Superposition Coding and Degraded BC Marton’s Coding Scheme and Semi-Deterministic BC Summary Lecture 8 Broadcast Channel I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 29, 2014 1 / 41 I-Hsiang Wang NIT Lecture 8
Superposition Coding and Degraded BC Marton’s Coding Scheme and Semi-Deterministic BC I-Hsiang Wang 2 / 41 K -receiver deterministic broadcast channel K -receiver Gaussian vector broadcast channel K -receiver degraded broadcast channel Capacity is characterized for specific classes of channel laws, including Unlike multiple access channels, characterization of the capacity region of hence a one-to-many one-hop topology. One-to-Many: Consisting of a single transmitter and multiple receivers; broadcast feature of shared wireless medium. Broadcast: The simplest one-hop channel model that captures the Broadcast Channel: Overview Summary NIT Lecture 8 a broadcast channel is in general an open problem. 2 -receiver semi-deterministic broadcast channel 3 -receiver less noisy broadcast channels 2 -receiver more capable broadcast channel
Superposition Coding and Degraded BC Marton’s Coding Scheme and Semi-Deterministic BC I-Hsiang Wang 3 / 41 . 2 Channel: NIT Lecture 8 Broadcast Channel: Problem Formulation Summary Y 1 DEC 1 c W 1 Y 2 DEC 2 c W 2 X W 1 , . . . , W K ENC p Y 1 ,...,Y K | X ...... ...... Y K DEC K c W K 1 K independent messages { W 1 , . . . , W K } , all accessible by the [ 1 : 2 NR k ] encoder. W k ∼ Unif , ∀ k ∈ [1 : K ] . ( ) X , p Y 1 ,..., Y K | X , Y 1 , . . . , Y K 3 Rate tuple: ( R 1 , . . . , R K ) .
Superposition Coding and Degraded BC Marton’s Coding Scheme and Semi-Deterministic BC I-Hsiang Wang 4 / 41 maps a channel output y N that K BC channel code consists of 4 A NIT Lecture 8 Summary Broadcast Channel: Problem Formulation Y 1 DEC 1 c W 1 Y 2 DEC 2 c W 2 X W 1 , . . . , W K ENC p Y 1 ,...,Y K | X ...... ...... Y K DEC K c W K ( ) 2 NR 1 , 2 NR 2 , . . . , 2 NR K , N an encoding function enc N : × [ 1 : 2 NR k ] → X N that maps k =1 message tuple ( w 1 , . . . , w K ) to a length N codeword x N . [ 1 : 2 NR k ] ∀ k ∈ [1 : K ] , a decoding function dec k , N : Y N k → k to a reconstructed message w k .
Superposition Coding and Degraded BC e I-Hsiang Wang 5 / 41 . e BC channel codes exist a sequence of . W K Marton’s Coding Scheme and Semi-Deterministic BC NIT Lecture 8 Summary Broadcast Channel: Problem Formulation Y 1 DEC 1 c W 1 Y 2 DEC 2 c W 2 X W 1 , . . . , W K ENC p Y 1 ,...,Y K | X ...... ...... Y K DEC K c W K { ( )} 5 Error probability P ( N ) � W 1 , . . . , � := Pr ( W 1 . . . , W K ) ̸ = 6 A rate tuple R := ( R 1 , . . . , R K ) is said to be achievable if there ( ) 2 NR 1 , 2 NR 2 , . . . , 2 NR K , N such that P ( N ) → 0 as N → ∞ . { } R ∈ [0 , ∞ ) K : R is achievable 7 The capacity region C := cl
Superposition Coding and Degraded BC W k I-Hsiang Wang 6 / 41 non-cooperating receivers, the above proposition holds. Remark : In fact, for any non-feedback multi-user system with individual e e Marton’s Coding Scheme and Semi-Deterministic BC NIT Lecture 8 , not on the conditional marginal distributions Summary Capacity Region Depends Only on Conditional Marginals Proposition 1 When there is no feedback, the capacity region C depends only on the { } p Y k | X : k = 1 , 2 , . . . , K conditional joint distribution p Y 1 ,..., Y K | X . { } pf : Define P ( N ) for k = 1 , . . . , K , observe that P ( N ) W k ̸ = � e , k := Pr e , k depends only on conditional marginal p Y k | X , and observe that ≤ ∑ K max k ∈ [1: K ] P ( N ) e , k ≤ P ( N ) k =1 P ( N ) e , k . Hence, lim N →∞ P ( N ) ⇒ lim N →∞ P ( N ) e , k = 0 , ∀ k ∈ [1 : K ] ⇐ = 0 .
