r|si), mi.

    (ML receiver)


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  • 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

pdf , p(n)

  • Gaussian ni nk

    :


Pdf p n 2

pdf , p(n) (2)

  • [nink]=0 ik, .

  • [nk2] =N0/2, (variance) 0/2. :

p(n)=


Pdf r p r s m

pdf r, p(r|sm)

  • O o ,

    nk=rk-sm,k :


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  • MAP:


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  • sm(t) r(t).

n'(t)

f2(t)

[s1,1, s1,2]

[r1, r2]

f1(t)


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()

  • :

  • 0/2 MAPreceiver


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  • (0/2)ln[pm] .

    • , pm

    • , pm .

  • .


Correlation receiver

Correlation Receiver

r(t)

-1/2

(0/2)ln(p1)

s1(t)

.

.

.

.

r(t)

-m/2

(0/2)ln(pm)

sm(t)

r(t)

-M/2

(0/2)ln(pM)

sM(t)


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r(t)

s1(t)

.

.

.

.

r(t)

sm(t)

r(t)

  • ML: (p1=p2==pM) pm .

  • (1=2=...=) .

  • :

Correlation Receiver

s(t)


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i)

r(t) ( rk)

r(t)

r1

f1(t)

r=[r1,r2,,rK]

r(t)

rK

fK(t)


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ii)

r=[r1,r2,,rK]

-1/2

(0/2)ln(p1)

S

-/2

(0/2)ln(pM)


Matched filter receiver

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

correlator

hk(t) = fk(T t)

t=T

h1(t)

r(t)

r1

r=[r1,r2,,rK]

t=T

hK(t)

r(t)

rK


Correlator 2

correlator (2)

t=T

s1(-t)

r(t)

-1/2

(0/2)ln(p1)

.

.

.

.

r(t)

sm(-t)

-m/2

hm(t)=sm(T-t)

(0/2)ln(pm)

r(t)

s1(-t)

-M/2

(0/2)ln(pM)


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  • , =4:

  • s1(t) s2(t)

  • s3(t) s4(t)

1

1

2

1

2

1

2

t

t

-1

1

1

1

1 2

1

2

1

2

t

t

T=2, E1=E2=E3=E4=2


Correlation rx

(Correlation Rx)

r(t)

-1/2

(0/2)ln(p1)

s1(t)

.

.

.

.

r(t)

-4/2

(0/2)ln(p4)

s4(t)


Matched filter rx

(Matched Filter Rx)

hk(t) = sk(2 t)

t=2

h1(t)

r(t)

(0/2)ln(p1)

t=2

h4(t)

r(t)

(0/2)ln(p4)


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  • 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

1

1 2

1 2

-1

-1


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r1

r(t)

r=[r1 r2]

f1(t)

r(t)

r2

f2(t)

hk(t) = fk(2 t)

h1(t)

r1

h1(t)

r(t)

t=2

1 2

r=[r1 r2]

h2(t)

h2(t)

r(t)

r2

1 2


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1r1+1r2

0ln(p1)/2

1r1-1r2

0ln(p2)/2

-1r1+1r2

0ln(p3)/2

-1r1-1r2

0ln(p4)/2

f2

s1

s3

f1

s4

s2

r=[r1,r2]

S


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( )

()

r(t)

f(t)=(1/T) 0tT

= 0

trigger at t=kT


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()

r(t)

h(t)=f(T-t)

h(t)=1/T 0tT

= 0

trigger at t=kT


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  • MAP sm(t) r(t) .

n'(t)

f2(t)

[s1,1, s1,2]

[r1, r2]

f1(t)


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  • (coherent) AWGN :

    • .

    • .

  • .

  • .


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  • -.

  • d.


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  • :

  • jm

  • Rm m- " " (= r m- )


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  • .

  • , .

  • , .


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  • Ps(e) = Pr[s] , :

    • Pr[si|s=si]=P(E|si) si si. :

      P(E|si)=

  • Ri pdf


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  • , 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)}


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BPSK

  • =2: ( = )

  • :

  • :

  • Pr[s1] = Pr[s2] = 0.5 ( )

Eb

-Eb

s2

s1


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BPSK

  • s1:

  • p(r|s1)Pr(s1) p(r|s2)Pr(s2)

R2

R1

0

s1 = Eb r

s2= - Eb


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BPSK

:


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BPSK

  • slide :

  • :

  • BPSK, .

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


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BPSK


Coherent fsk

(coherent) FSK

  • (=2):

    • f1-f2=k/2T, k = . (??).

  • :

  • :


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FSK

f2(t)

R2

s2

R1

s1

f1(t)

:

R2

R1

s2'= - Eb/2 s1'= Eb/2


Coherent fsk1

(coherent) FSK

  • , , .

  • BFSK b Eb/2 :


Ber bit error rate bpsk fsk

BER (bit error rate) BPSK FSK

To FSK 3db BPSK ( bit )

3db


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ASK

  • (=2):

  • :

  • :

  • FSK :

=>

R2 R1

R2 R1

X

X

s2=0 Eb/2 s1=2Eb

s2=-Eb/2 0 s1=Eb/2


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  • m(t)

  • : m(t) {0,1}

  • M-ary : m(t) {0, 1,,M-1}

    • To

    • =2k

    • k= bits/symbol.

  • :

    • (M-ary PSK)

    • (M-ary ASK)

    • (Quadrature Amplitude Modulation QAM)


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:

  • :

    • b bit

    • Ts

    • Rb = 1/Tb bits

    • Rs = 1/Ts

  • bits


M ary psk mpsk

M-ary PSK (MPSK)

  • :

    • m(t) {0, 1,,M-1}

    • To Ac .

  • : =2 BPSK


M 4 quadrature psk qpsk

M=4: Quadrature PSK (QPSK)

  • QPSK:

  • I/Q:


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QPSK

y(t)

c

- c

c

x(t)

- c

:

:


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MPSK

  • MPSK M ASK.


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QPSK

  • 4 QPSK :

  • :

  • :

  • s = PT=


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QPSK

f2

s2

s1

s3

f1

s4


To qpsk 45 0

To QPSK 450

s1=[Es/2, Es/2 ]

f2

s2

s1

s2=[-Es/2, Es/2 ]

f1

s3=[-Es/2, -Es/2 ]

s4=[Es/2, -Es/2 ]

s3

s4


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QPSK


Qpsk 2

QPSK (2)


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QPSK

  • H - 4 , :

  • QPSK BPSK:

  • BER (Bit Error Rate) BPSK QPSK BER.


R qpsk bpsk

R QPSK BPSK


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    • , ,

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