Multiple Access Channels

8/8/2019 Multiple Access Channels

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Network Information Theory

Multiple access channels

Broadcasting channel

Capacity region


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Network Information TheoryCommunication problems:

interference, cooperation and feedback

Many senders and receivers and channel transition matrix.

Distributed source coding (data compression), distributed

Communication (capacity region).

Multiple access channels

Broadcasting channel


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Examples of large communication network: computer

networks, satellite network and the phone system.

Other channels: relay channel, interference channel andChannel.

Relay channel: there is one source and one destination, but one or more

intermediate sender-receiver pairs that act as relays to facilitate the

communication between the source and one destination.

Network Information Theory


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Gaussian Multiple User Channels

The channel with input power P and additive white Gaussian

noise channel and noise variance is modeled by

Yi = Xi + Zi , i = 1, 2, 3, where Zi are i.i.d. Gaussian

random variables with mean zero and variance N.

The signal X = (X1, X2,, Xn) has a constrain

For a single userGaussian channel

Y = X+Z

PXn

n

i

i e!1

21

issionper transmbits),1log(2

1);(max

][:)( 2 N

PYXIC

PXEXp!!

e

)1log(2

1

N

PR


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The Gaussian multiple access channel with m users

)NP(

NPCZXY

m

i

i !!!

1log21)(and

1

capacity of a single user

The achievable rate region for the Gaussian channel is given

)(

)3

(

)2

(

)(

1 N

mPCR

N

PCRRR

N

PCRR

N

PCR

m

i

i

jji

ji

i

!

Here we have m codebooks, the i-th codebook

having codeword of power P.

Each of the independent transmitters chooses

an arbitrary codeword from the own codebook

and simultaneously send these vectors. The

Gaussian noise is added. Y = X + Z

The receive looking for the m codewords.

If (R1, R2, . . , Rm) is in the capacity region,

then the probability of error goes to 0 as n

tends to infinity.

inR2


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The Gaussian broadcasting channel

The model of channel: Y1

= X + Z1

Y2

= X + Z2

where Z1 and Z2 are arbitrarily correlated Gaussian random

variables with variance N1 and N2. The sender wishes to

send independent message at rate R1 and R2 to receivers Y1and Y

2.

The capacity region:

10,)1(

and2

2

1

1 ee

E

EE

N

PCR

N

PCR

The transmitter generates two codebooks: P, R1, (1- )P, R2.

The transmitter send the sum of the codewords X(i) +X(j).

The receivers decodes their message.

}2...,2,1,{and}2...,2,1,{ 21nRnR ji


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The Gaussian relay channel

For relay channel, we have a sender X and an ultimate intended receiver

Y. Also present is the relay channel intended solely to help the receive.

X Y

Y1:X1

Y1= X + Z1Y = X + Z1 + X1 + Z2

where Z1 and Z2 are independent zero mean

Gaussian random variables with varianceN1 and N2, respectively.

X1i = fi(Y1i, Y2i, , Y1(i-1))

X: power P; X1: power P1; Capacity

!

ee121

11

10,

)1(2minmax

N

PC

NN

PPPPCC

EE

E

1N

PCC

,if

1

12

1

!

!

u

E

N

P

N

P


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The channel appears to be noise-free after relay, and the

capacity C(P/N1) from X to the relay can be achieved. Thus

the rate C(P/(N1+N2)) without the relay is increased by the

presence of the relay to C(P/N1). For large N2, and for

we see that the increment in the rate is from

12

1

N

P

N

P

u

)C(P/NNNPC 121 to0))/(( }

The Gaussian relay channel


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The Gaussian Two-way Channel

Very similar to interference channel, with the additional provision thatsender 1 is attached to receiver 2 and the sender 2 is attached to

receiver 1.

