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
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
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
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
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
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
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
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
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions Network Model Two-user Gaussian interference channel: = h 11 x 1 + h 21 x 2 + ¯ y 1 z 1 h 12 x 1 + h 22 x 2 + ¯ y 2 = z 2
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions Network Model Two-user Gaussian interference channel: = h 11 x 1 + h 21 x 2 + ¯ y 1 z 1 h 12 x 1 + h 22 x 2 + ¯ y 2 = z 2 Normalized symmetric channel | h 11 | = | h 22 | , | h 12 | = | h 21 | , a = | h 21 | 2 | h 11 | 2 = | h 12 | 2 | h 22 | 2 , 0 < a < 1 , z i ∼ C N ( 0 , 1 ) , P 1 = P 2 = P = SNR √ √ y 1 = x 1 + ax 2 + z 1 , y 2 = ax 1 + x 2 + z 2 Channel fully characterized by interference coefficient a and P
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions Virtual 3-user MAC Channel Private/common power splitting ratio λ i (0 ≤ λ i ≤ 1) ⇒ private message u i : P u i = λ i P ⇒ common message w i : P w i = ( 1 − λ i ) P = ¯ λ i P
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions Virtual 3-user MAC Channel Private/common power splitting ratio λ i (0 ≤ λ i ≤ 1) ⇒ private message u i : P u i = λ i P ⇒ common message w i : P w i = ( 1 − λ i ) P = ¯ λ i P Symmetric Gaussian interference channel + rate splitting: √ √ = u 1 + w 1 + + au 2 + z 1 y 1 aw 2 � �� � � �� � signals noise ⇒ R 1 u , R 1 w , R 2 w ∈ C MAC-1 ( λ 1 , λ 2 ) u 2 + w 2 + √ + √ y 2 = aw 1 au 1 + z 2 � �� � � �� � signals noise ⇒ R 2 u , R 2 w , R 1 w ∈ C MAC-2 ( λ 1 , λ 2 )
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions HK Rate Optimization For fixed λ 1 and λ 2 , the maximum sum-rate with rate splitting is: � R HK ( λ 1 , λ 2 ) R 1 u , R 1 w , R 2 u , R 2 w ( R 1 u + R 1 w + R 2 u + R 2 w ) max � �� � � �� � R 1 R 2 � � � � λ 1 P λ 2 P = + γ + γ 1 + a λ 2 P 1 + a λ 1 P � � � � � a ¯ a ¯ λ 2 P λ 1 P + γ min γ , 1 + λ 1 P + a λ 2 P 1 + λ 2 P + a λ 1 P � � � �� ¯ λ 1 P + a ¯ λ 2 P + a ¯ ¯ 1 λ 2 P + 1 λ 1 P 2 γ 2 γ 1 + λ 1 P + a λ 2 P 1 + λ 2 P + a λ 1 P where γ ( x ) � log 2 ( 1 + x )
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions HK Rate Optimization For fixed λ 1 and λ 2 , the maximum sum-rate with rate splitting is: � R HK ( λ 1 , λ 2 ) R 1 u , R 1 w , R 2 u , R 2 w ( R 1 u + R 1 w + R 2 u + R 2 w ) max � �� � � �� � R 1 R 2 � � � � λ 1 P λ 2 P = + γ + γ 1 + a λ 2 P 1 + a λ 1 P � � � � � a ¯ a ¯ λ 2 P λ 1 P + γ min γ , 1 + λ 1 P + a λ 2 P 1 + λ 2 P + a λ 1 P � � � �� λ 1 P + a ¯ ¯ λ 2 P + a ¯ ¯ 1 λ 2 P + 1 λ 1 P 2 γ 2 γ 1 + λ 1 P + a λ 2 P 1 + λ 2 P + a λ 1 P where γ ( x ) � log 2 ( 1 + x ) The maximum HK sum-rate without time sharing is the solution to the following optimization problem: R RS ( a , P ) � 0 ≤ λ 1 ,λ 2 ≤ 1 R HK ( λ 1 , λ 2 ) max
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., λ 1 = λ 2 = λ sym ), the maximum symmetric sum rate achievable with rate splitting is: R sym ( a , P ) = 0 ≤ λ 1 = λ 2 ≤ 1 R HK ( λ 1 , λ 2 ) max � � P P ≤ 1 − a 2 γ if 1 + aP a 2 � � ( a 2 P + a − 1 )( 1 − a )+ aP 1 − a 3 1 − a = 2 γ if < P ≤ 1 + a ( a 2 P + a − 1 ) a 2 a 3 ( a + 1 ) � � � � ( 1 + a ) 2 P − ( 1 − a ) 1 − a 3 1 − a γ + γ if P > 2 a 2 a 3 ( a + 1 ) and the corresponding optimal power split ratio is: P ≤ 1 − a 1 if a 2 a 2 P + a − 1 1 − a 3 λ ∗ 1 − a sym = if < P ≤ a 2 a 3 ( a + 1 ) P 1 − a 3 1 − a if P > ( 1 + a )( aP ) a 3 ( a + 1 )
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., λ 1 = 0), the corresponding maximum sum rate is: � ( 1 + λ ∗ � 2 P + aP )( 1 + aP ) R asym = 0 ≤ λ 2 ≤ 1 R HK ( λ 1 = 0 , λ 2 ) = log 2 max 1 + a λ ∗ 2 P where λ ∗ 2 is the solution to the following equation: � 1 + λ ∗ 2 P ( 1 + P + aP ) = ( 1 + λ ∗ 2 P + aP )( 1 + aP ) 2 P . 1 + a λ ∗ 1 + a λ ∗ 2 P
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., λ 1 = 0), the corresponding maximum sum rate is: � ( 1 + λ ∗ � 2 P + aP )( 1 + aP ) R asym = 0 ≤ λ 2 ≤ 1 R HK ( λ 1 = 0 , λ 2 ) = log 2 max 1 + a λ ∗ 2 P where λ ∗ 2 is the solution to the following equation: � 1 + λ ∗ 2 P ( 1 + P + aP ) = ( 1 + λ ∗ 2 P + aP )( 1 + aP ) 2 P . 1 + a λ ∗ 1 + a λ ∗ 2 P 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., R RS = max { R sym , R asym } )
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions Summary Symmetric Rate R sym = 0 ≤ λ 1 = λ 2 ≤ 1 R HK ( λ 1 , λ 2 ) max
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions Summary Symmetric Rate R sym = 0 ≤ λ 1 = λ 2 ≤ 1 R HK ( λ 1 , λ 2 ) max Asymmetric Rate R asym = 0 ≤ λ 2 ≤ 1 R HK ( λ 1 = 0 , λ 2 ) max
Introduction Optimized HK Sum-Rate Numerical Results High SNR Time Sharing Conclusions Summary Symmetric Rate R sym = 0 ≤ λ 1 = λ 2 ≤ 1 R HK ( λ 1 , λ 2 ) max Asymmetric Rate R asym = 0 ≤ λ 2 ≤ 1 R HK ( λ 1 = 0 , λ 2 ) max Etkin, Tse and Wang R ETW = R HK ( λ 1 = λ 2 = 1 aP ) ≤ R sym
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