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Rate Splitting Approachesfor Noncoherent Fading Channels
Adriano Pastore
Ecole Polytechnique Federale de LausanneIC/ISC/LINX Laboratory
Lausanne, [email protected]
Workshop on Recent Advances in Network Information TheoryINSA Lyon
November 24, 2015
Outline
1 Noncoherent fading with LOSCapacity lower bounds via rate splittingA mismatched decoding perspective on rate splitting
2 Noncoherent fading without LOSCapacity lower bounds via rate splitting
3 The multiple-access channel
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Outline
1 Noncoherent fading with LOSCapacity lower bounds via rate splittingA mismatched decoding perspective on rate splitting
2 Noncoherent fading without LOSCapacity lower bounds via rate splitting
3 The multiple-access channel
2 / 35
Channel model
Yt = (H + Ht)Xt + Zt (t ∈ Z)
Assumptions:
• {Xt} are symbols from an i.i.d. Gaussian codebook
C ={(
X1(m), . . . ,Xn(m))
: m ∈[1:enR
]}• {Ht} is complex zero-mean i.i.d. ergodic
• {Zt} is complex zero-mean i.i.d. ergodic (not necessarily Gaussian)
• {Xt}, {Ht} and {Zt} are mutually independent
• Tx and Rx know distributions but ignore realizations of Ht and Zt
Channel coding theorem
The supremum of rates that support reliable communication is
limn→∞
1
nI (X n;Y n) = I (X ;Y )
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Medard’s lower bound
Y =(H + H
)X + Z
= HX + HX + Z︸ ︷︷ ︸effective noise
“Worst-case noise bound” (Medard 2000)
I (X ;Y ) ≥ log
(1 +
|H|2Pσ2 P + σ2
), ILB
M. Medard, “The effect upon channel capacity in wireless communications of perfectand imperfect knowledge of the channel,” IEEE Trans. on Inf. Theory, May 2000.
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The rate-splitting approach
Fix a power allocation P = (P1,P2) with P1 + P2 = P.
Split X into independent Gaussian signals:
X = X1 + X2 P = P1 + P2
Y =(H + H
)(X1 + X2
)+ Z
Chain rule
I (X ;Y ) = I (X1,X2;Y ) = I (X1;Y ) + I (X2;Y |X1)
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The rate-splitting approach1 “Decode” X1 while treating effective noise as AWGN:
Y = HX1 + HX2 + H(X1 + X2) + Z︸ ︷︷ ︸effective noise
I (X1;Y ) ≥ log
(1 +
|H|2P1
|H|2P2 + σ2P + σ2
), R1(P)
2 Decode X2 knowing X1:
Y ′ = Y − HX1
= HX2 + H(X1 + X2) + Z︸ ︷︷ ︸effective noise
I (X2;Y |X1) ≥ EX1
[log
(1 +
|H|2P2
σ2(|X1|2 + P2) + σ2
)], R2(P)
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The rate-splitting approach
• Medard’s bound: ILB
• Rate-splitting bound: ILB,RS(P) , R1(P) + R2(P)
ILB,RS(P) = R1(P) + EX1
[log
(1 +
|H|2P2
σ2(|X1|2 + P2) + σ2
)]Jensen≥ R1(P) + log
(1 +
|H|2P2
σ2(P1 + P2) + σ2
)= ILB
I (X ;Y ) ≥ ILB,RS ≥ ILB(P)
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Infinitesimal rate splitting
Fix a power allocation P = (P1, . . . ,PL) with P1 + . . .+ PL = P.
