TEMA 4 RECEPTORES DE COMUNICACIONESTEMA 4 RECEPTORES DE COMUNICACIONES MMC (UC3M) Comunicaciones...

TEMA 4

RECEPTORES DE COMUNICACIONES

MMC (UC3M) Comunicaciones Digitales Receptores 1 / 81

Indice

Deteccion optima de secuencias en canales con ISIDeteccion suboptima: Igualadores de canal + decisor sinmemoria (sımbolo a sımbolo)

ClasificacionesIgualacion no ciega

Conocimiento del canalSecuencia de entrenamiento

Igualacion ciega


Deteccion de secuencias - Planteamiento

Receptor: f (t) = g(−t), y rg(t) cumple Nyquist

z[n] blanco y gausiano: σ2z =

{N0/2, A[n] ∈ IRN0, A[n] ∈ C

Secuencia de sımbolos A[n]: constelacion de M sımbolosSecuencia blanca

E {A[n]} = 0, RA[n] = E {A[k + n] · A∗[k]} = Es · δ[n], SA(ejω)

= Es

Respuesta p(t) causal limitada en el tiempo (Tp segundos)p[n] de longitud K + 1, ⇒ K = $Tp/T%Salida sin ruido del canal discreto equivalente

o[n] =K∑

k=0

p[k] · A[n− k]

Observacion a la salida del demodulador

q[n] = o[n] + z[n]


Deteccion de secuencias de maxima verosimilitud (ML)

Secuencia a detectar: L sımbolos (ML posibles secuencias)

A = [A[0], A[1], · · · , A[L− 1]]T

Canal: p = [p[0], p[1], · · · , p[K]]T

Estadıstico suficiente: Nq = K + L observaciones

q = [q[0], q[1], · · · , q[Nq − 1]] , Nq = K + L

o[0] =p[0] · A[0] + p[1] · A[−1] + p[2] · A[−2] + · · · · p[K] · A[−K]o[1] =p[0] · A[1] + p[1] · A[0] + p[2] · A[−1] · · · + p[K] · A[−K]

· · ·o[K + L− 1] =p[0] · A[K + L− 1] + p[1] · A[K + L− 2] + · · ·

+ p[K − 1] · A[L] + p[K] · A[L− 1]

Informacion adicional necesaria:

A[−1], A[−2], · · · , A[−K] y A[L], A[L + 1], · · · , A[L− 1 + K]MMC (UC3M) Comunicaciones Digitales Receptores 4 / 81

Secuencia de maxima verosimilitud (ML)

ML posibles secuencias

ai = [ai[0], ai[1], · · · , ai[L− 1]]T , i = 0, 1, · · · , ML − 1

Secuencia mas verosimil:

A = ai = [ai[0], ai[1], · · · , ai[L− 1]]T

para la que se cumple que la verosimilitud es maxima

fq|A(q|ai) ≥ fq|A(q|aj), j = 0, 1, · · · , ML − 1, ∀j (= i.


Obtencion de la secuencia de maxima verosimilitud

Verosimilitud:

fq|A(q|ai) =Nq−1∏

n=0

fq[n]|A(q[n]|ai)

Distribucion condicional de cada observacion

fq[n]|A(q[n]|ai) = N(oi[n], σ2

z)

=1√

2πσz·exp

−1

2σ2z

∣∣∣∣∣q[n]−K∑

k=0

p[k] · ai[n− k]

∣∣∣∣∣

2

Verosimilitud total

fq|A(q|ai) =1

(2πσ2z )

Nq/2 · exp

{− 1

2σ2z

Nq−1∑

n=0

|q[n]− oi[n]|2}

Secuencia mas verosımil

A = arg mınai

Nq−1∑

n=0

|q[n]− oi[n]|2 , oi[n] =K∑

k=0

p[k] · ai[n− k]


Ejemplo: 2-PAM K = 1, L = 3

Constelacion de sımbolos: A[n] ∈ {±1}Canal: p[n] = δ[n] + 0,5 · δ[n− 1], K = 1Secuencia a estimar: A = [A[0], A[1], A[2]], L = 3Estadıstico para la decision: q = [q[0], q[1], q[2], q[3]]

q[−1] = A[−1] + 0,5 · A[−2] + z[−1]

q[0] = A[0] + 0,5 · A[−1] + z[0]

q[1] = A[1] + 0,5 · A[0] + z[1]

q[2] = A[2] + 0,5 · A[1] + z[2]

q[3] = A[3] + 0,5 · A[2] + z[3]

q[4] = A[4] + 0,5 · A[3]

Premisa: Se conoce el valor de A[−1] = A[3] = +1Problema: decidir la secuencia de 3 sımbolos cuandoq[0] = 1,4 - q[1] = −0,4 - q[2] = 0,6 - q[3] = 1,6


