[PPT] - The Approximate Sum Capacity of the Symmetric Gaussian K -User PowerPoint Presentation

SLIDE 1

The Approximate Sum Capacity of the Symmetric Gaussian K-User Interference Channel

Or Ordentlich Joint work with Uri Erez and Bobak Nazer July 5th, ISIT 2012 MIT, Cambridge, Massachusetts

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 2

The symmetric Gaussian 2-user IC : channel model

w1 E1 x1 1 g w2 E2 x2 1 g z1 y1 z2 y2 D1 ˆ w1 D2 ˆ w2 yk = xk + gx¯

k + zk

Channel is static and real valued. Gaussian noises zk are of zero mean and variance 1. All users are subject to the power constraint xk2 ≤ nSNR. Define INR g2SNR and α log(INR)

log(SNR).

Channel is symmetric: sum capacity = 2 × symmetric capacity

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 3

GDoF of symmetric Gaussian 2-user IC

Symmetric capacity is known to within 1/2 bit (Etkin et al. 08). DoF for each user is 1/2. GDoF gives more refined view α 2 1

2 3 1 2

d(α) 1

2 3 1 2

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 4

Symmetric Gaussian 2-user IC

Noisy interference regime Treat interference as noise α 2 1

2 3 1 2

d(α) 1

2 3 1 2

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 5

Symmetric Gaussian 2-user IC

Weak interference regime Jointly decode intended message and part of interference (Han-Kobayashi). α 2 1

2 3 1 2

d(α) 1

2 3 1 2

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 6

Symmetric Gaussian 2-user IC

Strong interference regime Jointly decode intended message and interference α 2 1

2 3 1 2

d(α) 1

2 3 1 2

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 7

Symmetric Gaussian 2-user IC

Very strong interference regime Decode interference and then successively decode intended message α 2 1

2 3 1 2

d(α) 1

2 3 1 2

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 8

The symmetric Gaussian K-user IC : channel model

w1 E1 x1 1 g g w2 E2 x2 1 g g

. . . . . .

wK EK xK 1 g g z1 y1 z2 y2 zK yK D1 ˆ w1 D2 ˆ w2 DK ˆ wK yk = xk + g

m=k

xm + zk INR g2SNR and α log(INR)

log(SNR).

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 9

The symmetric Gaussian K-user IC: what do we know?

DoF is discontinuous at the rationals (Etkin and E. Ordentlich 09, Wu et al. 11). GDoF of the symmetric K-user IC is independent of K, except for discontinuity at α = 1 (Jafar and Vishwanath 10). α 2 1

2 3 1 2

d(α) 1

2 3 1 2 1 K

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 10

The symmetric Gaussian K-user IC: what do we know?

What about finite SNR? Adding interference cannot increase capacity → Outer bounds for K = 2 remain valid for K > 2.

10

−2

10 10

2

1 2 3 4 5 6 3−user IC @ SNR=35dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 11

The symmetric Gaussian K-user IC: what do we know?

What about finite SNR? Can always use time-sharing → CSYM >

1 2K log(1 + KSNR). 10

−2

10 10

2

1 2 3 4 5 6 3−user IC @ SNR=35dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 12

The symmetric Gaussian K-user IC: what do we know?

What about finite SNR? Can treat interference as noise → achieves the approximate capacity for noisy interference regime

10

−2

10 10

2

1 2 3 4 5 6 3−user IC @ SNR=35dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 13

The symmetric Gaussian K-user IC: what do we know?

For the other regimes lattice codes are useful. Closed under addition = ⇒ K − 1 interferers folded to one effective interferer. Each receiver sees a K-user MAC yk = xk + g

m=k

xm + zk,

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 14

The symmetric Gaussian K-user IC: what do we know?

For the other regimes lattice codes are useful. Closed under addition = ⇒ K − 1 interferers folded to one effective interferer. Assume x1, . . . , xK ∈ Λ. = ⇒ Effective 2-user MAC at each receiver yk = xk + gxint,k + zk, where xint,k =

m=k

xm ∈ Λ.

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 15

The symmetric Gaussian K-user IC: what do we know?

For the other regimes lattice codes are useful. Closed under addition = ⇒ K − 1 interferers folded to one effective interferer. Assume x1, . . . , xK ∈ Λ. = ⇒ Effective 2-user MAC at each receiver yk = xk + gxint,k + zk, where xint,k =

m=k

xm ∈ Λ. How to decode xk?

