The Gaussian interference channel with lack of codebook ...adytso/presentations/ITWv2.pdf ·...

Alex Dytso, Daniela Tuninetti, Natasha Devroye

The Gaussian interference channel with lack of codebook

knowledge at one receiver: symmetric capacity to within a

gap with a PAM input

Monday, April 27, 2015

Motivation

W1

W2

Xn2 (W2)

Xn1 (W1)

Encoder 1

Encoder 2 Decoder 2

Decoder 1

p(y1, y2|x1, x2)

Y n1

Y n2

W1

W2

Legacy ReceiverNo Joint-Decoding


Outline• Relevant Channel Models

• Past Work

• Generalized Ozarow-Wyner Bound

• Discrete inputs are ‘good’ inputs and ‘good’ interferers

• Capacity and Approximate Capacity Results


Codebook Knowledge

Encoder Decoderp(y|x)Channel

W WXn Y n

F

F

F

W → Xn

1 → X1, X2, . . . , Xn

2 → X1, X2, . . . , Xn

.

.

.|W | → X1, X2, . . . , Xn

Point-to-Point Channel


Codebook Knowledge

Encoder Decoderp(y|x)Channel

W WXn Y n

F

F

F

W → Xn

1 → X1, X2, . . . , Xn

2 → X1, X2, . . . , Xn

.

.

.|W | → X1, X2, . . . , Xn

ChannelEncoder

EncoderDecoder

F1

W1

W2

F2

F1

F2

Decoder

F1 F2p(y1, y2|x1, x2)

Xn1

Xn2

Y n2

Y n1

W1

W2

Interference Channel (IC)Point-to-Point Channel


Past WorkA. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer, “Communication via decentralized processing,” IT July 2008.

1. Upper and lower bounds, which coincide for deterministic channels2. Gaussian noise: optimizing input unknown3. Gaussian noise: example where BPSK outperforms Gaussian inputs

PY1,Y

2|X

Y n1

Y n2

Xn

W WEncoder Decoder

Relay

RelayF F

F

F

F1

F2

F2

W1Xn

1Y n

1

W2Xn

2

PY1,Y

2,Y

3|X

1,X

2,X

3(y

1,y

2,y

3|x

1,x

2,x

3)

Encoder

EncoderRelay

Decoder

Decoder

W1

W2

Y n3

Y n2

F1

F2

F1

F2

F2F1

F1 F2

O. Simeone, E. Erkip, and S. Shamai, “On codebook information for interference relay channels with out-of-band relaying,” IT May 2011.

1. Primitive relay channel: capacity with compress forward2. IC+R+Oblivious receivers: capacity with compress forward and TIAN3. Gaussian noise: optimizing input unknown


Considered Channel Models

IC with one oblivious Rx

A. Dytso, N. Devroye, and D. Tuninetti, “On the capacity of interference channels with partial codebook knowledge,” ISIT 2013

A. Dytso, D. Tuninetti and N. Devroye, “On Discrete Alphabets for the Two-user Gaussian Interference Channel with One Receiver Lacking Knowledge of the Interfering Codebook,” ITA 2014, arXiv:1405.1117, submitted IT May 2014, revised Sept 2014

A. Dytso, D. Tuninetti and N. Devroye. “On Gaussian Interference Channels with Mixed Gaussian and Discrete Inputs,” ISIT 2014

O. Simeone, E. Erkip, and S. Shamai, “On codebook information for interference relay channels with out-of-band relaying,” IT May 2011

ChannelEncoder

EncoderDecoder

F1

W1

W2

F2

F1

F2

Decoder

F1 F2p(y1, y2|x1, x2)

Xn1

Xn2

Y n2

Y n1

W1

W2

ChannelEncoder

EncoderDecoder

F1

W1

W2

F2

F1

F2

Decoder

F1 F2p(y1, y2|x1, x2)

Xn1

Xn2

Y n2

Y n1

W1

W2

IC with twooblivious Rx


Question We Ask?

• What is the loss in performance with the loss due to lack of codebook knowledge?

• Are there inputs that do better than Gaussian? Can we provide analytical results?