I (X ;Y ) =L∑`=1
I (X`;Y |X1, . . . ,X`−1)
≥L∑`=1
E
[log
(1 +
P`|H|2
σ2∣∣∑i<`
Xi
∣∣2︸ ︷︷ ︸residual interf.from decoded
signals
+ σ2P`︸︷︷︸fading noise
+(|H|2 + σ2
) ∑i>`
Pi︸ ︷︷ ︸signals yet to be decoded
+σ2
)]
,L∑`=1
R`(P)
= ILB,RS(P)
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Infinitesimal rate-splitting
Theorem
The best rate-splitting bound is
I ?LB , supP : P1+...+PL=P
ILB,RS(P)
= limmax` P`↓0
ILB,RS(P)
= EU
[∫ 1
0
|H|2
(|H|2 + σ2)(1− λ) + σ2Uλ+ ρ−1dλ
]= EU
[|H|2
σ2(U − 1)− |H|2log
(1 +
σ2(U − 1)− |H|2
|H|2 + σ2 + ρ−1
)]where U is unit-mean exponential.
[1] A. Pastore, T. Koch, J.R. Fonollosa, “A rate splitting approach to fading channels withimperfect channel-state information,” IEEE Trans. on Inf. Theory, Jul. 2014
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Numerical results
−10 −5 0 5 10 15 20 25 300
0.5
1
1.5
2
P/σ2 [dB]
MI
[bit
s/ch
an
nel
use
]
upper bound on I (X ; Y )
I?LB,RS
ILB
Figure : Channel parameters: |H|2 = σ2 = 1/2.
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Nearest-neighbor decoding
Maximum-likelihood (matched) decoding:
m = argmaxm∈[1:enR ]
n∏t=1
p(Yt |Xt(m))
Nearest-neighbor (mismatched) decoding:
m = argminm∈[1:enR ]
n∑t=1
∣∣Yt − HXt(m)∣∣2
Generalized mutual information
The GMI (associated to NND and to i.i.d. Gaussian codes) is the
supremum of rates R such that Pr{m 6= m} n→∞−−−→ 0 and satisfies
GMI ≤ I (X ;Y )
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Medard’s bound and NND
Theorem [Lapidoth and Shamai 2002]
The GMI under nearest-neighbor decoding and Gaussian codes satisfies
GMI = ILB
Can this insight be extended to rate splitting?
A. Lapidoth and S. Shamai, “Fading channels: how perfect need perfect side information be?,”IEEE Trans. on Inf. Theory, May 2002
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The naive extension: successive NND
m1 = argminm1∈[1:enR1 ]
n∑t=1
∣∣Yt − HX1,t(m1)∣∣2
m2(m1) = argminm2∈[1:enR2 ]
n∑t=1
∣∣∣Yt − HX1,t(m1)− HX2,t(m2)∣∣∣2
This decoding rule achieves a sum rate
supθ<0
{T1θ − Λ1(θ)
}+ supθ<0
EX1
[T2θ − Λ2(θ)
]= ILB
where
T1 = |H|2P2 + σ2P + σ2
Λ1(θ) = (|H|2+σ2)P+σ2
1−|H|2P1θθ − log
(1− |H|2P1θ
)T2 = σ2P + σ2
Λ2(θ) = |H|2P2+σ2(|X1|2+P2)+σ2
1−|H|2P2θθ − log
(1− |H|2P2θ
)13 / 35
The naive approach: why does it fail?
