The ε -Capacity Region of AWGN Multiple Access Channels with Feedback Vincent Y. F. Tan (Joint work with Lan V. Truong and Silas L. Fong) National University of Singapore (NUS) SPCOM 2016, Bangalore Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 1 / 27
Information Transmission Shannon Centenary: INFORMATION TRANSMITTER RECEIVER DESTINATION SOURCE RECEIVED SIGNAL SIGNAL MESSAGE MESSAGE NOISE SOURCE Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 2 / 27
Information Transmission Shannon Centenary: INFORMATION TRANSMITTER RECEIVER DESTINATION SOURCE RECEIVED SIGNAL SIGNAL MESSAGE MESSAGE NOISE SOURCE For a channel { p ( y | x ) : x ∈ X , y ∈ Y} , we can transmit information with rates up to the capacity [Shannon (1948)] C = P ∈P ( X ) I ( X ; Y ) max Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 2 / 27
Information Transmission Shannon Centenary: INFORMATION TRANSMITTER RECEIVER DESTINATION SOURCE RECEIVED SIGNAL SIGNAL MESSAGE MESSAGE NOISE SOURCE For a channel { p ( y | x ) : x ∈ X , y ∈ Y} , we can transmit information with rates up to the capacity [Shannon (1948)] C = P ∈P ( X ) I ( X ; Y ) max “Feedback doesn’t increase capacity” [Shannon (1956)] Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 2 / 27
AWGN Channel Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 3 / 27
AWGN Channel Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 3 / 27
AWGN Channel At time i = 1 , 2 , . . . , n , the channel input and output are related by Y i = gX i + Z i , Z i ∼ N ( 0 , 1 ) Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 3 / 27
AWGN Channel At time i = 1 , 2 , . . . , n , the channel input and output are related by Y i = gX i + Z i , Z i ∼ N ( 0 , 1 ) Send M messages encoded as codewords { X n ( m ) : m = 1 , . . . , M } Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 3 / 27
AWGN Channel At time i = 1 , 2 , . . . , n , the channel input and output are related by Y i = gX i + Z i , Z i ∼ N ( 0 , 1 ) Send M messages encoded as codewords { X n ( m ) : m = 1 , . . . , M } Peak power constraint n 1 � X 2 i ( m ) ≤ P , ∀ m ∈ { 1 , . . . , M } n i = 1 Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 3 / 27
AWGN Channel At time i = 1 , 2 , . . . , n , the channel input and output are related by Y i = gX i + Z i , Z i ∼ N ( 0 , 1 ) Send M messages encoded as codewords { X n ( m ) : m = 1 , . . . , M } Peak power constraint n 1 � X 2 i ( m ) ≤ P , ∀ m ∈ { 1 , . . . , M } n i = 1 Expected or Long-Term power constraint M n � 1 � 1 � � X 2 i ( m ) ≤ P . M n m = 1 i = 1 Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 3 / 27
AWGN Channel : Non-Asymptotic Fundamental Limits Let the channel gain g = 1 wlog. Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 4 / 27
AWGN Channel : Non-Asymptotic Fundamental Limits Let the channel gain g = 1 wlog. The average probability of error is P ( n ) := Pr ( ˆ M � = M ) . e Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 4 / 27
AWGN Channel : Non-Asymptotic Fundamental Limits Let the channel gain g = 1 wlog. The average probability of error is P ( n ) := Pr ( ˆ M � = M ) . e Define � M ∗ PP ( n , P , ε ) := max M ∈ N : ∃ length- n code with � M codewords and P ( n ) ≤ ε under the PP constraint e Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 4 / 27
AWGN Channel : Non-Asymptotic Fundamental Limits Let the channel gain g = 1 wlog. The average probability of error is P ( n ) := Pr ( ˆ M � = M ) . e Define � M ∗ PP ( n , P , ε ) := max M ∈ N : ∃ length- n code with � M codewords and P ( n ) ≤ ε under the PP constraint e Define � M ∗ LT ( n , P , ε ) := max M ∈ N : ∃ length- n code with � M codewords and P ( n ) ≤ ε under the LT constraint e Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 4 / 27
First-Order Results Let C ( x ) := 1 2 log ( 1 + x ) , nats per ch. use Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 5 / 27
First-Order Results Let C ( x ) := 1 2 log ( 1 + x ) , nats per ch. use If we demand that the avg error prob. vanishes [Shannon (1948)] , 1 n log M ∗ PP ( n , P , ε ) = C ( P ) , lim ε ↓ 0 lim n →∞ 1 n log M ∗ LT ( n , P , ε ) = C ( P ) . lim ε ↓ 0 lim n →∞ Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 5 / 27