Deteccion: Comparacion con las salidas sin ruido

q[0] = 1,4 - q[1] = −0,4 - q[2] = 0,6 - q[3] = 1,6

Salidas sin ruido, o[n], generadas por las 8 posiblessecuencias

A[0] A[1] A[2] o[0] o[1] o[2] o[3] Metrica−1 −1 −1 −0,5 −1,5 −1,5 +0,5 10,44−1 −1 +1 −0,5 −1,5 +0,5 +1,5 4,84−1 +1 −1 −0,5 +0,5 −0,5 +0,5 6,84−1 +1 +1 −0,5 +0,5 +1,5 +1,5 5,24+1 −1 −1 +1,5 −0,5 −1,5 +0,5 5,64+1 −1 +1 +1,5 −0,5 +0,5 +1,5 0,04+1 +1 −1 +1,5 +1,5 −0,5 +0,5 6,04+1 +1 +1 +1,5 +1,5 +1,5 +1,5 4,44

Secuencia “mas parecida” (maxima verosimilitud):+1 −1 +1


Estado ψ[n]

La salida sin ruido es una maquina de estados finitos

o[n] = A[n] · p[0] +K∑

k=1

p[k] · A[n− k]

Definicion de estado

ψ[n] = [A[n− 1], A[n− 2], · · · , A[n− K]]T

Numero de posibles estados es MK .

Dependenciaso[n] = f (A[n], ψ[n])

o[n] = g(ψ[n], ψ[n + 1])

ψ[n + 1] = f (ψ[n], A[n])


Diagrama de estados

Representacion de la evolucion de los estados en un sistema con ISI

ψ[n] = [A[n− 1], A[n− 2], · · · , A[n− K + 1], A[n− K]]T

ψ[n + 1] = [A[n], A[n− 1], A[n− 2], · · · , A[n− K + 1]]T

Hay MK estados

De cada estado salen M flechas, una por cada posible valor deA[n]

A cada estado llegan M flechas, todas generadas por el mismovalor de A[n]

Cada flecha se etiqueta con la siguiente informacion

A[n]|o[n],

es decir, con el valor del sımbolo actual que fuerza la transicion alnuevo estado, y con la salida sin ruido en ese caso


Diagrama de estados - Ejemplo A

A[n] ∈ {±1}, p[n] = δ[n] + 12δ[n− 1]

Salida sin ruido

o[n] = A[n] +12

A[n− 1]

Estadoψ[n] = A[n− 1], ψ[n + 1] = A[n]

Diagrama de estados

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Diagrama de estados - Ejemplo B

A[n] ∈ {±1}, p[n] = δ[n] + 12δ[n− 1] + 1

4δ[n− 2]

o[n] = A[n] + 12 A[n− 1] + 1

4 A[n− 2]

ψ[n] = [A[n− 1], A[n− 2]]T , ψ[n + 1] = [A[n], A[n− 1]]T

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+1|1,25−1| − 1,25

+1|1,75

−1| − 0,75

+1|0,75


Diagrama de rejilla - Ejemplo A

Ejemplo: A[n] ∈ {±1}, p = [1 0,5]T

Estado: ψ[n] = A[n− 1]

Etiquetas: A[n], o[n] =K∑

k=0

p[k] · A[n− k]

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Diagrama de rejilla - Ejemplo B

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ψ[n] ψ[n + 1][−1,−1]

[+1,−1]

[−1,+1]

[+1,+1]


Diagrama de rejilla - Secuencia A

Camino a traves de la rejillaEjemplo A = [−1,+1,−1,−1,+1]Estado inicial = ψ[0] = +1

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Diagrama de rejilla - Deteccion ML


A = arg mınai

Nq−1∑

n=0

∣∣∣∣∣∣∣∣∣∣

q[n]−N∑

k=0

p[k] · ai[n− k]

︸︷︷︸oi[n]

∣∣∣∣∣∣∣∣∣∣

2

Metrica de rama |q[n]− oi[n]|2

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q[0] = +0,5 q[1] = −0,4 q[2] = +0,1 q[3] = −1,7 q[4] = +0,3

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1,96

0,36

0,16

2,56

10,24

1,44

4,84

0,04

1,44

0,04


Diagrama de rejilla - Deteccion ML


A = arg mınai

Nq−1∑

n=0

∣∣∣∣∣∣∣∣∣∣

q[n]−N∑

k=0

p[k] · ai[n− k]

︸︷︷︸oi[n]

∣∣∣∣∣∣∣∣∣∣

2

Metrica de rama |q[n]− oi[n]|2

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q[0] = +0,5 q[1] = −0,4 q[2] = +0,1 q[3] = −1,7 q[4] = +0,3

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Obtencion de la secuencia ML - Algoritmo de Viterbi

Calculo de la distancia euclıdea para las ML posiblessecuencias

Analıticamente, o mediante ML caminos a traves de la rejillaCostoso computacionalmente

Algoritmo de ViterbiCalculo eficiente del camino mas corto a traves de la rejilla

Fundamentos del algoritmo de ViterbiMetrica de ramaMetrica de un caminoMetrica acumulada de un nodoCamino superviviente en un nodo


Algoritmo de Viterbi

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Algoritmo de Viterbi truncado

