rate splitting approaches for noncoherent fading channels€¦ · rate splitting approaches for...

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




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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




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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

26 / 35


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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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




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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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