nearly optimal non-gaussian codes for the gaussian
TRANSCRIPT
Alex Dytso, Daniela Tuninetti, Natasha Devroye
Nearly Optimal Non-Gaussian Codes for the Gaussian Interference Channel
Channel Model
Symmetric:
�
�
Z1
Z2
W1 W1
W2W2
h11
h21
h12
h22Dec.
Dec.Enc.
Enc.
Xn1
Xn2 Y n
2
Y n1
|h11|2 = |h22|2 = S|h12|2 = |h21|2 = I
with the usual assumptions and capacity region definition.
Motivation
Treat Interference as Noise Inner Bound:RTIN+TS
in = co
✓SPX1X2=PX1PX2
⇢0 R1 I(X1;Y1)
0 R2 I(X2;Y2)
�◆
RTINnoTS
in
=S
PX1X2=PX1PX2
⇢0 R
1
I(X1
;Y1
)0 R
2
I(X2
;Y2
)
�
With Time Sharing
No Time Sharing
How far away is TINnoTS from the Capacity?
Capacity: C = limn!1 co
✓SPXn
1 Xn2
=PXn1
PXn2
⇢0 R1 1
nI(Xn1 ;Y n
1 )
0 R2 1nI(Xn
2 ;Y n2 )
�◆
R. Ahlswede, “Multi-way communication channels,” in Proc. IEEE Int. Symp. Inf. Theory, March 1973, pp. 23–52.
Outline
• Definitions and relevant past work
• Useful tools:
- Mutual information lower bounds
- Minimum Distance lower bounds
• Main result
Mixed InputsX =
p1� � XD +
p� XG,
� 2 [0, 1],
XD ⇠ PAM(N) ,
XG ⇠ N (0, 1)
x
PAM
X =p1� �XD1 +
p�XD2
XD1 ⇠ PAM(N1), XD2 ⇠ PAM(N2)
Capacity Strong Interference|h11|2 |h21|2 and |h22|2 |h12|2 Sato, H., "The capacity of the Gaussian interference channel under strong
interference," IEEE Trans. Inf. Theory, vol. 27, no. 6, pp. 786,788, Nov 1981.
Sum-Capacity in Very Weak Interferencer
SI
(1 + I) 12
X. Shang, G. Kramer, and B. Chen, “A new outer bound and the noisy-interference sum rate capacity for Gaussian interference channels,” IEEE Trans. Inf. Theory, vol. 55, no. 2, pp. 689–699, 2009.
Motahari, A.S.; Khandani, A.K., "Capacity Bounds for the Gaussian Interference Channel," IEEE Trans. Inf. Theory, vol.55, no.2, pp.620,643, Feb. 2009
Annapureddy, V.S.; Veeravalli, V.V., "Gaussian Interference Networks: Sum Capacity in the Low-Interference Regime and New Outer Bounds on the Capacity Region," IEEE Trans. Inf. Theory, vol.55, no.7, pp.3032,3050, July 2009
Known Results
Approximate Capacity
Universal Gap
R1
R2
RI
RO
Avestimehr, A. Salman, Suhas N. Diggavi, Chao Tian, and N. C. David Tse. "An Approximation Approach to Network Information Theory."
Approximate CapacityHK+Gaussian Inputs
1/2 bitR. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5534–5562, Dec. 2008.
Encoder 1
Encoder 2 Decoder 2
Decoder 1
p(y1, y2|x1, x2)
Y n1
Y n2
Xn1
Xn2
(W1c, W1p)
(W2c, W2p)
(W1c, W1p, W2c)
(W2c, W2p, W1c)
A. Dytso, D. Tuninetti, and N. Devroye, “On the two-user interference channel with lack of knowledge of the interference codebook at one receiver,” IEEE Trans. Inf. Theory, vol. 61, no. 3, pp. 1257–1276, March 2015.