Decision del sımbolo A[n] requiere la fusion de los caminossupervivientes en ψk[n + 1]

Retardo de decodificacionNecesidad de almacenamiento de informacion

Algoritmo truncado con profundidad dEn el instante n se decide el sımbolo A[n− d]

Seleccion del sımbolo asociado al camino superviviente conmınima distancia acumulada


Probabilidad de error - Suceso erroneo

Una secuencia puede modelarse como una sucesion deestadosSuceso erroneo: secuencia de estados distinta e = (ψ, ψ)

Asociado a un suceso erroneo aparecen dos cosas:Longitud del suceso erroneo (%(e))Numero de errores que implica (w(e))

Suceso erroneo de longitud #(e)ψ[m] = ψ[m]ψ[m + % + 1] = ψ[m + % + 1]ψ[n] (= ψ[n] para m < n ≤ m + %

El numero de errores: w(e), cumple 1 ≤ w(e) ≤ #


Probabilidad de secuencia erronea

La probabilidad de una secuencia erronea es

P{secuencia erronea} ≈ k · Q

(Dmin/2√

N0/2

)

Dmin: distancia euclıdea mınima entre las salidas sin ruidode dos secuencias distintas {oi[n], oj[n]}, j (= ik: maximo numero de secuencias cuyas salidas sin ruido sehallan a una distancia Dmin de la salida sin ruido de unasecuencia

A medida que L aumenta, tambien lo hace k, de forma quela probabilidad de error de secuencia tiende a infinito amedida que aumenta L


Probabilidad de error de sımbolo

Se trata de estimar Pe = P{A[n] (= A[n]}

Pe =1L·∑

e∈Ew(e) · P{e}

donde

P{e} es la probabilidad del suceso erroneo eE es el conjunto de todos los sucesos erroneos que puedendarse en la rejilla

Podemos escribir un suceso erroneo como e = (ψ, ψ)

P{e} = P{ψ|ψ} · P{ψ}

Difıcil de calcular → Cotas para Pe


Cotas de la probabilidad de error

Cotas de la probabilidad de error

k2 · Q

(Dmin

2√

N0/2

)≤ Pe ≤ k1 · Q

(Dmin

2√

N0/2

)

k2: fraccion de caminos en la rejilla que tienen asociado unsuceso erroneo a distancia Dmin. Siempre verifica k2 ≤ 1k1 pondera los errores que se cometen con los sucesoserroneos de mınima distancia k1 =

∑

e∈Emin

w(e) · P{ψ}

Normalmente se utiliza una aproximacion, que es

Pe ≈ k0 · Q

(Dmin

2√

N0/2

)

k0: constante tal que k2 ≤ k0 ≤ k1. Tanto k1 como k2 sonindependientes de la varianza de ruido


Distancia euclıdea mınima respecto a la salida sin ruido deuna secuencia dada

Secuencia de referencia A = ai

Dmin(ai) = arg mınajj"=i

√√√√√√√√√√

Nq−1∑

n=0

∣∣∣∣∣∣∣∣∣∣∣

oi[n]−K∑

k=0

p[k] · aj[n− k]

︸︷︷︸oj[n]

∣∣∣∣∣∣∣∣∣∣∣

2

Se puede encontrar con el Algoritmo de Viterbi

Metrica de rama: |oi[n]− oj[n]|2Referencia: oi[n]Se busca la secuencia que se separa de la secuencia dereferencia y vuelve a la misma con mınima distanciaeuclıdea


Distancia mınima Dmin

Es el mınimo valor de Dmin(ai), para i = 0, 1, · · · , ML − 1

Dmin = arg mınai,ajj "=i

√√√√Nq−1∑

n=0

∣∣∣∣∣

K∑

k=0

p[k] · (ai[n− k]− aj[n− k])

∣∣∣∣∣

2

que se puede obtener para dos secuencias de sımbolos distintas

Si el diagrama de rejilla es simetrico

Viterbi con respecto a una secuencia de referencia

En general: aplicando el algoritmo de Viterbi para la secuenciade errores

ξ[n] = ai[n]− aj[n]

Referencia: camino de ceros (secuencia sin error)


Cota del filtro adaptado

Cota para la probabilidad de error

Pe ≥ k · Q

(dmin

2· ||p||√

N0/2

)

k: maximo numero de sımbolos a distancia dmin de otro dado

La norma del canal es ||p|| =

√√√√K∑

k=0

|p[k]|2

Esto permite acotar la distancia mınima Dmin

Dmin ≤ dmin · ||p||

El incremento en relacion senal a ruido para conseguir la mismaPe que un canal sin ISI es

∆SNR = 20 log10dmin||p||

Dmin


Igualadores de canal

Complejidad exponecial del algoritmo de Viterbi

Hay MK estadosDe cada estado salen M flechas, una por cada posible valorde A[n]A cada estado llegan M flechas, todas generadas por elmismo valor de A[n]

Solucion sub-optima: Igualador de canal + decisor sımbolo asımbolo (sin memoria)