We would wish to swap sup and EX1 :
ILB = supθ<0
{T1θ − Λ1(θ)
}+ supθ<0
EX1
[T2θ − Λ2(θ)
]
≤ supθ<0
{T1θ − Λ1(θ)
}+ EX1
[supθ<0
{T2θ − Λ2(θ)
}]= ILB,RS(P)
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The smart approach: successive NND + multiplexing
Thresholds: 0 = ξ0 < ξ1 < ξ2 < . . . < ξK =∞Time partitions: Tk =
{t ∈ N : ξk−1 ≤ |X1,t | < ξk
}Time-multiplexing of K virtual users on the second layer:
supθ<0
EX1 [T2θ − Λ2(θ)]
K>1−−−→K∑
k=1
supθ<0
EX1
[T2θ − Λ2(θ)
∣∣∣ ξk−1 ≤ |X1| < ξk
]· µ|X1|
([ξk−1; ξk )
)K�1≈
K∑k=1
supθ<0
{T2θ − Λ2(θ)
}∣∣∣∣|X1|=ξk
· µ|X1|
([ξk−1; ξk )
)K→∞−−−−→
∫ ∞0
supθ<0
{T2θ − Λ2(θ)
}∣∣∣∣|X1|=ξ
· dµ|X1|(ξ)
= EX1
[supθ<0
{T2θ − Λ2(θ)
}]15 / 35
The smart approach: in-layer multiplexing
ξ1
ξ2|X1[t]|
|X2,1[t]|
|X2,2[t]|
|X2,3[t]|
time index t
|X2[t]|
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Power allocation diagrams
(1,1)
(2,1) (2,2) (2,3)
Figure : Example of a power allocation diagram
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Power allocation diagrams
(1,1)
(2,1) (2,2) (2,3) (2,4)
(3,1) (3,2)
(4,1) (4,2) (4,3) (4,4) (4,5)
Figure : Example of a power allocation diagram for L = 4.
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Power allocation
(1,1)
(2,1) (2,2)
Figure : Allocation diagram for L = 2
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Power allocation
00.2 0.4 0.6 0.8 1 0
0.5
10.85
0.86
0.87
P0/P11− e−ξ
sum
-GM
I
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Outline
1 Noncoherent fading with LOSCapacity lower bounds via rate splittingA mismatched decoding perspective on rate splitting
2 Noncoherent fading without LOSCapacity lower bounds via rate splitting
3 The multiple-access channel
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Looseness of previous bounds
Y = (H + H)X + Z
Lower bounds on I (X ;Y ) (reminder)
ILB = log
(1 +
|H|2Pσ2P + σ2
)I ?LB,RS = EU
[∫ 1
0
|H|2P(1− λ)(|H|2 + σ2)P + λσ2UP + σ2
dλ
]
For H = 0 we get ILB = I ?LB,RS = 0 but we know I (X ;Y ) > 0 /
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Failure of Medard’s bounding approach
Medard’s bound (reminder)
ILB = log
(1 +
|H|2Pσ2P + σ2
)≤ I (X ;Y )
Proof:
I (X ;Y ) = h(X )− h(X |Y )
= log(πeP)− h(X − X (Y )|Y )
≥ log(πeP)− h(X − X (Y ))
≥ log(P)− log(var(X − X (Y ))
)= log(P)− log
(P + var(X (Y ))− 2 Re{E[X ∗X (Y )]}
)should be 6= 0
Choosing X (Y ) = αY does not do the trick since E[X ∗X (Y )] = αHP
This approach fails for H = 0
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A fix: biased signalling + non-linear input estimator
Suggestion:
{X ∼ NC(
√P0,P1) with P0 + P1 = P
X (Y ) = α|Y |2
E[X ∗X (Y )] = ασ2P1
√P0
After optimizing α:
Lower bound on I (X ;Y ) for noncoherent fading without LOS
I (X ;Y ) ≥ log
(var(|Y |2)
var(|Y |2)− σ4P0P1
)
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Bias/signal power tradeoff
Theorem
The optimal power tradeoff is attained at a ratio
P0
P1
∣∣∣∣opt
=η(snr)−
√η(snr)
(η(snr)− snr2κH
)snr2κH
where
• κA = E[|A|4]/E[|A|2]2 denotes the kurtosis of A
• snr , P/σ2
• η(snr) , (2κH − 1)snr2 + 2snr + κZ − 1
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Bias/signal power tradeoff
limP/σ2→0
P0
P1
∣∣∣∣opt
=1
2
limP/σ2→∞
P0
P1
∣∣∣∣opt
= 2− 1
κH
−
√(2− 1
κH
)(1− 1
κH
)
−20 −10 0 10 20 30
0.50
0.60
0.70
P/σ2 [dB]
P0
P1
∣ ∣ ∣ op
t
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Infinitesimal rate splitting
Consider allocations P = (P0,P1, . . . ,PL) with
P0 + P1 + . . .+ PL = P
X0 + X1 + . . .+ XL = X
with X0 =√P0 = const.
