Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity Alexander Fengler Joint work with Peter Jung and Giuseppe Caire | Technische Universit¨ at Berlin
Communications and Information Theory Chair CommI Multiple Access - Uplink Typical mMTC specifications: – (very) large no. of potential users K tot ∼ ∞ – random access with sparse activity: K a ≪ K tot , – short messages of B bits – central BS – no cooperation between users Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 2
Communications and Information Theory Chair CommI Setting Current solutions – Coordination by the BS: Identification by a unique pilot sequence and subsequent resource allocation (can be very wasteful for short messages and a large no. of inactive users) – Packet based communication with content resolution (Aloha and co.) (simple but suboptimal, ignores the nature of the channel) Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 3
CommI Communications and Information Theory Chair Setting → ”Unsourced” (Polyanskiy 2017): – Each users employs the same codebook – Decoder recovers a list of codewords, up to permutation – Closer to the mMTC requirements but still information theoretic Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 4
CommI Communications and Information Theory Chair Previous work Real AWGN channel without fading – Random coding achievability (Polyanskiy 2017) → existing schemes like TIN or ALOHA perform poorly compared to this bound – Several practical approaches – Reed-Muller code based: (Calderbank and Thompson 2018) – LDPC based: (Vem et al. 2017; Ustinova et al. 2019) – Polar code based: (Marshakov et al. 2019; Pradhan et al. 2019) Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 5
Communications and Information Theory Chair CommI Previous work Real AWGN channel without fading – Random coding achievability (Polyanskiy 2017) → existing schemes like TIN or ALOHA perform poorly compared to this bound – Several practical approaches – Reed-Muller code based: (Calderbank and Thompson 2018) – LDPC based: (Vem et al. 2017; Ustinova et al. 2019) – Polar code based: (Marshakov et al. 2019; Pradhan et al. 2019) – Concatenated scheme: NNLS + Outer Tree Code (Amalladinne et al. 2018) Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 5
Communications and Information Theory Chair CommI This work Real AWGN channel without fading – Last year: Sparse Regression based Unsourced Random Access (Fengler, Jung, and Caire 2019a) (with the outer tree code of (Amalladinne et al. 2018)) – This work: Refined analysis and closed form limits Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 6
Communications and Information Theory Chair CommI Sparse Regression Coding (Barron and Joseph 2011) Each user encodes his LJ -bit message m into n real symbols in the following way: 1. Choose a codebook A = ( A 1 | ... | A L ) with A l = ( a l, 1 | ... | a l, 2 J ) ∈ R n × 2 J 2. Split m in L parts (sections): m = ( m 1 | ... | m L ) 3. Integer representation: → ( i m 1 | ... | i m L ) with i j ∈ [1 : 2 J ] (PPM) 4. Transmit: t = � L l =1 a l,i mi Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 7
Communications and Information Theory Chair CommI Sparse Regression Coding (Barron and Joseph 2011) Each user encodes his LJ -bit message m into n real symbols in the following way: 1. Choose a codebook A = ( A 1 | ... | A L ) with A l = ( a l, 1 | ... | a l, 2 J ) ∈ R n × 2 J 2. Split m in L parts (sections): m = ( m 1 | ... | m L ) 3. Integer representation: → ( i m 1 | ... | i m L ) with i j ∈ [1 : 2 J ] (PPM) 4. Transmit: t = � L l =1 a l,i mi In matrix form: t = Ax with x = ( x 1 | ... | x L ) ⊤ , where x l ∈ { 0 , 1 } 2 J indicate the chosen column in section l . The columns of A are normalized such that E � t � 2 2 = nP Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 7
Communications and Information Theory Chair CommI Channel Model Let K a active users transmit their messages in this way over an AWGN-Adder-MAC: � K a K a � Ax ( k ) + z = A � � x ( k ) y = + z (1) k =1 k =1 Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 8
