T (CODIGOS CONVOLUCIONALES - tsc.uc3m.es · TEMA 5 TECNICAS DE PROTECCI´ ON FRENTE A ERRORES´...

TEMA 5

TECNICAS DE PROTECCION FRENTE A ERRORES(CODIGOS CONVOLUCIONALES)

MMC (UC3M) Comunicaciones Digitales Codificacion de Canal 1 / 27

Codigos convolucionales

Introduccion de la redundancia mediante filtradoIntroduccion de memoria

Tasa R = k/n: Banco de filtros conk entradasn salidas

!B(0)[!]D ! D ! D

!"#$

!"#$

!C(0)[!]

!C(1)[!]

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Notacion:Entradas: B(i)[!], con i = 0, 1, · · · , k ! 1Salidas: C(j)[!], con j = 0, 1, · · · , n! 1


Representaciones

Esquematica

!B(0)[!]D ! D ! D

%%

!C(0)[!]

!C(1)[!]

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Relacion entrada salidas

C(0)[!] = B(0)[!] + B(0)[!! 1] + B(0)[!! 3]

C(1)[!] = B(0)[!] + B(0)[!! 1] + B(0)[!! 2] + B(0)[!! 3]

Notacion mediante polinomios en D

B(i)(D) =!

!

B(i)[!] · D!

PropiedadB(i)[!! d]" B(i)(D) · Dd


Representaciones (II)

Notacion mediante polinomios en D

C(0)(D) = B(0)(D)"

1 + D + D3#

C(1)(D) = B(0)(D)"

1 + D + D2 + D3#

Notacion matricial (polinomios):C(D)1!n = B(D)1!k · G(D)k!n

Matriz generadora de tamano k # nElemento fila i columna j: contribucion a la salida j-esima dela entrada i-esimaEjemplos

Ejemplo anterior (A): k = 1, n = 2

G(D) =$1 + D + D3, 1 + D + D2 + D3%

1!2

Otro ejemplo (B): k = 2,

G(D) =&

1 + D D DD 1 D

'

2!3


Paso a representacion esqumatica - Ejemplo B

G(D) =

&1 + D D D

D 1 D

'

2!3

Numero de entradas: k = 2Numero de salidas: n = 3

!B(0)[!]D

!B(1)[!]D

!"#$

!"#$

!"#$

!C(0)[!]

!C(1)[!]

!C(2)[!]

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Parametros de interes

Memoria total del codigo: Mt

Numero total de unidades de retardo (memorias)

Mt =k"1!

i=0

M(i)

Memoria de la entrada i-esima: M(i) = maxj

grado(gi,j(D))

Longitud de restriccion: KMaxima longitud de la respuesta al impulso del codificador(maximo numero de instantes de tiempo en los que un bitafecta a la salida del codificador)

K = 1 + maxi,j

grado(gi,j(D))

En general las prestaciones aumentan con K


Codigos sistematicos

Matriz de generacion

G(D) = [Ik P(D)]

Las entradas se “copian” en algunas de las salidasEjemplo (C)

G(D) =$1, 1 + D + D2%

!B(0)[!]D ! D

!"#$

!C(0)[!]

!C(1)[!].

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Diagrama de rejilla

Definicion del estadoContenido de las memorias del codificador

"[!] = [B(0)[!!1], · · · , B(0)[!!M(0)], · · · , B(k"1)[!!1], · · · , B(k"1)[l!M(k"1)]]

Etiquetado de la rejillaBits a la entrada del codificadorBits a la salida del codificador

B(0)[!], B(1)[!], · · · B(k"1)[!](((( C(0)[!], C(1)[!], · · · C(n"1)[!]


Ejemplo - Convolucional D

Estado: "[!] =$B(0)[!! 1], B(0)[!! 2]

%

Estados: "0 = [0, 0], "1 = [1, 0], "2 = [0, 1], "3 = [1, 1]

! D ! D

! &! &

"

"

!

#

!

B(0)[!]