Superposition Coding and Degraded BC Marton’s Coding Scheme and Semi-Deterministic BC I-Hsiang Wang 7 / 41 N 4 Average power constraint: k NIT Lecture 8 Summary (Scalar) Gaussian Broadcast Channel: Model Z 1 g 1 Y 1 DEC 1 c W 1 X ENC W 1 , W 2 Z 2 g 2 Y 2 DEC 2 c W 2 ( 0 , σ 2 ) 1 Channel law: Y k = g k X + Z k . Z k ∼ N ⊥ ⊥ ( X ) , k = 1 , 2 . 2 White Gaussian: { Z k [ t ] } is an i.i.d. Gaussian r.p. for k = 1 , 2 . ( ) W 1 , W 2 , X t − 1 , Z t − 1 3 Memoryless: Z k [ t ] ⊥ ⊥ , k = 1 , 2 . ∑ N t =1 | x [ t ] | 2 ≤ P . 1 5 Signal-to-noise ratio: SNR k := | g k | 2 P σ 2 , k = 1 , 2 .
Superposition Coding and Degraded BC Capacity region of scalar Gaussian broadcast channel I-Hsiang Wang 8 / 41 be discussed if time allows. Remark : Advanced topics such as Gaussian vector broadcast channel will Capacity region of semi-deterministic broadcast channel Marton’s coding scheme Capacity region of degraded broadcast channel Superposition coding scheme Marton’s Coding Scheme and Semi-Deterministic BC investigate a few important topics, including the general case remains open. information theory – the reason is that the capacity characterization for The research on broadcast channel spans a broad area in network Covered in this Lecture Summary NIT Lecture 8 Hence in this lecture, we only focus on 2 -receiver broadcast channels, and
Superposition Coding and Degraded BC Key challenge lies in encoding – how to embed two independent I-Hsiang Wang 9 / 41 Marton’s Coding Scheme and Semi-Deterministic BC Straightforward thinking: NIT Lecture 8 Summary Encode Two Independent Messages into One Codeword Y N 1 DEC 1 c W 1 X N p Y 1 ,Y 2 | X ENC W 1 , W 2 Y N DEC 2 2 c W 2 messages (data) W 1 and W 2 into a single codeword X N ( W 1 , W 2 ) ? U N 1 ( W 1 ) ENC 1 W 1 Independently encode W k into U N k , k = 1 , 2 , and combine them into X N x ( u 1 , u 2 ) X N via a deterministic map x ( u 1 , u 2 ) . U N 2 ( W 2 ) ENC 2 W 2 ex: X N = U N 1 + U N 2 . ENC
Superposition Coding and Degraded BC Let us use the scalar Gaussian broadcast channel as an example: I-Hsiang Wang 10 / 41 and hence not better than simple time sharing. Unfortunately, the union of these rate tuples are not even convex, Marton’s Coding Scheme and Semi-Deterministic BC NIT Lecture 8 Does this scheme work well? Summary The naïve coding approach gives the following inner bound: for some satisfies the following, then it is achievable: ( U 1 , U 2 ) ∼ p U 1 · p U 2 and deterministic map x ( u 1 , u 2 ) , if ( R 1 , R 2 ) R 1 ≤ I ( U 1 ; Y 1 ) , R 2 ≤ I ( U 2 ; Y 2 ) . U 1 ∼ N (0 , α P ) , U 2 ∼ N (0 , (1 − α ) P ) , α ∈ [0 , 1] . X = U 1 + U 2 . Capacity inner bound: for α ∈ [0 , 1] , ( ) ( ) 1 + (1 − α ) SNR 2 R 1 ≤ 1 α SNR 1 , R 2 ≤ 1 1 + 2 log 1+(1 − α ) SNR 1 2 log 1+ α SNR 2