Communications and Information Theory Chair CommI Channel Model Let K a active users transmit their messages in this way over an AWGN-Adder-MAC: � K a K a � Ax ( k ) + z = A � � x ( k ) y = + z (1) k =1 k =1 Inner Channel: s → As + z (Sparse Recovery Problem) ( x (1) , ..., x ( K a ) ) → � x ( k ) Outer Channel: (Binary Input MAC) Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 8
Communications and Information Theory Chair CommI Channel Model Let K a active users transmit their messages in this way over an AWGN-Adder-MAC: � K a K a � Ax ( k ) + z = A � � x ( k ) y = + z (1) k =1 k =1 Inner Channel: s → As + z (Sparse Recovery Problem) ( x (1) , ..., x ( K a ) ) → � x ( k ) Outer Channel: (Binary Input MAC) → MAC in the sparse domain, e.g. (Cohen, Heller, and Viterbi 1971) Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 8
Communications and Information Theory Chair CommI Outer Channel Assume that the inner decoder recovers the support with symbol-wise error probabilities p fa = P (”0 → 1”) and p md = P (”1 → 0”) . This leads to the following OR-MAC model for the support: Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 9
CommI Communications and Information Theory Chair Outer Channel Assume that the inner decoder recovers the support with symbol-wise error probabilities p fa = P (”0 → 1”) and p md = P (”1 → 0”) . This leads to the following OR-MAC model for the support: Assuming uniform iid messages the output entropy is well approximated (for the typical case K a ≪ 2 J ) by s ) = 2 J H 2 ((1 − p 0 )(1 − p md − p fa ) + p fa ) H (ˆ (2) Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 9
Communications and Information Theory Chair CommI Outer Channel – Assume K a , J → ∞ with J = α log 2 K a for some α > 1 – and that p fa ≤ cK a / 2 J for some constant c , i.e. the false alarm rate does not dominate the sparsity asymptotically (o.w. the achievable rates go to zero), then: � � I ( x 1 , ..., x K a ;ˆ s ) 1 − 1 = (1 − p md ) lim (3) JK a α K a ,J →∞ – For p md = 0 this is achievable by the tree code of (Amalladinne et al. 2018), at exponential complexity, or up to a constant with a polynomial complexity. Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 10
Communications and Information Theory Chair CommI Inner Channel 1. Approximate Message Passing (AMP)(Donoho, Maleki, and Montanari 2009) s t +1 = η t ( s t + A ⊤ z t ) (4) z t +1 = y − As t +1 + 2 J S n z t � t ( s t + A ⊤ z t ) � η ′ (5) (6) where η t ( r ) is applied componentwise and given by (Fengler, Jung, and Caire 2019b) � ˆ ��� − 1 � � � ˆ p 0 1 P P η t ( r i ) = 1 + exp − 2 r i (7) τ 2 1 − p 0 2 τ t t t = || z t || with τ 2 and p 0 = (1 − 2 − J ) K a . n 2. SBS-MAP i = 1 , ..., L 2 J MAP s ˆ = arg max P ( s i | y ) (8) i s i ∈{ 0 ,...,K a } Both can be analysed asymptotically by the RS-potential Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 11
Communications and Information Theory Chair CommI Inner Channel 1. Approximate Message Passing (AMP)(Donoho, Maleki, and Montanari 2009) s t +1 = η t ( s t + A ⊤ z t ) (4) z t +1 = y − As t +1 + 2 J S n z t � t ( s t + A ⊤ z t ) � η ′ (5) (6) where η t ( r ) is applied componentwise and given by (Fengler, Jung, and Caire 2019b) � ˆ ��� − 1 � � � ˆ p 0 1 P P η t ( r i ) = 1 + exp − 2 r i (7) τ 2 1 − p 0 2 τ t t t = || z t || with τ 2 and p 0 = (1 − 2 − J ) K a . n 2. SBS-MAP i = 1 , ..., L 2 J MAP s ˆ = arg max P ( s i | y ) (8) i s i ∈{ 0 ,...,K a } Both can be analysed asymptotically by the RS-potential Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 11
Communications and Information Theory Chair CommI Inner Channel 1. Approximate Message Passing (AMP)(Donoho, Maleki, and Montanari 2009) s t +1 = η t ( s t + A ⊤ z t ) (4) z t +1 = y − As t +1 + 2 J S n z t � t ( s t + A ⊤ z t ) � η ′ (5) (6) where η t ( r ) is applied componentwise and given by (Fengler, Jung, and Caire 2019b) � ˆ ��� − 1 � � � ˆ p 0 1 P P η t ( r i ) = 1 + exp − 2 r i (7) τ 2 1 − p 0 2 τ t t t = || z t || with τ 2 and p 0 = (1 − 2 − J ) K a . n 2. SBS-MAP i = 1 , ..., L 2 J MAP s ˆ = arg max P ( s i | y ) (8) i s i ∈{ 0 ,...,K a } Both can be analysed asymptotically by the RS-potential Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity | A.Fengler Page 11
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