),/,( 2121 xxyyp

X1 X2

Y1 Y2 1W2W

W1 W2

Let P1 and P2 : the power of transmitters 1 and 2

N1 and N2: the noise variances of the two channel

R1 < C(P1/N1) and C(P2/N2)


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Jointly Typical Sequences

Let (X1, X2, , Xk) a finite collection of discrete random variables with some

fixed joint distribution,p(x1,x2, ,xk). Let S the sub-set of these randomvariables and considern independent copies of S. Thus

By the law of the large numbers, for any subset of random variables,

Definition: The set de -typical n-sequence (x1,x2, ... ,xk) is defined by

!

!

!!!!!

!!!

n

k

jkikjijiji

n

i

ii

xxpPsSPS

sSPsSP

1

1

),()],(),[()(),,(iexampleFor

)()(

xxXXXX

)()(log1

),...,,(log1

1

21 SHSp

n

SSSp

n

n

i

in p! !)(nAI

!

!

],...,,[,)()(logn

1:)(

),...,,(

21

)(

21

)(

kk21

n

k

n

XXXSSHp

AXXXA

I

II

sx,...,x,x


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If S=(X1,X2), we have

Notation

Theorem: For any > 0, for sufficiently large n

})()(log1

,)()(log1

,),()(log1:){(),(

21

2121

)(

II

II

!

XHpn

XHpn

XXHpn

XXA

21

2121

n

xx

x,xx,x

l rls ffici tf rt log1

2 )( II ! s ban

a nbn

n

)2)/((

21

)(

2121

)2)(()(

))(()(21

)(

212)(then

),,(),(If}.,...,,{,Let.4

2)(.3

2)()(.2

},...,,{,1)]([.1

I

I

I

I

I

I

I

I

s

s

s

!

!

!

u

SSHn

n

k

SHnn

SHnnk

n

p

SSAXXXSS

SA

pSA

XXXSSAP

21

21

/ss

xx

ss


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Theorem:

Let S1,S2 be two subset of X1, X2, , Xk. For any >0, defineto be the set ofs1 sequences that are jointly -typical with a particulars2sequence.

If then for sufficiently large n, we have.

Let denote the typical set for the probability mass functionp(s1,s2,s3),

and let

)/( 1)( 2sSA nI

),( 2)( SA n

I

2s

ee

2

2121 |)/(|)(2)1(and2)/( 1)()2)/(()2)/((1)(s

nSSnSSnn SApSA 222 sss IIII I

)(nA

I

)6)/;(()(

21

3323

1

1221

2212),,{(

)()/()/(),,(

I

I

s

!

!ddd

!!d!d!d SSSInn

iiii

n

i

i

ASSSPthen

spsspsspSSSP

2

321sss


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The multiple access channel

Definition: A discrete memoryless multiple access channel consists of

three alphabets, X1, X2and Y, and a probability transition matrixp(y/x1,x2).

A Code for multiple access channel consists of two set of

integers W1 = and W2 = called the message

sets, two encode function

X1: W

1 X

1n, ; X2 :W2 X2

n

Decode function g: Yn W1

xW2

),2,2( 21 nnRnR

p(y/x1,x2)

W1

W2

X1

X2

Y ),( 11 WW

}2...,2,1,{ 1nR

}2...,2,1,{ 1nR

send}),(|),()({2

1

21),(

21)(

)(

21

21

wwwwYgPPxww

n

RRn

n

e

{!

21


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The multiple access channel

The capacity region of the multiple access channel is the closure of the set

of achievable (R1, R2).Theorem:The capacity of a multiple access channel (X1, X2,p(y/x1,x2), Y)

is the closure of the convex hull of all (R1, R2) satisfying

R1 < I(X1; Y/X2) ; R2 < I(X2; Y/X1) ; R1 + R2 I( (X1;X2); Y)

for some distributionp1(x1)p2(x2) on X1 xX1

Example of the capacity region for a multiple access channel

R2

C2

C1 R1

I(X2; Y)

I(X1; Y)


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Independent binary symmetric channel

1-p1

1-p1

0

1

0

1

X1 Y1p1

p1

1-p2

1-p2

0

1

0

1

X2 Y2

p2

p2

R2

C2=1-H(p2)

C1=1-H(p1) R10

Binary multiplier channel : Y = X1 X2

Setting X2=1, we can send at a rate of 1

bit per transmission from sender 1

to receiver. If X1=1, we can achieve R2=1.