Inner Bound

ChannelEncoder

EncoderDecoder

W1

F2

F1

F2

Decoder

F1 F2p(y1, y2|x1, x2)

Xn1

Xn2

Y n2

Y n1

W2 = (W2c, W2p)

(W1, W2c)

W2 = (W2c, W2p)

R1 ≤ I(X1;Y1|U2, Q)R2 ≤ I(X2;Y2|Q)

R1 + R2 ≤ I(X1, U2;Y1|Q) + I(X2;Y2|U2, Q)

Theorem (Region Ri). The following rate region is achievable

for all distributions in PQ,X1,X2,U2 = PQPX1|QPX2|QPU2|X2

Special Case of HK


Gap (Con’t)Theorem. If (R1, R2) ∈ Ro, then

(R1, R2) ∈ RO =⇒ (R1 − I(X2;T2|U2, Q), R2) ∈ Ri.

Gap

G-ICDeterministic

I(X2;T2|U2, Q) = 0 I(X2;T2|U2, Q) ≤ 12

Do we know maximizing distribution?


G-IC

α1

2

2

31 2

1

1

2

4

3

d(α)

1

2log

�1 +

S

I + 1

�+

1

2log

�(I + 1)2 + S

1 + I

��.

resulting in

d(GG)(α) =1

2+

�1

2− α

�+.

For future reference, with Time Division (TD) and Gaussiancodebooks we can achieve

(R1 +R2)(TD) =

1

2log (1 + S) ⇐⇒ d(TD)(α) =

1

2.

We plot the achievable gDoF vs. α in Fig. 2, together with thegDof of the classical IC given by d(W)(α) [8], which formsan outer bound to the gDoF of the G-IC-OR. We note thatGaussian inputs are indeed optimal for 0 ≤ α ≤ 1/2, i.e.,d(GG)(α) = d(W)(α), where interference is treated as noiseeven for the classical IC (which is also achievable by the G-IC-OR). For α ≥ 1/2 we have d(GG)(α) = d(TD)(α), thatis, Gaussian inputs perform as time division. Gaussian inputsare sub-optimal in general as we show next.

Consider α = 4/3: with Gaussian inputs or with timedivision we only achieve d(GG)(4/3) = d(TD)(4/3) = 1/2.Notice the similarity with the LD-IC-OR: the input distributionthat is optimal for the non-oblivious IC performs as timedivision for the G-IC-OR. Inspired by the LD-IC-OR weexplore now the possibility of using a non-Gaussian input. Inparticular, we choose an input distribution that allows the obvi-ous receiver to soft-estimate the interfering codeword symbols(even though it is not able in general to decode the interferingmessage). By following [1, Section VI.A], which demonstratedthat binary signaling outperforms Gaussian signaling for afixed finite SNR, we consider a uniform PAM constellationwith N points. Fig. 3 shows the achievable normalized sum-rate R1+R2

2· 12 log(1+S)as a function of S for the case where X1 (the

input of the non-oblivious pair) is a PAM constellation withN = �S1/6� points and X2 (the input of the oblivious pair)is Gaussian. Notice that the number of points in the discreteinput is a function of the direct link channel gain S. We alsoreport the achievable normalized sum-rate with time divisionand Gaussian inputs. Fig. 3 shows that for sufficiently large Susing a discrete input outperforms time division; moreover, forthe range of simulated S, it seems that the proposed discreteinput achieves a gDoF of d(DG)(α) = α/2 = 4/6 as for theclassical IC with full codebook knowledge.

We conjecture that a strategy with one discrete input out-performs Gaussian signaling for all α > 1, which appears tobe the case from extensive numerical evaluations and is thesubject of ongoing work. Proving the validity of our conjecturecould also help the settle the open question whether Gaussianinputs exhaust the outer bound in related oblivious channelmodels – see [2, Section III.A] and [3, Remark 5].

VI. CONCLUSION

We focused on an IC in which one of the decoders onlypossesses one of the two transmitting codebooks (in contrast toclassical ICs where all nodes are aware of all codebooks). We

50 100 150 200 250 300 350

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

10 log10S

Fig. 3. Achievable normalized sum-rate for the symmetric G-IC-OR withα = 4/3: (1) time division: solid blue line; (ii) Gaussian inputs at bothtransmitters: red stars; (3) X1 is a uniform PAM with N = �S1/6� pointsand X2 is Gaussian: dash-dotted black line.

characterized the capacity of the injective semi deterministicIC to within a constant gap and specialized it to the Gaussianchannel and to the Linear Deterministic approximation ofthe Gaussian channel at high SNR; in the former case weestablished capacity to within 1/2 bit, even though we couldnot determine the optimal input distribution; in the latter, weshowed the exact capacity region and that the sum-capacitywith partial codebook knowledge is the same as that of theclassical IC with full codebook knowledge. An important nextstep is to identify optimal input distributions for the Gaussiannoise channel. In this direction, we are currently investigatingthe usage of discrete inputs for the non-oblivious user and ofGaussian input for the oblivious transmitter, which numericallyseems to outperform Gaussian signaling and time division.