#A[n]p[n] # &

!

z[n]

#q[n]Igualador #u[n] #A[n− d]


Decision sımbolo a sımbolo - Retardo d

Canal ideal: p[n] = C · δ[n− d]

Observacion para decidir A[n− d]

q[n] = p[d] · A[n− d]︸︷︷︸termino deseado

+∑

k $=d

p[k] · A[n− k]

︸︷︷︸ISI

+ z[n]︸︷︷︸ruido

Eleccion del retardo d

q[n]

p[d]= A[n− d] +

∑

k $=d

p[k]p[d]

· A[n− k] +z[n]

p[d]

Elegir d tal que |p[d]| ≥| p[n]| para todo n


Nivel de ISI

Cuantificacion de la distorsion introducida por el canal

γISI =Dpico

η≥ 0

Distorsion de pico Dpico

Dpico =∑

k $=d

|p[k]||p[d]| ≥ 0

Eficiencia de la constelacion η

η =(dmin/2)|A|max

≥ 0

En ausencia de ruidoγISI < 1: detector sımbolo a sımbolo no comete erroresγISI > 1: detector sımbolo a sımbolo comete errores


Estructuras de igualacion

Igualador linealLTE: Linear Transversal Equalizer

Igualador con realimentacion de decisionesDFE: Decision Feedback Equalizer

Otras estructuras no linealesIgualador bayesianoRedes neuronales (MLP. RBF, etc.)Maquinas de vectores soporte. . .


Igualacion no ciega / ciega

Igualacion no ciegaSe conoce el canal, oSe dispone de una secuencia de referencia

Igualacion ciegaNo se conoce el canalNo se dispone de una secuencia de referenciaSe dispone de informacion estadıstica sobre A[n]


Igualacion no ciega lineal

#A[n]p[n] # &

!

z[n]

#q[n]w[n] #u[n] #A[n− d]

u[n] =Kw∑

k=0

w[k] · q[n− k] = wT · qn

Se asume que se conoce el canal p[n]

Criterios de igualacion

Forzador de ceros (ZF)Mınimo error cuadratico medio (MMSE)


Igualador lineal

Longitud del igualador Kw + 1

u[n] =Kw∑

k=0

w[k] · q[n− k] =Kw∑

k=0

(K∑

"=0

p[%] · A[n− k − %] + z[n− k]

)

Respuesta conjunta de canal e igualador (K + Kw + 1 coeficientesno nulos)

c[n] = w[n] ∗ p[n], 0 ≤ n ≤ K + Kw

u[n] =K+Kw∑

k=0

c[k] · A[n− k] +Kw∑

k=0

w[k] · z[n− k]

Interferencia entre sımbolos

u[n] = c[d] · A[n− d]︸︷︷︸termino deseado

+K+Kw∑

k=0k "=d

c[k] · A[n− k]

︸︷︷︸ISI residual

+Kw∑

k=0

w[k] · z[n− k]

︸︷︷︸ruido filtrado


Igualador lineal ZF sin limitacion de coeficientes

Respuesta ideal (en el dominio del tiempo)

c[n] = p[n] ∗ w[n] = δ[n− d]

Respuesta ideal (en el dominio de la frecuencia)

C(ejω) = P(ejω) · W(ejω) = e−jωd → W(ejω) =ejωd

P(ejω)

Seleccion del retardo

P(z) = P0 ·K1∏

k=1

(1− αk · z−1)

︸︷︷︸Pmin(z)

·K2∏

"=1

(1− β" · z−1)

︸︷︷︸Pmax(z)

|αk| < 1, para 1 ≤ k ≤ K1

|β"| > 1, para 1 ≤ % ≤ K2

Elegir d para obtener una inversa estable del canal causalMMC (UC3M) Comunicaciones Digitales Receptores 42 / 81

Inconveniente del igualador ZF

Densidad de potencia del ruido filtrado

SZFz (ejω) = Sz(ejω) ·

∣∣W(ejω)∣∣2 =

σ2z

|P(ejω)|2

Potencia de ruido

σ2z |ZF =

12π

∫ π

−πSZF

z (ejω) dω =σ2

z2π

∫ π

−π

1|P(ejω)|2

dω

Amplificacion de ruido para canales con nulos espectrales


Igualador lineal ZF con Kw + 1 coeficientes

Sistema de ecuaciones

c[n] =K∑

k=0

p[k] · w[n− k] = δ[n− d]

Sistema matricial

2

66664

c[0]c[1]

.

.

.c[K + Kw]

3

77775=

2

666666666666664

p[0] 0 0 · · · 0p[1] p[0] 0 · · · 0p[2] p[1] p[0] · · · 0

.

.

....

.

.

.. . .

.

.

.p[K] p[K − 1] p[K − 2] · · · 0

0 p[K] p[K − 1] · · · p[0]...

.

.

....

. . ....

0 0 0 · · · p[K]

3

777777777777775

·

2

66664

w[0]w[1]

.

.