Encoder 1
Encoder 2 Decoder 2
Decoder 1
p(y1, y2|x1, x2)
Y n1
Y n2
Xn1
Xn2
(W1c, W1p)
(W2, W1c)
(W1c, W1p)
W2
“One-sided” HK+ Mixed Inputs
1.75 bits
R. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5534–5562, Dec. 2008.
D(↵) :=
8<
:(d1, d2) 2 R2+ : di := lim
I = S↵,S ! 1
Ri12 log(1 + S)
, i 2 [1 : 2], (R1, R2) is achievable
9=
; .
d1
d2
1
11 ↵ 2
d1 + d2 = �
d1
d2
1
2/3 � 1 1
d1 + 2d2 = 2
d1 + d2 = 2� �
2d1 + d2 = 2
gDoF Region
Sum-gDoF
1 2↵
2
15/3
2/31/2
dW (�)
TIN�G
RTINnoTS
in
=S
PX1X2=PX1PX2
⇢0 R
1
I(X1
;Y1
)0 R
2
I(X2
;Y2
)
�
Achievability
RTINnoTS
in
=[
N1,N2,�1,�2
⇢0 R
1
I(X1
;Y1
)0 R
2
I(X2
;Y2
)
�with
Xi =p1� �i XiD +
p�i XiG,
�i 2 [0, 1],
XiD ⇠ PAM(Ni) ,
XiG ⇠ N (0, 1),
i = 1, 2.
I(X2;Y2) = I(X2;h21X1 + h22X2 + ZG)
= I
X1D, X2D;
p1� �1h21X1D +
p1� �2h22X2Dp
1 + |h21|2�1 + |h22|2�2
+ ZG
!
� I
X1D;
p1� �1p
1 + |h21|2�1
h21X1D + ZG
!
+
1
2
log
�1 + |h21|2�1 + |h22|2�2
�� 1
2
log(1 + |h21|2�1).
Analytical rate expressions not trivial
Bounds
I(XD;
pSXD + Z) � H(XD)� gap,
gap = ⇠ log
1
⇠+ (1� ⇠) log
1
1� ⇠+ ⇠ log(N � 1) ,
⇠ := 2Q
pSdmin(XD)
2
!,
Ozarow-Wyner-AI(XD;
pSXD + Z) � H(XD)� gap,
gap =
12 log
�⇡e6
�+
12 log
✓1 +
12Sd2
min(XD)
◆.
Ozarow-Wyner-B
DTD-ITA`14-AI(XD;
pSXD + Z)
�
2
4� log
0
@X
(i,j)2[1:N ]2
pipj1p4⇡
e
� (si�sj)2
4
1
A� 1
2
log (2⇡e)
3
5+
I(XD;
pSXD + Z) � log(N)� gap,
gap =
1
2
log
⇣e
2
⌘+
1
2
log
1 + (N � 1)e
�Sd2
min(XD)4
!.
DTD-ITA`14-B
Dytso, A.; Tuninetti, D.; Devroye, N., "On discrete alphabets for the two-user Gaussian interference channel with one receiver lacking knowledge of the interfering codebook," ITA, 2014 , vol., no., pp.1,8, 9-14 Feb. 2014
Bounds
I(XD;
pSXD + Z) � H(XD)� gap,
gap = ⇠ log
1
⇠+ (1� ⇠) log
1
1� ⇠+ ⇠ log(N � 1) ,
⇠ := 2Q
pSdmin(XD)
2
!,
Ozarow-Wyner-AI(XD;
pSXD + Z) � H(XD)� gap,
gap =
12 log
�⇡e6
�+
12 log
✓1 +
12Sd2
min(XD)
◆.
Ozarow-Wyner-B
DTD-ITA`14-AI(XD;
pSXD + Z)
�
2
4� log
0
@X
(i,j)2[1:N ]2
pipj1p4⇡
e
� (si�sj)2
4
1
A� 1
2
log (2⇡e)
3
5+
I(XD;
pSXD + Z) � log(N)� gap,
gap =
1
2
log
⇣e
2
⌘+
1
2
log
1 + (N � 1)e
�Sd2
min(XD)4
!.