Acknowledgment The work of the authors was partiallyfunded by NSF under award 1017436. The contents of thisarticle are solely the responsibility of the authors and do notnecessarily represent the official views of the NSF.

REFERENCES

[1] A. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer, “Communi-cation via decentralized processing,” IEEE Trans. Inf. Theory, vol. 54,no. 7, pp. 3008 –3023, Jul. 2008.

[2] O. Simeone, E. Erkip, and S. Shamai, “On codebook information forinterference relay channels with out-of-band relaying,” IEEE Trans. Inf.

Theory, vol. 57, no. 5, pp. 2880 –2888, May 2011.[3] Y. Tian and A. Yener, “Relaying for multiple sources in the absence of

codebook information,” in Proc. ASILOMAR, Nov. 2011, pp. 1845 –1849.[4] A. El Gamal and Y.-H. Kim, Network Information Theory. Camrbidge

University Press, 2012.[5] E. Telatar and D. Tse, “Bounds on the capacity region of a class of

interference channels,” in Proc. IEEE Int. Symp. Inf. Theory, Jun. 2007,pp. 2871 –2874.

[6] T. Han and K. Kobayashi, “A new achievable rate region for theinterference channel,” IEEE Trans. Inf. Theory, vol. IT-27, no. 1, pp.49–60, Jan. 1981.

[7] G. Bresler and D. Tse, “The two-user gaussian interference channel:A deterministic view,” European Transactions in Telecommunications,vol. 19, pp. 333–354, Apr. 2008.

[8] R. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacityto within one bit,” IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5534–5562, Dec. 2008.

Point under consideration

g(α) := limS→+∞

Csum

2 12 log(1 + S)

Csum

2 12 log(1 + S)

PAM +Gaussian

Both Users use Gaussian or TDMA

α =log I

log Sα =

log I

log S

gDoF generalized Degrees of freedom


Discrete Inputs• Discrete Input

• PAM Input

• Mixed Input

x

XD ∼ PAM(N), |X| = N,pi =1N

for all i ∈ [1, ...,N]

XD ∼ P (XD) =|X|�

i=1

piδ(xi)

X =√

1− δXD +√

δXG, δ ∈ [0, 1]


Main Tool• A generalization of Ozarow-Wyner

PAM-on-AWGN boundId(XD) :=

�H(XD)− gap(dmin(XD))

�+

≤ I(XD;XD + Z) ≤ H(XD),

gap(d) ≤ 1

2log

�2πe

12

�+

1

2log

�1 +

12

d2

�

valid for any discrete input onany additive noise channel

H(XD) = log(N)

For PAM

dmin(XD) =�

12EXD

N2 − 1


Examples (Con’t)1.Point-to-point Gaussian noise Channel

Y =√

snrX + ZG :

E[X2] ≤ 1, ZG ∼ N (0, 1)

Capacity

C =12

log(1 + snr) := Ig(snr)

Achievable with PAMN = �

√1 + snr�

R =12

log(1 + snr)− gap

gap =12

log�

4πe3

�We have performance guarantee.Discrete inputs are approximately

optimal.


Example (Con’t)

Y =√

snrX + hT + ZG :

E[X2] ≤ 1, ZG ∼ N (0, 1),

T ∼ discrete: |T | = N and d2min(T ) > 0

2.Point-to-point Gaussian noise Channel with State

R = I(XG;√

snrXG + hT + ZG)

≥ 12

log(1 + snr)− gap

gap :=12

log

�2πe12

�1 +

12d2

min(T)

|h|2ET

|h|2ET + 1 + snr

��.

Interference Free CapacityC =

12

log(1 + snr) := Ig(snr)

If the ‘gap’ is bonded then have approximately interference free capacity


Inner BoundR1 ≤ I(Y1;X1|U2, Q)R2 ≤ I(Y2;X2|Q)

R1 + R2 ≤ I(Y1;X1, U2, Q)+ I(Y2;X2|U2, Q)

X1D, X1G, X2Gc, X2Gp independent and distributed as

X1D ∼ PAM (N) , N ∈ N,all the others are N (0, 1),

X1 =�

1− δ1X1D +�

δ1X1G, δ1 ∈ [0, 1],

X2 =�

1− δ2X2Gc +�δ2X2Gp, δ2 ∈ [0, 1].