C(0)[!]

C(1)[!]

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"[n] "[n + 1]"0

"1

"2

"3


Ejemplo - Convolucional E

Estado: "[!] =$B(0)[!! 1], B(0)[!! 2]

%

Estados: "0 = [0, 0], "1 = [1, 0], "2 = [0, 1], "3 = [1, 1]

!B(0)[!] D ! D

! &"

!

!C(0)[!]

C(1)[!]

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"[n] "[n + 1]"0

"1

"2

"3


Secuencia de bits - Camino a traves de la rejilla

Secuencia de datos: B(0)[!] = [11010]

Estado inicial: "0 (Se fija con el envıo de una cabecera debits)

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"[0] "[1] "[2] "[3] "[4] "[5]

"0

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"3

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Secuencia codificada: C[k] = [11 10 10 00 01]


Decodificacion - Algoritmo de Viterbi

Recuperacion de la secuencia mas verosımilEstados inicial/final

Cabecera de referencia (habitualmente ceros “bit flushing”)Salida dura: observacion de bits decididos

Secuencia con el menor numero de bits codificados distintosa la observacionMetrica de rama: distancia de Hamming con la observacion

Salida dura: observacion de q[n]Secuencia cuyos sımbolos asociados estan a la menordistancia euclıdea de la observacionMetrica de rama: |q[n]! A[n]|2

Hay que tener en cuenta la constelacion que se utiliza parahacer la conversion de las etiquetas de la rejilla basica asımbolos de la constelacion (A[n])

Mejores prestaciones que con salida blanda


Metricas de rama - Salida dura

Secuencia recibida: R[k] = [11 10 10 00 01]

"[n] "[n + 1]

"0

"1

"2

"3

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0

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Metricas acumuladas en cada estado (Viterbi)

"[1] "[2] "[3] "[4] "[5]"3 – 0 2/3 3/4 3/2"2 – 2 0/5 3/4 0/0"1 0 3 3/4 0/5 4/3"0 2 3 3/4 2/3 4/3

Si no hay errores: camino con metrica acumulada nula


Metricas de rama - Salida blanda

Secuencia recibida: q[n] =)

1,10,9

* )1,1"0,8

* )0,75"0,6

* )"1,2"1,1

* )"0,71,2

*

"[n] "[n + 1]

"0

"1

"2

"3

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1|110|01

1|100|11

1|000|10

1|01

(

(

(

(+1!1

+1

!1

00

01

10

11

"[0] "[1] "[2] "[3] "[4] "[5]

"0

"1

"2

"3

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8,02

0,024,42

3,620,02

8,023,62

4,42

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4,45

3,257,65

0,053,25

4,450,05

7,65

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3,22

2,625,62

0,222,62

3,220,22

5,62

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0,05

9,254,45

4,859,25

0,054,85

4,45

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4,93

2,930,13

7,732,93

4,937,73

0,13

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Metricas acumuladas en cada estado (Viterbi)

"[1] "[2] "[3] "[4] "[5]"3 – 0,07 5,69/11,49 10,14/15,74 10,27/8,07"2 – 7,67 0,29/16,89 10,54/15,34 17,87/0,47"1 0,02 11,27 10,89/15,09 0,34/19,54 15,47/12,47"0 8,02 12,47 10,29/15,69 9,54/10,34 13,47/14,47


Prestaciones

Salida dura

Pe $ c ·n·z!

i=t

+n · z

i

,· #i · (1! #)n·z"i

DHmin: mınima distancia de Hamming entre salidas para

secuencias distintasz: longitud del evento erroneo de distancia mınimat =

-DH

min"12

.(capacidad de correccion sobre n · z bits)

#: probabilidad de error de bit del sistema (BER)Salida blanda

Pe $ c · Q

/De

min

20

N0/2

1

Demin: mınima distancia euclıdea entre salidas para

secuencias distintasMMC (UC3M) Comunicaciones Digitales Codificacion de Canal 17 / 27

