TEMA 3 MODULACIONES MULTIPULSO OFDM Y ESPECTRO...

TEMA 3

MODULACIONES MULTIPULSO:OFDM Y ESPECTRO ENSANCHADO

MMC (UC3M) Comunicaciones Digitales Modulaciones multipulso 1 / 35

Indice

Modulaciones multiportadora (OFDM)OFDM en tiempo continuoOFDM en tiempo discreto

Modulaciones de espectro ensanchadoEspectro ensanchado por secuencia directaEspectro ensanchado por salto de frecuencia


Modulacion con multiples portadoras - FDM

Division en N subcanales

Tasa de cada subcanal: Rs = 1T baudios

Tasa total: Rs!total = 1Ts

= NT baudios

Ancho de banda de cada subcanal W/N = 2!T · (1 + !) rad/s

Transmision de una senal en cada subcanal (p.e. PAM)

Transmisor

Conversion serie / paralelo: A[m] ! {A0[n], · · · , AN!1}N ramas con senales PAM

Filtro transmisor "k(t), k = 0, · · · , N " 1

Receptor

N filtros adaptados al transmisorConversion paralelo serie: {A0[n], · · · , AN!1}! A[m]


Modulador FDM

!A[m]

S/P

!A0[n]

!0(t)

!A1[n]

!1(t)

!AN!1[n]

!N!1(t)

......

...

!x0(t)

!x1(t)

!xN!1(t)

! !x(t)

TN

T


Demodulador FDM

!y(t)

! !"N!1("t) "" !qN!1[n]

! !! !! !! !"1("t) "" !q1[n]

! !"0("t) "" !q0[n]

#

t = nT

P/S ! Decisor !A[m]q[m]

TN

TN


Inconvenientes solucion FDM

Se necesitan filtros idealesIntervalos de guarda

Perdida de eficiencia espectral

Complejidad del sistemaN filtros (complejos)N muestreadores sıncronos

Solucion:N pulsos ortogonales (con solapamiento)


Modulacion OFDM en tiempo continuo

Senal transmitida

x(t) =#

2 · Re{s(t) · ej"ct}

Senal compleja en banda base

s(t) =!

n

AT [n] · !(t " nT) =!

n

N!1!

k=0

Ak[n] · "k(t " nT).

N pulsos: Filtro prototipo $ N portadoras distintas

"k(t) =1#T

· ej 2!kT ·t · wT(t)

wT(t) =

"1 0 % t < T0 resto


Pulsos ortonormales

Base ortonormal

&!k,!#' =1T

# T

0ej 2!k

T ·t · e!j 2!"T ·t dt

=1T

# T

0cos

$2#(k " $)

T· t

%dt + j

1T

# T

0sen

$2#(k " $)

T· t

%dt

=%[k " $]

Relacion entre los pulsos

"k(t " nT) = "0(t " nT) · e2!k

T ·(t!nT) = "0(t " nT) · e2!k

T ·t


Espectro de OFDM en tiempo continuo

Respuesta en frecuencia de los pulsos

|!k(j&)|2 = T · sinc2

&'& " 2!k

T

(T

2#

), k = 0, · · · , N " 1.

Ak[n] y A#[n] incorreladas y Ak[n] blanca (k

Ss(j&) =1T

·N!1!

k=0

Es,k · |!k(j&)|2

Potencia de la senal transmitida

P =1

2#

# !

"!Ss(j&) d& =

1T

N"1!

k=0

Es,k ·1

2#

# !

"!|!k(j&)|2d& =

1T

N"1!

k=0

Es,k

Constelaciones identicas: P = EsT · N


Espectro OFDM en tiempo continuo - N = 8

-10 -5 0 5 10 15 20 250

0.10.20.30.40.50.60.70.80.91

" · T!