.w[Kw]

3

77775


Igualador lineal ZF con Kw + 1 coeficientes (II)

Matriz de convolucion de canal P

cd = P · w

cd = [00 · · · 0︸︷︷︸d

10 · · · 0]T

Sistema sobre-determinado

K + Kw + 1 ecuacionesKw + 1 incognitas

Solucion de mınimos cuadrados

wd|ZF = arg mınw

||cd − P · w||2 = P# · cd

Pseudo-inversaP# = (PH · P)−1 · PH


Igualador lineal MMSE sin limitacion de coeficientes

#q[n]w[n] #'()*

!A[n− d]

u[n] #ed[n](−)

Filtrado lineal optimo MMSE: minimizar E[|ed[n]|2

]

Principio de Ortogonalidad:Salida ortogonal al errorError ortogonal a la entrada

E[(A[n− d]− u[n]︸︷︷︸ed[n]

) · q∗[%]] = 0, ∀%

Esta ecuacion puede reescribirse como

E[A[n− d] · q∗[%]] = E[u[n] · q∗[%]], ∀%


Principio de ortogonalidad - Primer termino

Suposiciones de partida

Secuencia de datos blanca: RA[n] = Es · δ[n]A[n] y z[n] son independientes

E[A[n− d] · q∗[%]] = E

[A[n− d] ·

(K∑

k=0

p[k] · A[%− k] + z[%]

)∗]

=K∑

k=0

p∗[k] · E [A[n− d] · A∗[%− k]]︸︷︷︸RA[n−d−"+k]

+E [A[n− d] · z∗[%]]

= Es · p∗[% + d − n]


Principio de ortogonalidad - Segundo termino

E [u[n] · q∗[%]] = E

[(Kw∑

k=0

w[k] · q[n− k]

)· q∗[%]

]

=Kw∑

k=0

w[k] · E[q[n− k · q∗[%]]︸︷︷︸Rq[n−k−!]

= (w[k] ∗ Rq[k])|k=n−!

Autocorrelacion de las observacionesRq[n] = E [q[% + n] · q∗[%]]

= E

(

K∑

k=0

p[k] · A[% + n− k] + z[% + n]

)·

K∑

j=0

p[j] · A[%− j] + z[%]

∗

=K∑

k=0

K∑

j=0

p[k] · p∗[j] E [A[% + n− k] · A∗[%− j]]︸︷︷︸RA[n−k+j]

+E [z[% + n] · z∗[%]]

= Es ·K∑

k=0

p[k] · p∗[k − n] + σ2z · δ[n] = Es · (p[n] ∗ p∗[−n]) + σ2

z · δ[n]


Principio de ortogonalidad - Igualador MMSE

Combinando ambos terminos

Es·p∗[# + d − n︸︷︷︸−(n−"−d)

] = w[n]∗[Es · (p[k] ∗ p∗[−k])|k=n−" + σ2

z · δ[n− #]]

Cambio de variable k = n− #, y division por Es

p∗[−(k − d)] = w[k] ∗[(p[k] ∗ p∗[−k]) +

σ2z

Es· δ[k]

]

Igualador en el dominio transformado

W(ejω) =P∗(ejω) · e−jωd

P(ejω) · P∗(ejω) + σ2z

Es


Igualador lineal MMSE con Kw + 1 coeficientes

Principio de ortogonalidad

RAq[n− d] =Kw∑

k=0

w[k] · Rq[n− k]

Sistema de Kw + 1 ecuaciones para las Kw + 1 incognitas

rdAq = Rq · w → wd|MMSE = (Rq)−1 · rd

Aq

Matriz de canal

wd|MMSE = (PH · P + λ · I)−1 · PH · cd

λ =σ2

zEs


Sistema de ecuaciones en forma matricial

Rq[0] R∗q [1] R∗q [2] · · · R∗q [KW ]Rq[1] Rq[0] R∗q [1] · · · R∗q [KW − 1]Rq[2] Rq[1] Rq[0] · · · R∗q [KW − 2]

......

......

...Rq[KW ] Rq[Kw − 1] Rq[Kw − 2] · · · Rq[0]

·

w[0]w[1]w[2]

...w[Kw]

=

RAq[−d]RAq[−(d − 1)]RAq[−(d − 2)]

...RAq[Kw − d]


Igualacion con realimentacion de decisiones

#A[n]p[n] # &

!

z[n]

#q[n]w[n] # &

!

