ee360 – lecture 3 outline

EE360 – Lecture 3 Outline

Announcements:Classroom Gesb131 is available, move on

Monday?Broadcast Channels with ISIDFT DecompositionOptimal Power and Rate AllocationFading Broadcast Channels

Broadcast Channels with ISI

ISI introduces memory into the channel

The optimal coding strategy decomposes the channel into parallel broadcast channelsSuperposition coding is applied to each

subchannel.

Power must be optimized across subchannels and between users in each subchannel.

Broadcast Channel Model

Both H1 and H2 are finite IR filters of length m. The w1k and w2k are correlated noise samples. For 1<k<n, we call this channel the n-block

discrete Gaussian broadcast channel (n-DGBC). The channel capacity region is C=(R1,R2).

w1kH1()

H2()w2k

xk

y h x wk ii

m

kk i1 11

1

y h x wk ii

m

kk i2 21

2

Circular Channel Model

Define the zero padded filters as:

The n-Block Circular Gaussian Broadcast Channel (n-CGBC) is defined based on circular convolution:

{~} ( ,..., , ,..., )h h hi in

m 1 1 0 0

~ ~(( ))y h x w x h wk i

i

n

k i i kk i1 10

1

1 1 1

~ ~(( ))y h x w x h wk i

i

n

k i i kk i2 20

1

2 2 2

0<k<n

where ((.)) denotes addition modulo n.

Equivalent Channel Model

Taking DFTs of both sides yields

Dividing by H and using additional properties of the DFT yields

~ ~Y H X Wj j j j1 1 1 ~ ~Y H X Wj j j j2 2 2

0<j<n

~

Y X Vj j j1 1

Y X Vj j j2 2

where {V1j} and {V2j} are independent zero-mean Gaussian random variables with lj l ljn N j n H l2 22 1 2 ( ( / )/| ~ | , , .

0<j<n

Parallel Channel Model

+

+X1

V11

V21

Y11

Y21

+

+Xn

V1n

V2n

Y1n

Y2n

Ni(f)/Hi(f)

f

Channel Decomposition

The n-CGBC thus decomposes to a set of n parallel discrete memoryless degraded broadcast channels with AWGN. Can show that as n goes to infinity, the circular and

original channel have the same capacity region

The capacity region of parallel degraded broadcast channels was obtained by El-Gamal (1980)

Optimal power allocation obtained by Hughes-Hartogs(’75).

The power constraint on the original channel is converted by Parseval’s theorem to on the equivalent channel.

E x nPii

n

[ ]2

0

1

E X n Pii

n

[( ) ]

0

12 2

Capacity Region of Parallel Set

Achievable Rates (no common information)

Capacity Region For 0< find {j}, {Pj} to maximize R1+R2+ Pj. Let (R1

*,R2*)n, denote the corresponding rate pair.

Cn={(R1*,R2

*)n, : 0< }, C=liminfn Cn .1n

PnP

PP

PR

PPP

R

jj

j j

jj

j jjj

jj

j jjj

jj

j j

jj

jjjj

jjjj

2: 2: 2

2

: 1: 11

,10

,)1(

1log5.)1(

1log5.

,)1(

1log5.1log5.

2121

2121

R1

R2

Limiting Capacity Region

PdffPf

NfHfPf

fHNfPffPfR

PP

NfHfPfR

fHfHffHfHf

fHfHf jjj

jj

fHfHf

)(,1)(0

,5.

|)(|)())(1(1log5.|)(|/5.)()(

)())(1(1log5.

,)1(

1log5.5.

|)(|)()(1log5.

)()(: 0

22

)()(:2

202

)()(: 1)()(: 0

21

1

2121

2121

Optimal Power Allocation:

Two Level Water Filling

Capacity vs. Frequency

Capacity Region

Fading Broadcast Channels

Broadcast channel with ISI optimally allocates power and rate over frequency spectrum.

In a fading broadcast channel the effective noise of each user varies over time.

If TX and all RXs know the channel, can optimally adapt to channel variations.

Fading broadcast channel capacity region obtained via optimal allocation of power and rate over time Consider CD, TD, and FD.

Two-User Channel Model

+

+X[i]

1[i]

2[i]

Y1[i]

Y2[i]

x

x

g1[i]

g2[i]

+

+X[i]

1[i]/g1[i]

2[i]/g2[i]

Y1[i]

Y2[i]

At each time i:n={n1[i],n2[i]}

CD with successive decoding

M-user capacity region under CD with successive decoding and an average power constraint is:

The power constraint implies

)()()( PPFP CDCD CwhereCPC

}1,][1)(

)(1log

1

MjnnnPBn

nPBER M

iijij

jnj

PnPEM

jjn

1

)(

Proof

Achievability is obviousConverse

Exploit stationarity and ergodicityReduces channel to parallel degraded

broadcast channelCapacity known (El-Gamal’80)Optimal power allocation known

(Hughes-Hartogs’75, Tse’97)

Capacity Region Boundary

By convexity, RM+, boundary vectors satisfy:

Lagrangian method:

Must optimize power between users and over time

})]([][1)(

)(1log{max

1

1

1)(

M

jjnM

iijij

jM

jjnnP

nPEnnnPBn

nPBE

RPCR

)(max

Water Filling Power Allocation Procedure

For each state n, define (i):{n(1)n(2)…n(M)}

If set P(i)=0 (remove some users)

Set power for cloud centers

Stop if ,otherwise remove n(i), increase noises n(i) by P(i), and return to beginning

)1(

)1(

)(

)(

i

i

i

i

nn

)1()(

)1()()()1(

)1(

)1(

1)1( min,min

i

iiBnBn

BnPi

][)1(

)1(

)1( BnP

Time Division For each fading state n, allocate power Pj(n) and

fraction of time j(n) to user j.

Achievable rate region:

Subject to

Frequency division equivalent to time-division

)(),()( PPFP TDTD CwhereCPC

}1,)(

1log)( MjBnnP

BnERj

jjnj

M

jjj

M

jj nPnPnandn

11

)()()(1)(

OptimizationUse convexity of region: boundary vectors

satisfy

Lagrangian method used for power constraint

Four step iterative procedure used to find optimal power allocationFor each n the channel is shared by at most 2 usersSuboptimal strategy: best user per channel state is

assigned power – has near optimal TD performance

RPCR

)(max

CD without successive decoding

M-user capacity region under CD with successive decoding and an average power constraint is:

The best strategy for CDWO is time-division

)()()( PPFP CDWOCDWO CwhereCPC

}1,][1)(

)(1log

)(1

MjnnnPBn

nPBER M

jiiijij

jnj