First-Order Results Let C ( x ) := 1 2 log ( 1 + x ) , nats per ch. use If we demand that the avg error prob. vanishes [Shannon (1948)] , 1 n log M ∗ PP ( n , P , ε ) = C ( P ) , lim ε ↓ 0 lim n →∞ 1 n log M ∗ LT ( n , P , ε ) = C ( P ) . lim ε ↓ 0 lim n →∞ In n channel uses, can send up to n C ( P ) nats over p ( y | x ) reliably. Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 5 / 27
First-Order Results If we do not demand that the avg error prob. vanishes [Yoshihara (1964), Polyanskiy-Poor-Verdú (2010)] , 1 n log M ∗ PP ( n , P , ε ) = C ( P ) lim n →∞ 1 P � � n log M ∗ ∀ ε ∈ ( 0 , 1 ) . lim LT ( n , P , ε ) = C , 1 − ε n →∞ Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 6 / 27
First-Order Results If we do not demand that the avg error prob. vanishes [Yoshihara (1964), Polyanskiy-Poor-Verdú (2010)] , 1 n log M ∗ PP ( n , P , ε ) = C ( P ) lim n →∞ 1 P � � n log M ∗ ∀ ε ∈ ( 0 , 1 ) . lim LT ( n , P , ε ) = C , 1 − ε n →∞ The above limits are known as the ε -capacities Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 6 / 27
First-Order Results If we do not demand that the avg error prob. vanishes [Yoshihara (1964), Polyanskiy-Poor-Verdú (2010)] , 1 n log M ∗ PP ( n , P , ε ) = C ( P ) lim n →∞ 1 P � � n log M ∗ ∀ ε ∈ ( 0 , 1 ) . lim LT ( n , P , ε ) = C , 1 − ε n →∞ The above limits are known as the ε -capacities Since for peak-power, the ε -capacity does not depend on ε , the strong converse holds Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 6 / 27
First-Order Results If we do not demand that the avg error prob. vanishes [Yoshihara (1964), Polyanskiy-Poor-Verdú (2010)] , 1 n log M ∗ PP ( n , P , ε ) = C ( P ) lim n →∞ 1 P � � n log M ∗ ∀ ε ∈ ( 0 , 1 ) . lim LT ( n , P , ε ) = C , 1 − ε n →∞ The above limits are known as the ε -capacities Since for peak-power, the ε -capacity does not depend on ε , the strong converse holds Since for long-term, the ε -capacity depends on ε , the strong converse does not hold Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 6 / 27
Strong Converse? 1 n →∞ P ( n ) ε = lim e , R = lim n log M n →∞ 1 0.9 0.8 0.7 0.6 Peak 0.5 ε 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 R C(P) Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 7 / 27
Strong Converse? 1 n →∞ P ( n ) ε = lim e , R = lim n log M n →∞ 1 0.9 0.8 0.7 0.6 Long Term Peak 0.5 ε 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 R C(P) Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 7 / 27
Higher-Order Results More refined asymptotic expansions. Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 8 / 27
Higher-Order Results More refined asymptotic expansions. Third-order [Polyanskiy-Poor-Verdú (2010), T.-Tomamichel (2015)] , n V ( P )Φ − 1 ( ε ) + 1 log M ∗ � PP ( n , P , ε ) = n C ( P ) + 2 log n + O ( 1 ) where the channel dispersion is V ( x ) := x ( x + 2 ) squared nats per ch. use 2 ( x + 1 ) 2 and � a 1 e − t 2 / 2 d t . Φ( a ) := √ 2 π −∞ Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 8 / 27
Higher-Order Results More refined asymptotic expansions. Third-order [Polyanskiy-Poor-Verdú (2010), T.-Tomamichel (2015)] , n V ( P )Φ − 1 ( ε ) + 1 log M ∗ � PP ( n , P , ε ) = n C ( P ) + 2 log n + O ( 1 ) where the channel dispersion is V ( x ) := x ( x + 2 ) squared nats per ch. use 2 ( x + 1 ) 2 and � a 1 e − t 2 / 2 d t . Φ( a ) := √ 2 π −∞ Second-order [Yang-Caire-Durisi-Polyanskiy (2015)] � n log n + o ( √ n ) . P P � � � �� log M ∗ LT ( n , P , ε ) = n C − V 1 − ε 1 − ε Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 8 / 27
Feedback Feedback helps to simplify coding schemes Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 9 / 27
Feedback Feedback helps to simplify coding schemes Long-term power constraint under feedback M n � 1 � � 1 � � i ( m , Y i − 1 ) � X 2 ≤ P . E M n m = 1 i = 1 Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 9 / 27
Feedback Feedback helps to simplify coding schemes Long-term power constraint under feedback M n � 1 � � 1 � � i ( m , Y i − 1 ) � X 2 ≤ P . E M n m = 1 i = 1 Non-asymptotic fundamental limit � M ∗ M ∈ N : ∃ length- n code with FB ( n , P , ε ) := max � M codewords and P ( n ) ≤ ε under the LT-FB constraint e Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 9 / 27
Feedback : Existing Results First-order [Shannon (1956)] 1 n log M ∗ lim ε ↓ 0 lim FB ( n , P , ε ) = C ( P ) . n →∞ Vincent Tan (NUS) AWGN MACs with Feedback SPCOM 2016 10 / 27
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