#u[n] #A[n− d]

b[n] "

u[n] =Kw∑

k=0

w[k] · q[n− k]−Kb∑

k=1

b[k] · A[n− d − k]

Estructura: dos filtros

Filtro lineal: Kw + 1 coeficientesRealimentacion de decisiones: Kb coeficientes

Criterios de igualacion

Forzador de ceros (ZF)Mınimo error cuadratico medio (MMSE)


Ecuaciones con realimentacion de decisiones

u[n] = c[d] · A[n− d]︸︷︷︸cursor

+d−1∑

k=0

c[k] · A[n− k]

︸︷︷︸ISI precursora

+K+Kw∑

d+1

c[k] · A[n− k]

︸︷︷︸ISI postcursora

−Kb∑

k=1

b[k] · A[n− d − k]

︸︷︷︸decisiones realimentadas

+Kw∑

k=0

w[k] · z[n− k]

︸︷︷︸ruido filtrado z′[n]

u[n] = c[d] · A[n− d]︸︷︷︸cursor

+d−1∑

k=0

c[k] · A[n− k]

︸︷︷︸ISI precursora

+d+Kb∑

k=d+1

c[k] · A[n− k]

︸︷︷︸ISI postcursora (a)

+K+Kw∑

k=d+Kb+1

c[k] · A[n− k]

︸︷︷︸ISI postcursora (b)

−Kb∑

k=1

b[k] · A[n− (d + k)]

︸︷︷︸decisiones realimentadas

+ z′[n]︸︷︷︸ruido


Coeficientes del igualador DFE

Coeficientes del filtro de realimentacion (Kb)

Objetivo: Cancelacion de la ISI postcursora (a)

b[k] = c[d + k], k = 1, 2, · · · , Kb

Coeficientes del filtro precursor: ZF

Sistema de ecuaciones reducido

c′d = P′ · w

c′d = [c[0], c[1], · · · , c[d], c[d + Kb + 1], · · · , c[K + Kw]]T

Coeficientes del filtro precursor: MMSE

wd|MMSE = (PH · Dd · P + λ · I)−1 · PH · cd

Dd = diag(11 · · · 1︸︷︷︸d+1

00 · · · 0︸︷︷︸Kb

11 · · · 1︸︷︷︸K+Kw−d−Kb

)


Prestaciones asintoticas de los igualadores linealesEstudio basado en la caracterizacion de ed[n] = A[n− d]− u[n]

Salida del igualador

u[n] = A[n] ∗ p[n] ∗ w[n] + z[n] ∗ w[n]

Error a la salida del igualador

ed[n] = A[n] ∗ (δ[n− d]− w[n] ∗ p[n])]− z[n] ∗ w[n]

Densidad espectral de potencia

Sed

(ejω)

= SA(ejω)

·∣∣e−jωd −W

(ejω)

· P(ejω)∣∣2 +Sz

(ejω)

·∣∣W

(ejω)∣∣2

Para secuencias blancas y ruido blanco

Sed

(ejω)

= Es ·∣∣e−jωd −W

(ejω)

· P(ejω)∣∣2 + σ2

z ·∣∣W

(ejω)∣∣2

Potencia del termino de error

σ2ed

=1

2π

∫ π

−πSed

(ejω)

dω


Prestaciones asintoticas de los igualadores lineales (II)

Criterio ZFW(ejω) =

ejωd

P(ejω)

σ2ed(ZF) =

σ2z

2π

∫ π

−π

1|P (ejω)|2

dω

Criterio MMSE

W(ejω) =P∗(ejω) · e−jωd

P(ejω) · P∗(ejω) + σ2z

Es

σ2ed(MMSE) =

σ2z

2π

∫ π

−π

1

|P (ejω)|2 + σ2z

Es

dω


Prestaciones asintoticas de los igualadores DFE

Densidad espectral de potencia de ruido

SDFEed

(ejω)

= SLINed

(ejω)

·∣∣1 + B

(ejω)∣∣2

Aproximaciones considerando decisiones correctasCriterio ZF

σ2ed(ZF − DFE) = σ2

z · exp(

12π

∫ π

−π

ln1

|P (ejω)|2dω

)

Criterio MMSE

σ2ed(MMSE − DFE) = σ2

z · exp

(1

2π

∫ π

−π

ln1

|P (ejω)|2 + σ2z

Es

dω

)


Prestaciones: igualadores lineales con Kw + 1 coeficientes

Salida del igualador

u[n] = c[d]︸︷︷︸ganancia

·A[n−d]+K+Kw∑

k=0k "=d

c[k] · A[n− k]

︸︷︷︸ISI residual

+Kw∑

k=0

w[k] · z[n− k]

︸︷︷︸ruido filtrado z′[n]

SuposicionesISI y ruido filtrado independientesDistribucion de ISI: gausiana

Aproximacion de la probabilidad de error

Pe ≈ k · Q

dmin · c[d]

2√

σ2z′ + σ2

ISI


Medias y varianza de z′[n]

Media de z′[n]

E[z′[n]

]=

Kw∑

k=0

w[k] · E[z[n− k]] = 0

Varianza de z′[n]

σ2z′ = E

( Kw∑

k=0

w[k] · z[n− k]

)·

Kw∑

j=0

w∗[j] · z∗[n− j]

=Kw∑

k=0

Kw∑

j=0

w[k] · w∗[j] · E [z[n− k] · z∗[n− j]]︸︷︷︸Rz[j−k]=σ2

z ·δ[j−k]