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.........................................................................................................................................................................................................""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

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Espectro asintoticamente plano

Indices infinitos y constelaciones identicas

Ss(j&) =Es

#!

k=!#sinc2

&'& " 2!k

T

(T

2#

)

=Es · sinc2$

&T2#

%)

#!

k=!#%

$& " 2#k

T

%

Solo es plano si se cumple

Es

T· ("0(t) ) "0("t)) ·

#!

k=!#%(t " kT) = C · %(t)


Espectro OFDM en tiempo continuo - N = 16

-20 -10 0 10 20 30 40 500

0.10.20.30.40.50.60.70.80.91

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" · T!


Modulacion OFDM en tiempo discreto

Ancho de banda limitado (aprox. 2#N/T rad./s)

Muestras s[m] cumpliendo Nyquist: a T/N segundosConversion digital analogica

Muestras en el intervalo 0 % t < T (Primeras N muestras)

s[m] =N!1!

k=0

Ak[0] · "k(mT/N), m = 0, · · · , N " 1

Expresion equivalente

s[m] =1#T

·N!1!

k=0

Ak[0] · ej 2!kN ·m, m = 0, · · · , N " 1

IDFT de Ak[0], con k = 0, 1, · · · , N " 1


Expresion general

Muestras de la senal OFDM

s[m] =!

n

N!1!

k=0

Ak[n] · "k(mT/N " nT)

=1#T

!

n

N!1!

k=0

Ak[n] · ej 2!kN ·(m!nN) · wN [m" nN]

wN [m] =

"1 0 % m % N " 10 resto

Filtros reconstructores: a perıodo T/N

g(t) = sinc$

NT

· t%

, G(j&) =TN

· "$

&T2#N

%


Modulador OFDM en tiempo discreto

!A[m]S/P

!A0[n]

!A1[n]

!AN!1[n]

...IDFT

N puntos

!s0[n]

!s1[n]

!sN!1[n]

...P/S !s[m]

g(t) !sr(t)

TN

TN

T T

sr(t) =!

m

s[m] · g(t " mT/N)


Base ortonormal en tiempo discreto

Funciones base

'k[m] =1#N

· ej 2!kN ·m · wN [m], k = 0, · · · , N " 1

Base ortonormal

&'k, '#' =!

m

'k[m] · '"# [m] =1N

·N!1!

m=0

ej 2!(k!")N ·m = %[k " $]

Muestras de la senal

s[m] =*

NT

·!

n

N!1!

k=0

Ak[n] · 'k[m" nN]


Funciones base en tiempo continuo

Senal reconstruida

sr(t) =*

NT

·!

n

N!1!

k=0

Ak[n] ·!

m

'k[m" nN] · g(t " mT/N)

sr(t) =!

n

N!1!

k=0

Ak[n] · "k(t " nT)

Funciones base en tiempo continuo

"k(t) =*

NT

·!

m

'k[m] · g(t " mT/N)

=1#T

·N!1!

m=0

ej 2!kN ·m · sinc

$NT

· (t " mT/N)%


Funciones base en tiempo discreto

-1 -0.5 0 0.5 1 1.5 2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

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.

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

tT

. ....................... #0(t). ....................... #0(t)

-1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

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

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

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

tT

. ....................... #1(t). ....................... #1(t)

-1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

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tT

. ....................... #2(t). ....................... #2(t)

-1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

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...........

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

tT

. ....................... #3(t). ....................... #3(t)


Funciones base en tiempo discreto

-1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

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tT

. ....................... #4(t). ....................... #4(t)

-1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

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tT

. ....................... #5(t). ....................... #5(t)

-1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

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tT

. ....................... #6(t). ....................... #6(t)

-1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

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tT

. ....................... #7(t). ....................... #7(t)


Base ortonormal

&!k, !#' =# #

!#"k(t) · ""#(t)dt

=1T

N!1!

m=0

N!1!

i=0

ej 2!kN ·me!j 2!"