Superposition Coding and Degraded BC Marton’s Coding Scheme and Semi-Deterministic BC Summary Naïve coding scheme turns out to be quite suboptimal 11 / 41 I-Hsiang Wang NIT Lecture 8 R 2 1 2 log (1 + SNR 2 ) Capacity Region Naive Superposition 1 2 log (1 + SNR 1 ) R 1
Superposition Coding and Degraded BC Marton’s Coding Scheme and Semi-Deterministic BC Summary We shall present two methods to improve the naïve coding scheme: 1 Superposition coding: improvement comes from the decoding part 2 Marton’s coding: improvement comes from the encoding part Before entering the main part, let us introduce two lemmas regarding typicality decoding, which extend from the achievability proofs in the point-to-point channel and the multiple access channel. These lemmas will help reduce time in deriving achievability based on typicality arguments. 12 / 41 I-Hsiang Wang NIT Lecture 8
Superposition Coding and Degraded BC Pr I-Hsiang Wang 13 / 41 Marton’s Coding Scheme and Semi-Deterministic BC Pr pf : NIT Lecture 8 Lemma 1 (Joint Typicality Lemma) Joint Typicality Lemma Summary Let ( U , Y , X ) ∼ p U , Y , X . Let ( � u n , � y n ) be a pair of arbitrary sequences, X n ∼ ∏ n and � x i | � i =1 p X | U ( � u i ) . Then, {( X n ) } y n , � ∈ T ( n ) ≤ 2 − n ( I ( X ; Y | U ) − δ ( ϵ )) � u n , � ( U , Y , X ) ϵ {( X n ) } ∑ y n , � ∈ T ( n ) � u n , � = y n ) p ( � x n | � u n ) ϵ x n ∈T ( n ) � ( X | � u n , � � � ϵ � � � 2 − n (1 − ϵ ) H ( X | U ) ≤ 2 n (1+ ϵ ) H ( X | U , Y ) 2 − n (1 − ϵ ) H ( X | U ) � T ( n ) ≤ ( X | � u n , � y n ) ϵ = 2 − n ( I ( X ; Y | U ) − δ ( ϵ )) , where δ ( ϵ ) = ϵ ( H ( X | U ) + H ( X | U , Y ))
Superposition Coding and Degraded BC be a pair of arbitrarily distributed I-Hsiang Wang 14 / 41 . Marton’s Coding Scheme and Semi-Deterministic BC U n . Furthermore, NIT Lecture 8 Lemma 2 (Packing Lemma) Packing Lemma Summary ( Y n ) � U n , � Let ( U , Y , X ) ∼ p U , Y , X . Let random sequences. Consider random sequences X n ( m ) , indexed by m ∈ M , |M| ≤ 2 nR , is pairwise conditionally independent of � ( U n ) Y n given u n ) · ∏ n X n ( m ) , � � � ∼ p ( � i =1 p X | U ( � x i | � u i ) . {( ) } Y n , X n ( m ) ∈ T ( n ) � U n , � Define event A m := ( U , Y , X ) ϵ {∪ } Then, ∃ δ ( ϵ ) → 0 as ϵ → 0 such that lim m ∈M A m = 0 if n →∞ Pr R < I ( X ; Y | U ) − δ ( ϵ ) .
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