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On Achievable Rates of the Two-user Symmetric Gaussian Interference - - PowerPoint PPT Presentation

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions On Achievable Rates of the Two-user Symmetric Gaussian Interference Channel Omar Mehanna, John Marcos and Nihar Jindal University of Minnesota Allerton

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

On Achievable Rates of the Two-user Symmetric Gaussian Interference Channel

Omar Mehanna, John Marcos and Nihar Jindal

University of Minnesota

Allerton 2010

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Interference Channel

  • Interference channel is one of the most fundamental models

for wireless communication systems

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Interference Channel

  • Interference channel is one of the most fundamental models

for wireless communication systems

  • Capacity region for the simplest two user symmetric Gaussian

interference channel is not yet fully characterized

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Interference Channel

  • Interference channel is one of the most fundamental models

for wireless communication systems

  • Capacity region for the simplest two user symmetric Gaussian

interference channel is not yet fully characterized

  • Best known achievability strategy was proposed by Han and

Kobayashi (1981)

  • Split each user’s transmitted message into private and

common portions

  • Time sharing between multiples of such splits
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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

State-of-the-Art

  • Low interference regime: [Annapureddy et al.] [Motahari et al.]

[Shang et al.]

⇒ Send only private message and treat interference as noise

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

State-of-the-Art

  • Low interference regime: [Annapureddy et al.] [Motahari et al.]

[Shang et al.]

⇒ Send only private message and treat interference as noise

  • Strong interference regime: [Han-Kobayasi] [Sato]

⇒ Send only common message

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

State-of-the-Art

  • Low interference regime: [Annapureddy et al.] [Motahari et al.]

[Shang et al.]

⇒ Send only private message and treat interference as noise

  • Strong interference regime: [Han-Kobayasi] [Sato]

⇒ Send only common message

  • Other interference regimes:

Etkin, Tse and Wang showed that Han-Kobayashi with Gaussian inputs, no time sharing and equal fixed power splitting ratios can achieve to within a single bit of the capacity

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

State-of-the-Art

  • Low interference regime: [Annapureddy et al.] [Motahari et al.]

[Shang et al.]

⇒ Send only private message and treat interference as noise

  • Strong interference regime: [Han-Kobayasi] [Sato]

⇒ Send only common message

  • Other interference regimes:

Etkin, Tse and Wang showed that Han-Kobayashi with Gaussian inputs, no time sharing and equal fixed power splitting ratios can achieve to within a single bit of the capacity

  • Our Contributions (Gaussian inputs):
  • Best power splitting ratios with no time sharing
  • The corresponding maximum achievable HK sum-rate
  • Comparison with orthogonal signaling (TDMA/FDMA)
  • Study of some time sharing schemes
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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Network Model

Two-user Gaussian interference channel: y1

=

h11x1 + h21x2 + ¯ z1 y2

=

h12x1 + h22x2 + ¯ z2

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Network Model

Two-user Gaussian interference channel: y1

=

h11x1 + h21x2 + ¯ z1 y2

=

h12x1 + h22x2 + ¯ z2 Normalized symmetric channel

|h11| = |h22|, |h12| = |h21|, a = |h21|2 |h11|2 = |h12|2 |h22|2 , 0 < a < 1,

zi ∼ CN(0, 1), P1 = P2 = P = SNR y1 = x1 +

√

ax2 + z1, y2 =

√

ax1 + x2 + z2 Channel fully characterized by interference coefficient a and P

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Virtual 3-user MAC Channel

Private/common power splitting ratio λi (0 ≤ λi ≤ 1)

⇒ private message ui: Pui = λiP ⇒ common message wi: Pwi = (1 − λi)P = ¯ λiP

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Virtual 3-user MAC Channel

Private/common power splitting ratio λi (0 ≤ λi ≤ 1)

⇒ private message ui: Pui = λiP ⇒ common message wi: Pwi = (1 − λi)P = ¯ λiP

Symmetric Gaussian interference channel + rate splitting: y1

=

u1 + w1 +

√

aw2

  • signals

+ √

au2 + z1

  • noise

⇒

R1u, R1w, R2w ∈ CMAC-1(λ1, λ2) y2

=

u2 + w2 + √ aw1

  • signals

+ √

au1 + z2

  • noise

⇒

R2u, R2w, R1w ∈ CMAC-2(λ1, λ2)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

HK Rate Optimization

For fixed λ1 and λ2, the maximum sum-rate with rate splitting is:

RHK (λ1, λ2)