R1 + R2 = 1

R2

C2=1

C1=1 R10

Capacity region

Capacity region


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Binary erasure multiple access channel

Capacity region

R2

C2=1

C1=1 R1

1/2

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Binary input X1= X

2= (0,1) and a ternary output Y = X1+ X2

1-p

CBEC = 1 - p

p = 1/2;

CBEC=1 bit per transmission

1-p

0

1

0

E

1

Yp

p

X

X1

X2

0

1

2

Y


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The capacity region of the multiple access channel

The closure of the set of achievable (R1, R2).

Theorem:The capacity of a multiple access channel (X1, X2,p(y/x1,x2), Y)is the closure of the convex hull of all (R1, R2) satisfying

R1 < I(X1; Y/X2) ; R2 < I(X2; Y/X1) ; R1 + R2 I( (X1;X2); Y)

for some distributionp1(x1)p2(x2) on X1 xX1

The point A correspond to the maximum

rate achievable from sender 1 to the

receiver 2 is not sending any

Information. This is

maxR1 < maxp(x1)p(x2) I(X1; Y /X2) ;

For any distributionp1(x1)p2(x2)

)/;(max

)/;()()/;(

221

2212221

2

xXYXI

xXYXIxpXYXIX

!e

!!

R2

I(X2;Y/X1)

I(X1;Y/X2) R1

I(X2; Y)

I(X1; Y)

D C

B

A

Achievable region of multiple

access channel or fixed input

Distribution.


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Multiple access channel Gaussian - MAC

Independent joint Gaussian input distributions achieves the capacity region

)1log(2

1)2log(

2

1))(2log(

2

1

)2log(21)(

)()(),/()/(

),/()/(

),/()/()/;(

11

1

12121

2121221

21221

N

PNeNPe

NeZXh

ZhZXhXXZhXZXh

XXZXXhXZXXh

XXYhXYhXYXI

!e

!

!!

!

!

TT

T


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Multiple access channel Gaussian capacity region

)();();(:)1log(

2

1)( 2121

22

11

N

PPCRR

N

PCR

N

PCRxxC

!


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Example:


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Distributed Source Coding

How to encoding a source X: A rate R > H(X) is sufficient.

Two sourceH

(X,Y) ~p(x,y): A rate R >H

(X,Y) is sufficient.

If X-source and Y-source are separately described: R=Rx+Ry>H(X)+H(Y)

is sufficient.

R=H(X,Y)


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Slepian-Wolf Theorem

Theorem: For the distributed source coding probability for the source (X,Y)

drawn i.i.d ~p(x,y), the achievable rate region is given by


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Broadcast Channel

One-to-many channel Downlink of cellular or satellite channels

TV, radio broadcasting, DMB, DVB


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General capacity region unknown

Capacity region known for degraded broadcast channel Physically degraded X Y1 Y2 Stochastically degraded

Same conditional marginal distributions as a

physically degraded channel

Broadcast capacity depends only on conditional

marginal distributions since users do not cooperate

Superposition coding is optimal

Example) Gaussian broadcast channel

Capacityregion: convex hull of the closure of all (R1, R2)

such that

Broadcast capacity region

222

22222

)1()1(

)(1)/()();(

ppp

pHUYHYHYUIR

FFF

F

!

!!e


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Gaussian Broadcast Channel


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Gaussian Broadcast Channel


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Example:


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Example:


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Gaussian Vector Broadcast Channel

Documents

Multiple Access Channels