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 16

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 17

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 18

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k g z y

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 19

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k g z y

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 20

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k g z y Decode xint,k

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 21

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k g z y Decode xint,k

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 22

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k z y Cancel xint,k

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 23

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k z y Decode xk

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 24

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k z y Decode xk

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 25

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k z y Decode xk

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 26

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k z y Decode xk

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 27

The symmetric Gaussian K-user IC: what do we know?

For large g, can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) xint,k xk xk 1 xint,k z y Decode xk

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 28

The symmetric Gaussian K-user IC: what do we know?

What about finite SNR? Successive decoding is optimal in the very strong interference regime.

10

−2

10 10

2

1 2 3 4 5 6 3−user IC @ SNR=35dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 29

The symmetric Gaussian K-user IC: strong interference

yk = xk + gxint,k + zk, xk, xint,k ∈ Λ Assume strong interference: g > 1 but not ≫ 1. For 2-user IC jointly decoding intended message and interference is

ptimal.

For K-user IC jointly decoding xk, xint,k seems like a good idea.

Question

What rates are achievable?

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 30

The symmetric Gaussian K-user IC: strong interference

yk = xk + gxint,k + zk, xk, xint,k ∈ Λ Assume strong interference: g > 1 but not ≫ 1. For 2-user IC jointly decoding intended message and interference is

ptimal.

For K-user IC jointly decoding xk, xint,k seems like a good idea. MAC capacity theorem does not hold when both transmitters use the same lattice codebook = ⇒ Need a new coding theorem.

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 31

MAC with same lattice code

What’s the problem with using the same lattice code? xint,k xk xk 1 xint,k 2 y Assume there is no noise at all

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 32

MAC with same lattice code

What’s the problem with using the same lattice code? xint,k xk xk 1 xint,k 2 y AMBIGUITY!

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 33

MAC with same lattice code: new decoder

Decoding the two lattice points directly is difficult. Instead...

New decoder based on compute-and-forward

Decode two equations with integer coefficients and solve for desired codeword. yk = xk + gxint,k + zk,

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 34

MAC with same lattice code: new decoder

Decoding the two lattice points directly is difficult. Instead...

New decoder based on compute-and-forward

Decode two equations with integer coefficients and solve for desired codeword. ˜ y1

k

˜ y2

k

=

a11 a12 a21 a22 xk xint,k

+

zeff,1 zeff,2

Ordentlich, Erez, Nazer
Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 35

MAC with same lattice code: new decoder

Decoding the two lattice points directly is difficult. Instead...

New decoder based on compute-and-forward

Decode two equations with integer coefficients and solve for desired codeword. ˜ y1

k

˜ y2

k

=

a11 a12 a21 a22 xk xint,k

+

zeff,1 zeff,2

Main result

We use this approach to obtain the approximate symmetric capacity region

f the K-user symmetric IC up to an outage set.

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 36

The symmetric Gaussian K-user IC: new inner bounds

10

−1

10 10

1

0.5 1 1.5 2 2.5 3 3.5 3−user IC @ SNR=20dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 37

The symmetric Gaussian K-user IC: new inner bounds

10

−2

10 10

2

1 2 3 4 5 6 3−user IC @ SNR=35dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 38

The symmetric Gaussian K-user IC: new inner bounds

10

−2

10 10

2

2 4 6 8 3−user IC @ SNR=50dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 39

The symmetric Gaussian K-user IC: new inner bounds

10

−2

10 10

2

2 4 6 8 10 3−user IC @ SNR=65dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 40

Main tool: compute-and-forward

Theorem - Nazer-Gastpar 11

For the channel y =

K

k=1

hkxk + z the equation

K

k=1

akxk with a = [a1 · · · aK] ∈ ZK can be decoded reliably as long as the rates of all users satisfy R < 1 2 log

SNR

SNRβh − a2 + β2

for some β ∈ R.

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 41

Main tool: compute-and-forward

Theorem - Nazer-Gastpar 11

For the channel y =

K

k=1

hkxk + z the equation

K

k=1

akxk with a = [a1 · · · aK] ∈ ZK can be decoded reliably as long as the rates of all users satisfy R < 1 2 log

SNR

SNRβh − a2 + β2

for some β ∈ R.

Use one channel output to decode two equations yk = xk + gxint,k + zk,

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 42

Main tool: compute-and-forward

Theorem - Nazer-Gastpar 11

For the channel y =

K

k=1

hkxk + z the equation

K

k=1

akxk with a = [a1 · · · aK] ∈ ZK can be decoded reliably as long as the rates of all users satisfy R < 1 2 log

SNR

SNRβh − a2 + β2

for some β ∈ R.