  • max

R1u,R1w ,R2u,R2w(R1u + R1w

  • R1

+ R2u + R2w

  • R2

) = γ

  • λ1P

1 + aλ2P

  • + γ
  • λ2P

1 + aλ1P

  • +

min

  • γ
  • a¯

λ2P 1 + λ1P + aλ2P

  • + γ
  • a¯

λ1P 1 + λ2P + aλ1P

  • ,

1 2γ

  • ¯

λ1P + a¯ λ2P 1 + λ1P + aλ2P

  • + 1

2γ

  • ¯

λ2P + a¯ λ1P 1 + λ2P + aλ1P

  • where γ(x) log2(1 + x)
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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

HK Rate Optimization

For fixed λ1 and λ2, the maximum sum-rate with rate splitting is:

RHK (λ1, λ2)

  • max

R1u,R1w ,R2u,R2w(R1u + R1w

  • R1

+ R2u + R2w

  • R2

) = γ

  • λ1P

1 + aλ2P

  • + γ
  • λ2P

1 + aλ1P

  • +

min

  • γ
  • a¯

λ2P 1 + λ1P + aλ2P

  • + γ
  • a¯

λ1P 1 + λ2P + aλ1P

  • ,

1 2γ

  • ¯

λ1P + a¯ λ2P 1 + λ1P + aλ2P

  • + 1

2γ

  • ¯

λ2P + a¯ λ1P 1 + λ2P + aλ1P

  • where γ(x) log2(1 + x)

The maximum HK sum-rate without time sharing is the solution to the following optimization problem: RRS(a, P) max

0≤λ1,λ2≤1 RHK(λ1, λ2)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Symmetric Power Split

Theorem If we only consider symmetric power splits (i.e., λ1 = λ2 = λsym), the maximum symmetric sum rate achievable with rate splitting is:

Rsym(a, P) = max

0≤λ1=λ2≤1 RHK (λ1, λ2)

=                2γ

  • P

1+aP

  • if

P ≤ 1−a

a2

2γ

  • (a2P+a−1)(1−a)+aP

1+a(a2P+a−1)

  • if

1−a a2

< P ≤

1−a3 a3(a+1)

γ

  • 1−a

2a

  • + γ
  • (1+a)2P−(1−a)

2

  • if

P >

1−a3 a3(a+1)

and the corresponding optimal power split ratio is: λ∗

sym =

       1 if P ≤ 1−a

a2 a2P+a−1 P

if

1−a a2

< P ≤

1−a3 a3(a+1) 1−a (1+a)(aP)

if P >

1−a3 a3(a+1)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Asymmetric Power Split

If we constrain one of the users to send only a common message (i.e., λ1 = 0), the corresponding maximum sum rate is:

Rasym = max

0≤λ2≤1 RHK(λ1 = 0, λ2) = log2

(1 + λ∗

2P + aP)(1 + aP)

1 + aλ∗

2P

  • where λ∗

2 is the solution to the following equation:

  • 1 + λ∗

2P

1 + aλ∗

2P (1 + P + aP) = (1 + λ∗ 2P + aP)(1 + aP)

1 + aλ∗

2P

.

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Asymmetric Power Split

If we constrain one of the users to send only a common message (i.e., λ1 = 0), the corresponding maximum sum rate is:

Rasym = max

0≤λ2≤1 RHK(λ1 = 0, λ2) = log2

(1 + λ∗

2P + aP)(1 + aP)

1 + aλ∗

2P

  • where λ∗

2 is the solution to the following equation:

  • 1 + λ∗

2P

1 + aλ∗

2P (1 + P + aP) = (1 + λ∗ 2P + aP)(1 + aP)

1 + aλ∗

2P

. Conjecture The maximum HK sum-rate is achieved either using symmetric power splits or constraining one of the users send only a common message (i.e., RRS = max{Rsym, Rasym})

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Summary

Symmetric Rate Rsym = max

0≤λ1=λ2≤1 RHK(λ1, λ2)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Summary

Symmetric Rate Rsym = max

0≤λ1=λ2≤1 RHK(λ1, λ2)

Asymmetric Rate Rasym = max

0≤λ2≤1 RHK(λ1 = 0, λ2)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Summary

Symmetric Rate Rsym = max

0≤λ1=λ2≤1 RHK(λ1, λ2)

Asymmetric Rate Rasym = max

0≤λ2≤1 RHK(λ1 = 0, λ2)

Etkin, Tse and Wang RETW = RHK(λ1 = λ2 = 1 aP ) ≤ Rsym

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Summary

Symmetric Rate Rsym = max

0≤λ1=λ2≤1 RHK(λ1, λ2)

Asymmetric Rate Rasym = max

0≤λ2≤1 RHK(λ1 = 0, λ2)