Calculo de DHmin

Comparacion con la secuencia de todo ceros

"[n] "[n + 1]

"0

"1

"2

"3

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"0

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"3

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1

1

0

2

4

4

3

5

DHmin = 5

z = 3


TEMA 5

TECNICAS DE PROTECCION FRENTE A ERRORES(CODIGOS DE REJILLA (TCM))


Codigos de rejilla

Conocidos tambien como codigos TCMTrellis Coded Modulation

Diseno conjunto de codigos convolucionales y delcodificador del transmisorUso eficiente del ancho de banda de un canal

La redundancia se introduce aumentando el orden de laconstelacion

Orden de la constelacion acorde al tamano de la salidacodificada

Posibilidad de tasas relativamente altas mediante las“transiciones paralelo”


Codigos de rejilla - Definiciones

Convolucionalknp, nnp

...knp ... nnp

...kp npConstelacionn = np + nnp

bits/sımbolo

k entradaskp transiciones “paralelo” (sin proteger)knp transiciones “no paralelo” (protegidas)

n salidasnp = kp transiciones “paralelo” (sin proteger)nnp transiciones “no paralelo” (protegidas)

Constelacion: 2n sımbolos (n = np + nnp bits/sımbolo)MMC (UC3M) Comunicaciones Digitales Codificacion de Canal 21 / 27

Codigos de rejilla - Asignacion de bits

Reglas de diseno: reglas de UngerboeckAsignacion para minimizar la probabilidad de errorDivision de la constelacion + asignacion de bits

Division de la constelacion en subconstelacionesDivision sucesiva aumentando la distancia mınimaSımbolos por subconstelacion: 2np

Numero de subconstelaciones: 2nnp

Asignacion de bitsTransiciones paralelo: seleccion de un sımbolo dentro de lasubconstelacion

Proteccion “fısica” sobre la constelacion (Gray)Transiciones no paralelo: seleccion de una subconstelacion

Proteccion mediante codigo convolucionalMaxima distancia: ramas que salen de o llegan a un mismoestadoEquiprobabilidad en la asignacion de sımbolos


Ejemplo: kp = 1, knp = 1, nnp = 2, 8-PSK

Codigo de tasa 2/3Una transicion en paraleloConvolucional de tasa 1/2 (Convolucional Ejemplo D)

Convolucional1/2

Constelacion3 bits/sımbolo

"[n] "[n + 1]

"0

"1

"2

"3

((((

((((

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1|01

Constelacion con 3 bits por sımbolo: 8 sımbolos8-PSK


Ejemplo: Division de la constelacion

( ( ((((( (

( * (*(*( *B0 * ( *(*(* (B1

( * **(** *C0 * * (***( *C1 * ( ***(* *C2 * * *(*** (C3


Ejemplo: Asignacion de las transiciones no paralelo

"[n] "[n + 1]

"0

"1

"2

"3

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++++

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C0

C1

C2

C3

C1

C0

C3

C2


Asignacion (convolucional D)

"[n] "[n + 1]

"0

"1

"2

"3

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"[n] "[n + 1]

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( 0|001|01

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0|11

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Prestaciones

Distancia mınima del codigo - La mınima de:Distancia mınima paraleloEjemplo anterior (constelacion normalizada): dp = 2

Mınima distancia entre puntos de las subconstelacionesDistancia mınima no paralelo

Distancia del convolucional medida sobre lassubconstelacionesEjemplo anterior

dnp =2

d2(C0, C1) + d2(C0, C2) + d2(C0, C1) = 2,14

Ganancia de codificacionComparacion con la constelacion equivalente sin codificar(misma velocidad de transmision efectiva)Ejemplo anterior: 4-PSK

G =ESC

bETCM

b·3dTCM

min42

3dSC

min42 = 2 (3 dB)


T (CODIGOS CONVOLUCIONALES - tsc.uc3m.es · TEMA 5 TECNICAS DE PROTECCI´ ON FRENTE A ERRORES´...

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