Use one channel output to decode two equations ˜ y1

k

˜ y2

k

=

a11 a12 a21 a22 xk xint,k

+

zeff,1 zeff,2

Ordentlich, Erez, Nazer
Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 43

Main tool: compute-and-forward

Decoding two equations is not very effective when channel gains are close to integers. This causes the notches in the achievable rate region. Fortunately, this rarely happens...

1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 g Normalized Computation Rate First Equation Second Equation Sum

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 44

Main tool: compute-and-forward

PROMO

To hear more about this come to ”The Compute-and-Forward Transform” tomorrow at 15:20.

1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 g Normalized Computation Rate First Equation Second Equation Sum

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 45

Compute-and-forward for the symmetric K-user IC

Transmit Equations Decoded by Receivers x1 a11x1 + a12

ℓ=1

xℓ a21x1 + a22

ℓ=1

xℓ x2 a11x2 + a12

ℓ=2

xℓ a21x2 + a22

ℓ=2

xℓ

. . . . . . . . .

xK a11xK + a12

ℓ=1

xℓ a21xK + a22

ℓ=1

xℓ From one real equation decode two linearly independent equations with integer coefficients. Corresponding computation rates are Rcomp,1, Rcomp,2.

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 46

Approx. symmetric capacity: strong interference regime

CSYM ≥ Rcomp,2 Rcomp,2 is the solution to an integer-least squares optimization problem. Inner bound can be found numerically and plotted.

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 47

Approx. symmetric capacity: strong interference regime

CSYM ≥ Rcomp,2 Rcomp,2 is the solution to an integer-least squares optimization problem. Inner bound can be found numerically and plotted.

10

1

1 2 3 4 5 3−user IC @ SNR=30dB g symmetric rate[bits/channel use] Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 48

Approx. symmetric capacity: strong interference regime

CSYM ≥ Rcomp,2 Rcomp,2 is the solution to an integer-least squares optimization problem. Inner bound can be found numerically and plotted.

Question

For c > 0 bits, what is the fraction of channel gains g for which

uter bound − inner bound > c bits?

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 49

Outage set

10

1

1 2 3 4 5 3−user IC @ SNR=30dB g symmetric rate[bits/channel use]

Strong interference regime

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 50

Outage set

10

1

1 2 3 4 5 3−user IC @ SNR=30dB g symmetric rate[bits/channel use]

c = 0.25 bits

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 51

Outage set

48% outage for c = 0.25 bits

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 52

Outage set

22% outage for c = 0.5 bits

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 53

Outage set

11% outage for c = 0.75 bits

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 54

Approx. symmetric capacity: strong interference regime

Theorem - inner bound for the strong interference regime

The symmetric capacity of the symmetric Gaussian K-user IC is lower bounded by CSYM ≥ 1 4 log+(INR) − c 2 − 3 for all values of 1 ≤ g2 < SNR except for an outage set whose measure is a fraction of 2−c of the interval 1 ≤ |g| < √ SNR, for any c > 0.

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 55

Approx. symmetric capacity: strong interference regime

Theorem - inner bound for the strong interference regime

The symmetric capacity of the symmetric Gaussian K-user IC is lower bounded by CSYM ≥ 1 4 log+(INR) − c 2 − 3 for all values of 1 ≤ g2 < SNR except for an outage set whose measure is a fraction of 2−c of the interval 1 ≤ |g| < √ SNR, for any c > 0. Outage set approach appeared first in Niesen and Maddah-Ali 11 (next talk) The outage set phenomena seems inherent to the problem (Etkin and

E. Ordentlich 09).

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 56

Weak interference regime: Lattice Han-Kobayshi

Similar approach works for the weak interference regime. Just choose public and private codewords from lattice codebooks. Decoding is done using compute-and-forward. Achievable rate is the solution to integer least-squares optimization problem. Can be shown to be within a constant gap from outer bound (except for an outage set).

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 57

Summary: new inner bounds

New inner bound for strong interference regime.

◮ Constant gap from outer bound except for outage set.

10

−2

10 10

2

2 4 6 8 3−user IC @ SNR=50dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 58

Summary: new inner bounds

New inner bound for moderately weak interference regime.

◮ Constant gap from outer bound except for outage set.

10

−2

10 10

2

2 4 6 8 3−user IC @ SNR=50dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC

SLIDE 59

Summary: new inner bounds

New inner bound for weak interference regime.

◮ Constant gap from outer bound for all channel gains.

10

−2

10 10

2

2 4 6 8 3−user IC @ SNR=50dB g symmetric rate[bits/channel use]

Ordentlich, Erez, Nazer

Approx. Sum Capacity of the Symmetric Gaussian K-User IC