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**Unformatted text preview: **Harvard SEAS ES250 – Information Theory Network Information Theory * 1 Gaussian Multiple-User Channels Definition The basic discrete-time AWGN channel with input power P and noise variance N is modeled by Y i = X i + Z i , i = 1 , 2 , ··· where i is the time index and Z i are i.i.d. Gaussian r.v., with a power constraint 1 n n X i =1 X 2 i ≤ P Definition For the convenience, define C ( x ) as : C ( x ) = 1 2 log(1 + x ) Theorem The capacity C of the basic AWGN channel is obtained by max E [ X 2 ] ≤ P I ( X ; Y ) and is given by C µ P N ¶ = 1 2 log µ 1 + P N ¶ Theorem (Gaussian Multiple-Access Channel) The achievable rate region for the Gaussian multiple access channel with m users is: X i ∈ S R i < C µ | S | P N ¶ for all S ⊂ { 1 , 2 , ··· , m } . Theorem (Gaussian Broadcast Channel) The capacity region of the Gaussian broadcast channel is R 1 < C µ αP N 1 ¶ R 2 < C µ (1- α ) P αP + N 2 ¶ where α may be arbitrarily chosen (0 ≤ α ≤ 1) to trade off rate R 1 for rate R 2 as the transmitter wishes. Theorem (Gaussian Relay Channel) The capacity region of the Gaussian relay channel is C = max ≤ α ≤ 1 min ‰ C µ P + P 1 + 2 √ ¯ αP P 1 N 1 + N 2 ¶ , C µ αP N 1 ¶¾ where sender X has power P , sender X 1 (relay) has power P 1 , and ¯ α = 1- α . * Based on Cover & Thomas, Chapter 15 1 Harvard SEAS ES250 – Information Theory 2 Jointly Typical Sequences Definition The set A ( n ) ² of ²-typical n-sequences ( x 1 , x 2 , ··· , x k ) is defined by A ( n ) ² ( X (1) , X (2) , ··· , X ( k ) ) = ‰ ( x 1 , x 2 , ··· , x k ) : fl fl fl fl- 1 n log p ( s )- H ( S ) fl fl fl fl < ², ∀ S ⊆ { X (1) , X (2) , ··· , X ( k ) } ¾ . Definition We will use the notation a n . = 2 n ( b ± ² ) to mean that fl fl fl fl 1 n log a n- b fl fl fl fl < ² for n sufficiently large. Theorem For any ² > 0, for sufficiently large n , 1. P ( A ( n ) ² ( S )) ≥ 1- ² , ∀ S ⊆ { X (1) , X (2) , ··· , X ( k ) } . 2. s ∈ A ( n ) ² ( S ) ⇒ p ( s ) . = 2 n ( H ( S ) ± ² ) . 3. | A ( n ) ² ( S ) | . = 2 n ( H ( S ) ± 2 ² ) . 4. Let S 1 , S 2 ⊆ { X (1) , X (2) , ··· , X ( k ) } . If ( s 1 , s 2 ) ∈ A ( n ) ² ( S 1 , S 2 ), then p ( s 1 , s 2 ) . = 2 n ( H ( S 1 | S 2 ) ± 2 ² ) Theorem Let S 1 , S 2 be two subsets of X (1) , X (2) , ··· , X ( k ) . For any ² > 0, define A ( n ) ² ( S 1 | s 2 ) to be the set of s 1 sequences that are jointly ²-typical with a particular s 2 sequence. If s 2 ∈ A ( n ) ² ( S 2 ), then for sufficiently large n , we have | A ( n ) ² ( S 1 | s 2 ) | ≤ 2 n ( H ( S 1 | S 2 )+2 ² ) and (1- ² )2 n ( H ( S 1 | S 2 )- 2 ² ) ≤ X s 2 p ( s 2 ) | A ( n ) ² ( S 1 | s 2 ) | ....

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