DTD-ITA`14-B
Dytso, A.; Tuninetti, D.; Devroye, N., "On discrete alphabets for the two-user Gaussian interference channel with one receiver lacking knowledge of the interfering codebook," ITA, 2014 , vol., no., pp.1,8, 9-14 Feb. 2014
Number of Points
SNRdB0 10 20 30 40 50 60 70 80 90 100 110
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Ozarow-Wyner-BOzarow-Wyner-ADTD-ITA14-ADTD-ITA14-Bshaping loss
SNRdB0 10 20 30 40 50 60 70 80 90 100 110
0
2
4
6
8
10
12
14
16
18
20
Ozarow-Wyner-BOzarow-Wyner-ADTD-ITA14-ADTD-ITA14-BCapacity
Bound ComparisonN = b
p1 + Sc
Minimum Distance
0 2 4 6 8 10 12 14 160
0.05
0.1
0.15
0.2
0.25
0.3
0.35
h1
d min
(h1X 1+h
2X 2)
Very Irregular
Example: h2=1, N1=N2=10
dmin(h1X1D+h2X2D)
I(XD;
pSXD + Z) � H(XD)� gap,
gap =
12 log
�⇡e6
�+
12 log
✓1 +
12Sd2
min(XD)
◆.
I(X1D, X2D;h1X1D + h2X2D + Z)
dmin(h1X1D+h2X2D) = min(h1dmin(X1D), h2dmin(X2D))
if either h2dmin(X2D)N2 h1dmin(X1D)
or h1dmin(X1D)N1 h2dmin(X2D)
dmindmin
h11X1
h11X1 + h12X2
0 2 4 6 8 10 12 14 160
0.05
0.1
0.15
0.2
0.25
0.3
0.35
d min(hxX+h
yY)
hx
Proposition 2 is not valid Proposition 2 is valid
No Overlaps
dmin: bound 1
0 2 4 6 8 10 12 14 160
0.05
0.1
0.15
0.2
0.25
0.3
0.35
d min(hxX+h
yY)
hx
Proposition 2 is not valid Proposition 2 is valid
Overlaps
dmin: bound 2
dmin
h11X1
h11X1 + h12X2
dmin(h1X1D+h2X2D) = ⇥ min
✓h1dmin(X1D), h2dmin(X2D),max
✓h1dmin(X1D)
N2,h2dmin(X2D)
N1
◆◆
⇥ =
�
(1 + log(max(N1, N2)))
for all (h1, h2) except an outage set of measure � for any � > 0.
Main Result Very Weak
↵
Weak I Weak II Strong VeryStrong
0 1/2 2/3 1 2
gap = 1/2 gap = 3.79 gap = 1.25gap = O(log log(min(S, I))
up to an outage
of controllable measure
Gaussian Mixed Mixed Discrete DiscreteXi =
p1� �i XiD +
p�i XiG, i 2 [1 : 2],
Closed-form expressionsfor number of points,power splits and gap
↵ =
log I
log S
Example
R1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
R 2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5Capacity RegionAchiev. Theorem 7Achiev. Ozarow-Wyner-BAchiev. DTD-ITA-AAchiev. Monte Carl
PAM+PAMXD := XD1 +XD2 ,
where XD1 ⇠ discrete : dmin(XD1 )> 0,
XD2 ⇠ discrete : dmin(XD2 )> 0,
XM := XD1 +XG,
where XD1 , XG and XD2 are mutually independent. Then, for ZG ⇠ N (0, 1)independent of everything else, we have
I(XD; gXD + ZG)� I(XM ; gXM + ZG) 1
2
log(2),
I(XM ; gXM + ZG)� I(XD; gXD + ZG) 1
2
log
⇣⇡e3
⌘+
1
2
log
1 +
12
g2 d2min(XD)
!.
Concluding Remarks
• Key idea: use non-Gaussian inputs
• Developed very general tools of use beyond 2-IC
• Applicable to Block Asynchronous Interference Channel and Codebook Oblivious Interference Channel