Etkin, Tse and Wang RETW = RHK(λ1 = λ2 = 1 aP ) ≤ Rsym Orthogonal signaling (TDMA/FDMA) Rorth = log2(1 + 2P)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Sum Rate vs. Interference Coefficient a

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3.5 4 4.5 5 5.5 6 6.5 SNR = 10 dB Interference Coefficient a Sum Rate RUB Rsym Rorth Rasym RETW

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Sum Rate vs. Interference Coefficient a

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 7 7.5 8 8.5 9 9.5 10 Interference Coefficient a Sum Rate SNR = 20 dB Rorth Rasym Rsym RUB RETW

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Rate maximizing strategy

10 20 30 40 50 60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a−P Diagram P (SNR) [dB] Interfernce Coefficient a

1 3 2

Rasym

4

Rsym Rorth Rall−private

(λ1 = λ2 = 1)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Rate maximizing strategy

10 20 30 40 50 60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a−P Diagram P (SNR) [dB] Interfernce Coefficient a

1 3 2

Rasym

4

Rsym Rorth Rall−private

(λ1 = λ2 = 1)

Low SNR: Rall-private, Rorth

  • High SNR: Rsym, Rasym
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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

SNR vs. INR

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 SNR−INR Diagram SNR (P) [dB] INR [dB]

1 3

Rorth

2

Rasym

4

Rsym Rall−private

INRdB = SNRdB + 10 log10 a

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

SNR vs. INR

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 SNR−INR Diagram SNR (P) [dB] INR [dB]

1 3

Rorth

2

Rasym

4

Rsym Rall−private

INRdB = SNRdB + 10 log10 a α

INRdB SNRdB

         all-private 0 < α < 1

2

  • rth

α = 1

2

sym

1 2 < α < 1

asym α = 1 α only explains behavior above 20 dB

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Asymptotic sum-rate offset

Fix the interference coefficient a and take P → ∞

∆R(a) lim

P→∞(R − log2(P))

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Asymptotic sum-rate offset

Fix the interference coefficient a and take P → ∞

∆R(a) lim

P→∞(R − log2(P))

∆Rsym(a) =

log2

(1 + a)3

4a

  • ∆Rasym(a)

=

log2

1 + a √a

  • ∆RETW(a)

=

log2

(2a + 1)(a + 1))

4a

  • ∆Rorth(a)

=

1

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

High SNR Behavior

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 a High SNR offset ∆ Rasym ∆ Rorth ∆ Rsym ∆ RETW

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

High SNR Behavior

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 a High SNR offset ∆ Rasym ∆ Rorth ∆ Rsym ∆ RETW

Rasym > Rsym for a > 0.087

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Time Sharing Schemes

Scheme I: 2 equal time slots (optimization over α1, α2, λ1 and λ2)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Time Sharing Schemes

Scheme I: 2 equal time slots (optimization over α1, α2, λ1 and λ2) Scheme II: 4 time slots (optimization over β, λ1 and λ2) - Sason (04)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Numerical Results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 7.6 7.65 7.7 7.75 7.8 7.85 7.9 7.95 8 a Sum Rate SNR = 20 dB RTS RNo−TS RSason

Similar behavior at other SNR’s

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Conclusions

  • Derived expressions for the maximum achievable HK

sum-rate with no time sharing and corresponding optimal power split ratios ⇒ tighter capacity lower bound

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Conclusions

  • Derived expressions for the maximum achievable HK

sum-rate with no time sharing and corresponding optimal power split ratios ⇒ tighter capacity lower bound

  • Despite the fact that the channel is symmetric, allowing for

asymmetric power split ratio at both users (i.e., asymmetric rates) provides larger sum rate for a wide range of a and P values (a > 0.087 at the high SNR regime)

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Conclusions

  • Derived expressions for the maximum achievable HK

sum-rate with no time sharing and corresponding optimal power split ratios ⇒ tighter capacity lower bound

  • Despite the fact that the channel is symmetric, allowing for

asymmetric power split ratio at both users (i.e., asymmetric rates) provides larger sum rate for a wide range of a and P values (a > 0.087 at the high SNR regime)

  • Orthogonal signaling is good for INRdB

SNRdB ≈ 1 2 and low SNR’s

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Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions

Conclusions

  • Derived expressions for the maximum achievable HK

sum-rate with no time sharing and corresponding optimal power split ratios ⇒ tighter capacity lower bound

  • Despite the fact that the channel is symmetric, allowing for

asymmetric power split ratio at both users (i.e., asymmetric rates) provides larger sum rate for a wide range of a and P values (a > 0.087 at the high SNR regime)

  • Orthogonal signaling is good for INRdB

SNRdB ≈ 1 2 and low SNR’s

  • Advantage of using time sharing schemes is quite small