N ·i ·# #

!#g(t " mT/N) · g(t " iT/N)dt

# #

!#g(( " mT/N) · g(( " iT/N)d( = (g(t) ) g("t))|t=(m!i)T/N

Como g(t) cumple el criterio de Nyquist# #

!#g(( " mT/N) · g(( " iT/N)d( =

TN

· %[m" i]

&!k, !#' =1N

N!1!

m=0

ej 2!(k!")N ·m = %[k " $]


Espectro de OFDM en tiempo discreto

Densidad espectral de potencia

Ssr(j&) =1T

·N!1!

k=0

Es,k ·+++!k(j&)

+++2

Respuesta en frecuencia de las funciones base discretas

++#k'ej"(++2 =

1N

sen2[(& " 2#k/N)N/2]sen2[(& " 2#k/N)/2]

Respuesta en frecuencia de las funciones base continuas+++!k(j&)

+++2

=NT

·+++#k

,ej" T

N

-+++2·$

TN

%2

· "$

&T2#N

%

+++!k(j&)+++2

=T

N2 · sen2[(& " 2#k/T)T/2]sen2[(& " 2#k/T)T/2N]

, |&| <#

T· N


Espectro OFDM en tiempo discreto - N = 8

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

" · T!

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"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

""""""""""""""""""""""""""""""""""""""""""""""""


Espectro OFDM en tiempo discreto - Periodicidad

-25 -20 -15 -10 -5 0 5 10 15 20 250

0.10.20.30.40.50.60.70.80.91

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" · T!


Receptor OFDM en tiempo discreto

Salida de cada muestreador

qk[n] =,

v(t) ) ""k("t)-+++

t=nT

=*

NT

·!

m

'"k [m] · (v(t) ) g("t " mT/N))|t=nT

=*

1T

·N!1!

m=0

e!j 2!kN ·m (v(t) ) g("t))|t=nT+mT/N

Secuencia v[m]: muestrear v(t) ) g("t) a perıodo T/N

qk[n] =1#T

N!1!

m=0

e!j 2!kN ·m · v[nN + m]


Receptor OFDM en tiempo discreto (II)

!v(t)g("t) ""

#

t = m · TN

!v[m]S/P

!q0[n]

!q1[n]

!qN"1[n]

...DFT

N puntos

!v0[n]

!v1[n]

!vN"1[n]

...P/SDec

!A[m]

T T

TN


Ruido en el receptor

Filtro receptor subcanal k:#

2 · fk(t)

Densidad espectral de potencia del ruido

Sz,k(ej") =2T

·!

i

Sn

$j&

T" j

&c

T" j

2#iT

%·++++Fk

$j&

T" j

2#iT

%++++2

fk(t) = ""k("t)

n(t): blanco, gausiano, Sn(j&) = N0/2

zk[n] blanco, circularmente simetrico

)2z,k = N0, k = 0, · · · , N " 1

Ortogonalidad de los pulsos en cada subcanal

E{zi[n] · z"k [n]} = 0, si i *= k


Canal discreto equivalente en banda base

Salida antes del muestreador

qk(t) =N!1!

i=0

!

#

Ai[$] · pk,i(t " $T) + zk(t), k = 0, · · · , N " 1

Respuesta conjunta: pk,i(t) = "i(t) ) heq(t) ) fk(t)

Salida muestreada

qk[n] =N!1!

i=0

!

#

Ai[$] · pk,i[n" $] + zk[n], k = 0, · · · , N " 1

qk[n] =N!1!

i=0

Ai[n] ) pk,i[n] + zk[n], k = 0, · · · , N " 1

N2 canales discretos equivalentes


Criterio de Nyquist generalizado

Interferencia entre sımbolos (ISI)

pi,i[n] = K · %[n]

Interferencia entre portadoras (ICI)

pk,i[n] = 0, para k *= i

Criterio de Nyquist generalizado en frecuencia

P(ej") = IN$N

Pk,i(ej"): transformada de Fourier de pk,i[n]P(ej"): matriz con elementos Pk,i(ej") (fila k, columna i)

Difıcil de satisfacer: N2 restricciones, N grados de libertad


Extension al caso discreto

Respuesta conjunta

pk,i(t) =NT

!

m

!

