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EE360 – Lecture 3 Outline. Announcements: Classroom Gesb131 is available, move on Monday? Broadcast Channels with ISI DFT Decomposition Optimal Power and Rate Allocation Fading Broadcast Channels. Broadcast Channels with ISI. ISI introduces memory into the channel - PowerPoint PPT Presentation
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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