U2 = X2Gc, Q = ∅.oblivious

ChannelEncoder

EncoderDecoder

W1

F2

F1

F2

Decoder

F1 F2p(y1, y2|x1, x2)

Xn1

Xn2

Y n2

Y n1

W2 = (W2c, W2p)

(W1, W2c)

W2 = (W2c, W2p)


Inner BoundR1 ≤ I(Y1;X1|U2, Q)R2 ≤ I(Y2;X2|Q)

R1 + R2 ≤ I(Y1;X1, U2, Q)+ I(Y2;X2|U2, Q)

R1 ≤ Id

�N,

|h11|2(1− δ1)

1 + |h11|2δ1 + |h12|2δ2

�+ Ig

�|h11|2δ1

1 + |h12|2δ2

�,

R2 ≤ Id

�N,

|h21|2(1− δ1)

1 + |h21|2δ1 + |h22|2

�+ Ig

�|h22|2

1 + |h21|2δ1

�

− Ig

�min

�N2 − 1,

|h21|2(1− δ1)

1 + |h21|2δ1

��,

R1 +R2 ≤ Id

�N,

|h11|2(1− δ1)

1 + |h11|2δ1 + |h12|2

�+ Ig

�|h11|2δ1 + |h12|2

�− Ig

�|h12|2δ2

�

+ Id

�N,

|h21|2(1− δ1)

1 + |h21|2δ1 + |h22|2δ2

�+ Ig

�|h22|2δ2

1 + |h21|2δ1

�

− Ig

�min

�N2 − 1,

|h21|2(1− δ1)

1 + |h21|2δ1

��.

How good is this inner bound with respect to full codebook capacity?


α =log(inr)

log(snr)

β =log(N2)

log(snr)

gap ≤ 1

2log (12πe) ≈ 3.34

Nd(x) :=�√

1 + x�

β = 1 : N = Nd(snr)

β = α− 1 : N = Nd

�inr

1 + snr

�

β = 2α− 1 : N = Nd

�inr2

1 + snr + 2inr

�

β = 1− α : N = Nd

�snr · inr

(1 + inr)2 + snr

�

Power split:either 0 (DI),

or “ETW” (DII)

R1 ≤ I(Y1;X1|U2, Q)R2 ≤ I(Y2;X2|Q)

R1 + R2 ≤ I(Y1;X1, U2, Q)+ I(Y2;X2|U2, Q)

δ1 = δ2 =1

1 + inr


Conclusion IC-OR

• Capacity to within 1/2 bit: do not know how to evaluate the outer bound

• Compared to outer bound with full codebook knowledge at all nodes:PAM input at non-oblivious Tx looses at most 3.34 bits


Channel ModelsChannelEncoder

EncoderDecoder

F1

W1

W2

F2

F1

F2

Decoder

F1 F2p(y1, y2|x1, x2)

Xn1

Xn2

Y n2

Y n1

W1

W2

C =�

R1 ≤ I(X1;Y1|Q)R2 ≤ I(X2;Y2|Q)

�

for some distribution p(q)p(x1|q)p(x2|q)

Capacity

O. Simeone, E. Erkip, and S. Shamai, “On codebook information for interference relay channels with out-of-band relaying,” IT May 2011.

• Maximizing Distribution is Unknown

• TIN is very robust. Industry standard is to treat interference as Gaussian Noise.

Treating Interference as Noise (TIN) +time sharing


TINC =

�R1 ≤ I(X1;Y1|Q)R2 ≤ I(X2;Y2|Q)

�

for some distribution p(q)p(x1|q)p(x2|q)

Look at inner bound with out time sharing RTIN

in =�

PX1X2=PX1PX2

�R1 ≤ I(X1;Y1)R2 ≤ I(X2;Y2)

�.

Question:

R1 ≤ Ig (snr) ,R2 ≤ Ig (snr) ,

R1 +R2 ≤�Ig (snr)− Ig (inr)

�++ Ig(inr + snr),

R1 +R2 ≤ 2Ig�inr + snr

1+inr

�,

2R1 +R2 ≤ Ig(snr + inr) + Ig�inr + snr

1+inr

�+

�Ig (snr)− Ig (inr)

�+,

R1 + 2R2 ≤ Ig(snr + inr) + Ig�inr + snr

1+inr

�+

�Ig (snr)− Ig (inr)

�+,

?=

up to a constant additive gap

RTINin =

�

PX1X2=PX1PX2

�R1 ≤ I(X1;Y1)R2 ≤ I(X2;Y2)

�.

Capacity outer bound with full codebook knowledge

with synchronization

No-synchronization required


Thank you


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