= σ2z ·

Kw∑

k=0

|w[k]|2


Medias y varianza de la ISI

Media de la ISI

E [ISI] =K+Kw∑

k=0k "=d

c[k] · E[A[n− k]] = 0

Varianza de la ISI

σ2ISI = E

K+Kw∑

k=0k "=d

c[k] · A[n− k]

·

K+Kw∑

j=0j"=d

c∗[j] · A∗[n− j]

=K+Kw∑

k=0k "=d

K+Kw∑

j=0j "=d

c[k] · c∗[j] · E [A[n− k] · A∗[n− j]]︸︷︷︸RA[j−k]=Es·δ[j−k]

= Es ·K+Kw∑

k=0k "=d

|c[k]|2


Igualdor DFE con limitacion de coeficientes

Asumiendo decisiones correctas

E [ISI] =K+Kw∑

k=0k /∈[d,d+Kb]

c[k] · E[A[n− k]] = 0

Varianza de la ISI

σ2ISI = E

K+Kw∑

k=0k /∈[d,d+Kb]

c[k] · A[n− k]

·

K+Kw∑

j=0j/∈[d,d+Kb]

c∗[j] · A∗[n− j]

=K+Kw∑

k=0k /∈[d,d+Kb]

K+Kw∑

j=0j/∈[d,d+Kb]

c[k] · c∗[j] · E [A[n− k] · A∗[n− j]]︸︷︷︸RA[j−k]=Es·δ[j−k]

= Es ·K+Kw∑

k=0k /∈[d,d+Kb]

|c[k]|2


Igualacion sımbolo a sımbolo optima

El igualador optimo minimiza la Perror

Minimizar ISI o MMSE no minimizan Perror

Problema: Dadas KW + 1 observaciones

qn = [q[n], q[n− 1], · · · , q[n− Kw]]T

encontrar la estima del sımbolo A[n− d]

A[n− d] = g(qn)

tal que la Perror sea mınimaSolucion: Igualador Bayesiano o de mınima Perror


Igualador Bayesiano (Mınima Perror)

Basado en la definicion de estados del canal

O+1 = {oin|A[n− d] = +1}, O−1 = {oi

n|A[n− d] = −1}

Modelo para cada observacion

qn = oin + nn

Asumiendo sımbolos equiprobables

g(qn) =∑

oi∈O+1

exp(− ||qn − oi

n||2

2σ2n

)−

∑

oi∈O−1

exp(− ||qn − oi

n||2

2σ2n

)

Funcion que minimiza la Pe

A[n− d] = sign(g(qn))


Estados del canal - Ejemplo

Ejemplo: p[n] = δ[n] + 12 · δ[n− 1], Kw = 1

Definicion de los estados[

o[n]o[n− 1]

]

︸︷︷︸on

=[

1 1/2 00 1 1/2

]·

A[n]

A[n− 1]A[n− 2]

︸︷︷︸An

Existen 8 estados

o0n =

[−3/2−3/2

], o1

n =[−3/2−1/2

], o2

n =[−1/2+1/2

], o3

n =[−1/2+3/2

]

o4n =

[+1/2−3/2

], o5

n =[

+1/2−1/2

], o6

n =[

+3/2+1/2

], o7

n =[

+3/2+3/2

]

A0n =

2

4−1−1−1

3

5 , A1n =

2

4−1−1+1

3

5 , A2n =

2

4−1+1−1

3

5 , A3n =

2

4−1+1+1

3

5 , A4n =

2

4+1−1−1

3

5 , A5n =

2

4+1−1+1

3

5 , A6n =

2

4+1+1−1

3

5 , A7n =

2

4+1+1+1

3

5


Estados del canal para d = 0

Estados asociados a A[n] = −1: O−1 ={

o0n, o1

n, o2n, o3

n

}

Estados asociados a A[n] = +1: O+1 ={

o4n, o5

n, o6n, o7

n

}

++

++

♣

♣

♣

♣

o0n

o1n

o2n

o3n

o4n

o5n

o6n

o7n


Regiones de decision para d = 0


o0n, o1

n, o2n, o3

n

}


o4n, o5

n, o6n, o7

n

}

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$




o0n, o1

n, o4n, o5

n

}


o2n, o3

n, o6n, o7

n

}

++

♣

♣

++

♣

♣

o0n

o1n

o2n

o3n

o4n

o5n

o6n

o7n




o0n, o1

n, o4n, o5

n

}


o2n, o3

n, o6n, o7

n

}

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$




o0n, o2

n, o4n, o6

n

}


o1n, o3

n, o5n, o7

n

}

+♣

+♣

+♣

+♣

o0n

o1n

o2n

o3n

o4n

o5n

o6n

o7n




o0n, o2

n, o4n, o6

n

}


o1n, o3

n, o5n, o7

n

}

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

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$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

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$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

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$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$


Igualacion/Clasificacion

El problema de igualacion puede interpretarse como unproblema de clasificacion entre estados del canalLa frontera de separacion optima es no lineal (incluso conun canal lineal)El retardo del igualador influye en la frontera de separacion

Ejemplo: Canal de fase no mınima y d = 0⇒ Los estados del canal no son linealmente separables