#

'i[m] · '"k [$] · (g(t"mT/N) ) heq(t) ) g("t" $T/N))

Canal discreto equivalente

pk,i[n] =1T

N!1!

m=0

N!1!

#=0

ej 2!iN ·m · e!j 2!k

N ·# · d[nN + $" m]

Secuencia d[m]

d[m] = (g(t) ) heq(t) ) g("t))|t=mT/N

Cumplimiento del criterio de Nyquist: d[n] = K · %[n]


Eliminacion de la ISI y la ICI

Canal d[n] causal y de duracion K + 1

Nuevas funciones base en tiempo discreto

'k[m] =1#N

· ej 2!kN ·m · wN+M[m + M], k = 0, · · · , N " 1

Valores no nulos en ["M, N " 1] (en lugar de en [0, N " 1])Hipotesis para el desarrollo: M + K

Senal en tiempo discreto

s[m] =*

NT

·!

n

N!1!

k=0

Ak[n] · 'k[m" n(N + M)]

Senal en el demodulador

qk[n] =1#T

·N!1!

m=0

'"k [m] · v[n(N + M) + m]


Canal discreto equivalente

pk,i[n] =1T

·N!1!

m=!M

N!1!

#=0

ej 2!iN ·m · e!j 2!k

N ·# · d[n(N + M) + $" m]

pk,i[n] =1T

·N!1!

#=0

#+M!

u=#!N+1

e!j 2!iN ·u · ej 2!(i!k)

N ·# · d[n(N + M) + u]

pk,i[n] =1T

·K!

u=0

e!j 2!iN ·u · d[u] · %[n] ·

N!1!

#=0

ej 2!(i!k)N ·#

=NT

· %[n] · %[k " i] ·K!

u=0

e!j 2!iN ·u · d[u]

ISI=ICI=0: siendo D[k] la DFT de N puntos de d[n]

qk[n] =NT

· Ak[n] · D[k]


OFDM como proceso en bloques

Muestras: s(n)[m] obtenidas de IDFTN(Ak[n]), m = 0, · · · , N " 1

Extension cıclica de las muestras

s(n)[m] =

"s(n)[m + N] m = "M, · · · ,"1s(n)[m] m = 0, · · · , N " 1

Transmision a traves de d[m]

q(n)[m] = s(n)[m] ) d[m] + z(n)[m]

Eliminar extension cıclica

q(n)[m] = q(n)[m] · wN [m]

Estima Ak[n] = DFTN(q(n)[m])


OFDM como proceso en bloques (II)

Q(n)[k]: DFT de N puntos de q(n)[m]

Q(n)[k] =N!1!

m=0

q(n)[m] · e!j 2!kN ·m =

N!1!

m=0

q(n)[m] · e!j 2!kN ·m

Q(n)[k] = DFTN(q(n)[m]) = DFTN(q(n)[k]) = DFTN(s(n)[m] ) d[m])

Convolucion lineal de s(n)[k] y d[m] equivale a la convolucioncircular de s(n)[m] y d[m]

Q(n)[k] =DFTN(s(n)[k]) · DFTN(d[m])=IDFTN(DFTN(Ak[n])) · DFTN(d[m])=Ak[n] · D[k]


Modulador/demodulador OFDM con prefijo cıclico

!A[m]S/P

!A0[n]

!A1[n]

!AN"1[n]

...IDFT

N puntos

!

!

!

...P/S !s[m] Extension

cıclica!s[m] g(t) !sr(t)

TN

TN

TN + M

T T

!v(t) g("t) ""

#

t = m · TN+M

! Eliminarprefijocıclico

!v[m]S/P

!q0[n]

!q1[n]

!qN"1[n]

...DFT

N puntos

!

!

!

...P/SDec

!A[m]

TN

TNT T


Cuestiones de interes practico

SincronizacionPortadorasMuestreoTrama

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