I (X ;Y ) = I(X0;Y
)+
L∑`=1
I (X`;Y |X1, . . . ,X`−1)
≥ 0 +L∑`=1
E
[log
(var(|Y |2 | X `)
var(|Y |2 | X `)− σ4P`|X `|2
)]
≈L∑`=1
E
[σ4P`|X `|2
var(|Y |2 | X `)
]L→∞−−−→
∫ 1
0
E
[σ4|Xλ|2
var(|Y |2 | Xλ)
]P dλ
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Infinitesimal rate splitting
Written out in full:
I ?LB,RS = snr2
∫ 1
0
∫ ∞0
λν
D (λν, 1− λ)e−ν dν dλ
where
D(x , y) = snr2(κH − 1)x2 + snr2(2κH − 1)y(2x + y)
+ 2snr(x + y) + κZ − 1
and
• κA = E[|A|4]/E[|A|2]2 denotes kurtosis
• snr , P/σ2
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Numerical results
0 2 4 6 8 10 12 140
0.1
0.2
0.3
P/σ2 [linear]
nat
s/
chan
nel
use
I (X ; Y ) (Rayleigh fading + AWGN)
ILB,RS
ILB
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Outline
1 Noncoherent fading with LOSCapacity lower bounds via rate splittingA mismatched decoding perspective on rate splitting
2 Noncoherent fading without LOSCapacity lower bounds via rate splitting
3 The multiple-access channel
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Multiple-access channel
Yt = (H1 + H1,t)X1,t + (H2 + H2,t)X2,t + Zt (t ∈ Z)
MAC rate region
For fixed (P1,P2) the rate region is the convex hull around[I (X1;Y |X2) I (X2;Y )
]and
[I (X1;Y ) I (X2;Y |X1)
]An inner bound is obtained using lower bounds
ILB,RS?(X1;Y |X2) ≤ I (X1;Y |X2)
ILB,RS?(X2;Y ))≤ I (X2;Y )
)ILB,RS?(X1;Y ) ≤ I (X1;Y )
ILB,RS?(X2;Y |X1))≤ I (X2;Y |X1)
)31 / 35
Multiple-access channel with LOS
0 0.2 0.4 0.60
0.2
0.4
0.6
R1 [bits/channel use]
R2
[bit
s/ch
ann
elu
se]
I ?LB,RS
ILB
Figure : MAC with P1/σ2 = P2/σ
2 = 1 and |H1|2 = |H2|2 = σ21 = σ2
2 = 1/2
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Multiple-access channel without LOS
0 5 · 10−2 0.10
5 · 10−2
0.1
R1 [nats/channel use]
R2
[nat
s/ch
ann
elu
se]
I ?LB,RS
ILB
Figure : MAC with P1/σ2 = P2/σ
2 = 10dB and σ21 = σ2
2 = 12
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Bounds put together
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
α
MI
bou
nd
s[n
ats
/ch
ann
elu
se] I ?LB,RS from Section 1
I ?LB,RS from Section 2
unified bound (not covered in this talk)
Figure : |H|2 = α and σ2 = 1− α. SNR is P/σ2 = 0dB.
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Conclusion
• With LOS• Rate splitting improves Medard’s bound• Improved bound achievable with a modified NND scheme
• Without LOS• Modified Medard-type lower bound with biased signalling• Rate splitting allows to
, remove the signal bias, greatly improve the bound, come close to exact MI for Rayleigh AWGN
• Both bounds applicable to MAC
Open questions:
• Non-LOS achievable with modified NND scheme?
• Does multiplexing allow analysis of non-Gaussian codes?
Thanks!
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