Igualador Bayesiano: estructura de RBFOtras aproximaciones universales como el MLP sonutilizablesEl numero de estados del canal crece exponencialmentecon la longitud del canal y la del igualador


ZF y MMSE vs. Maximo Margen

Los igualadores lineales ZF y MMSE pueden ser una malaaproximacion del clasificador optimoAlternativa: hiperplano de maximo margen


Igualadores ZF y MMSE para d = 0


o0n, o1

n, o2n, o3

n

}


o4n, o5

n, o6n, o7

n

}

++

++

♣

♣

♣

♣

o0n

o1n

o2n

o3n

o4n

o5n

o6n

o7n

ZF

MMSE (10 dB)

MMSE (0 dB)

.

.....................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................ .

...............................................................................................................................................................................................................................................................................................................................................................................................

. .............................................

. .............................................

. .............................................




o0n, o1

n, o4n, o5

n

}


o2n, o3

n, o6n, o7

n

}

++

♣

♣

++

♣

♣

o0n

o1n

o2n

o3n

o4n

o5n

o6n

o7n

ZF

MMSE (10 dB)

MMSE (0 dB)

.

..........................................................................................................................................................................................................................................................................

..........................................................................................................

..................................................................................................................................................................................................................

.....................................................................................................................................................................

.

.................................................................................................................

..................................................................................................................

.................................................................................................................

.....................................................

. .............................................

. .............................................

. .............................................




o0n, o2

n, o4n, o6

n

}


o1n, o3

n, o5n, o7

n

}

+♣

+♣

+♣

+♣

o0n

o1n

o2n

o3n

o4n

o5n

o6n

o7n

ZF

MMSE (10 dB)

MMSE (0 dB)

.

.....................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................................................................................................................................................. . .............................................

. .............................................

. .............................................


Hiperplano optimo / SVM

Hiperplano optimo: margen

ρ(w) = mınci∈C+1

< w, ci >

||w|| − maxci∈C−1

< w, ci >

||w||

Problema de minimizacion con restriccionesVectores soporte: w =

∑i αisici

Utilizando una funcion nucleo: SVM

K(ci, cj) = exp(− ||ci − cj||22

σ2

)

f (q) = sign

(∑

i

αisiK(ci, qn)

)


Igualacion adaptativa

Metodos anteriores: estiman el igualador empleando unbloque de N datosIgualacion adaptativa (on-line)

Permite seguir canales variantes en el tiempoNo introduce retardosConduce a algoritmos computacionalmente sencillos


Igualacion adaptativa MMSE con LTE

Adaptacion muestra a muestra

Requieren una senal de referencia

Secuencia de entrenamiento Ar[n]

Definicion de error

ed[n] = Ar[n− d]− wTqn

LMS: Minimizacion estocastica de E{|ed[n]|2}

wn+1 = wn + µ · q∗n · ed[n]

RLS: minimizacion determinista den∑

i=1

λn−1 · |ed[n]|2

wn+1 = wn + K∗n+1 · αn+1


Igualacion adaptativa MMSE con DFE

Definicion de la senal de error

ed[n] = Ar[n− d]−(

Kw∑

k=0

w[k] · q[n− k]−Kb∑

k=1

b[k] · A[n− d − k]

)

Durante entrenamiento: A[n− d − k] = Ar[n− d − k]

Agrupacion de filtros

w′ = [w[0], w[1], · · · , w[Kw],−b[1],−b[2], · · · ,−b[Kb]]T

Redefinicion de la senal de entrada

q′n = [q[n], · · · , q[n− Kw], A[n− d − 1], · · · , A[n− d − Kb]]T

Senal de errored[n] = Ar[n− d]− w′T · q′n


Modo dual: entrenamiento + guiado por decision

Modo de trabajo en igualacion no ciegaSecuencia inicial de entrenamiento (hasta convergencia)Conmutacion a modo guiado por decision

La salida del decisor actua como senal de entrenamiento

#A[n]Canal #q[n]

w[n] #u[n]

Decisor

#bA[n − d]

$AlgoritmoAdaptativo

," $-"+

Secuencia de Entrenamiento!

%%

&&'

-.

"

"

Entrenamiento ed[n] = Ar[n− d]− u[n]

Guiado por decision ed[n] = A[n− d]− u[n]


Fractionally Spaced Equalizer (FSE)

El muestreo al perıodo de sımbolo a la salida del filtroadaptado produce estadısticos suficientes para ladeteccion, pero introduce aliasingUn error de muestreo puede generar nulos espectrales enel canal muestreado al perıodo de sımbolo que el igualadordebe compensarSolucion: muestrear a un submultiplo del perıodo demuestreoExtension de T a T/P

Mismas estructuras (LTE, DFE)Mismos criterios (MMSE, min Perror, ZF)


TEMA 4 RECEPTORES DE COMUNICACIONESTEMA 4 RECEPTORES DE COMUNICACIONES MMC (UC3M) Comunicaciones...

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