RateDiversity Optimal Multiblock SpaceTime Codes via Sum-Rank Codes - PowerPoint PPT Presentation

Rate–Diversity Optimal Multiblock Space–Time Codes via Sum-Rank Codes Mohannad Shehadeh and Frank R. Kschischang Department of Electrical and Computer Engineering University of Toronto { mshehadeh, frank } @ece.utoronto.ca 2020 IEEE International Symposium on Information Theory June 21–26, 2020

Channel Model MIMO Rayleigh block-fading channel: ◮ n r receive antennas, n t transmit antennas ◮ L fading blocks of duration T Y i = ρ H i X i + W i for i = 1 , 2 , . . . , L ◮ H i are n r × n t with iid CN (0 , 1) entries ◮ X i is n t × T transmission in the i th fading block ◮ W i are n r × T with iid CN (0 , 1) entries ◮ ρ related to SNR in the usual way L � � ρ X i � 2 E F = L · T · SNR i =1

Multiblock Space–Time Codes ◮ Fix a constellation which is a finite A ⊂ C ◮ An L -block n t × T code is a finite X ⊆ A n t × LT ◮ Any n t × LT codeword X ∈ X partitions as � · · · � X = X 1 X 2 X L ◮ X i for i = 1 , 2 , . . . , L are n t × T sub-codewords ◮ Perfect channel knowledge at receiver with ML decoding ◮ T ≥ n t ◮ T = n t referred to as minimal-delay

Definition (Transmit Diversity Gain) An L -block n t × T code X ⊆ A n t × LT will be said to achieve a transmit diversity gain of d if L � rank( X i − X ′ d = min i ). X , X ′ ∈X i =1 X � = X ′ [Guey et al., 1996], [Tarokh et al., 1998] ◮ If this is the case, then P e (SNR) = O (SNR − n r d ) as SNR → ∞ ◮ d ≤ Ln t ◮ d = Ln t referred to as full diversity

Rate–Diversity Tradeoff Definition (Rate) The rate R of an L -block n t × T code X ⊆ A n t × LT is R = 1 LT log |A| |X| . Theorem (Rate–Diversity Tradeoff) Let X ⊆ A n t × LT be an L-block n t × T code achieving transmit diversity gain d and rate R, then R ≤ n t − d − 1 . L [El Gamal and Hammons, 2003], [Lu and Kumar, 2005] Proof. True by a Singleton bound argument.

Codes over Extension Fields ◮ F q m is an m -dimensional vector space over F q ◮ Fix an ordered basis B = { β 1 , β 2 , . . . , β m } of F q m over F q ◮ Any c ∈ F s q m can be written as �� m � m � m · · · � c = i =1 β i c i 1 i =1 β i c i 2 i =1 β i c is m � � � = β i c i 1 c i 2 · · · c is i =1 Definition (Matrix Representation) The matrix representation of any c ∈ F s q m is given by the map M B : F s → F m × s q m − defined by q   c 11 c 12 · · · c 1 s c 21 c 22 · · · c 2 s   M B ( c ) =  .  . . .  ... . . .   . . .  c m 1 c m 2 · · · c ms

Sum-Rank Metric [N´ obrega and Uchˆ oa-Filho, 2016] ◮ Fix a sum-rank length partition N = r 1 + r 2 + · · · + r L ◮ Any c ∈ F N q m partitions as c (1) c (2) c ( L ) � � c = · · · where c ( i ) ∈ F r i q m for i = 1 , 2 , . . . , L ◮ Define sum-rank distance between c , d ∈ F N q m L � rank( M B ( c ( i ) ) − M B ( d ( i ) )) d SR ( c , d ) = i =1 ◮ r 1 = r 2 = · · · = r L = 1 , L = N : Hamming distance ◮ r 1 = N , L = 1: Rank distance [Gabidulin, 1985] ◮ Define minimum sum-rank distance of a code C ⊆ F N q m d SR ( C ) = min d SR ( c , d ) c , d ∈C c � = d

◮ If C is a k -dimensional linear code over F q m , then k ≤ N − d SR ( C ) + 1 ◮ Maximum sum-rank distance (MSRD) codes meet above with equality ◮ L = N (Hamming distance): Classical Reed–Solomon codes ◮ L = 1 (Rank distance): Gabidulin codes [Gabidulin, 1985] Arbitrary sum-rank length partition: ◮ Gabidulin codes [Gabidulin, 1985] are MSRD for m ≥ N ◮ m ≥ N translates to T ≥ Ln t in construction of [Lu and Kumar, 2005] based on Gabidulin codes ◮ Linearized Reed–Solomon codes [Mart´ ınez-Pe˜ nas, 2018] are MSRD for m ≥ max i r i ◮ m ≥ max i r i translates to T ≥ n t

Definition (Rank-Metric-Preserving Map) → A n t × T be an → A be a bijection and ˜ φ : F n t × T Let φ : F q − − q element-wise version. φ is rank-metric-preserving if, for all C , D ∈ F n t × T , C � = D , q rank(˜ φ ( C ) − ˜ φ ( D )) ≥ rank( C − D ) with the first rank over C and the second over F q . Example (Gaussian Integer Map [Bossert et al., 2002]) ◮ Z [ ı ] = { a + ı b | a , b ∈ Z } form a Euclidean domain ◮ If Π is prime in Z [ ı ], then Z [ ı ] / Π Z [ ı ] is isomorphic to F | Π | 2 ◮ The map � a � φ : a �→ a − round Π Π with a interpreted as an integer is rank-metric-preserving

Rate–Diversity Optimal Multiblock Space–Time Codes ◮ Let r 1 = r 2 = · · · = r L = n t and N = Ln t ◮ Let m = T ◮ Let C ⊆ F Ln t q T be a linear code with generator matrix of the form ∈ F k × Ln t � � G = G 1 G 2 · · · G L q T with G 1 , G 2 , . . . , G L ∈ F k × n t referred to as sub-codeword q T generators

Linearized Reed–Solomon Codes [Mart´ ınez-Pe˜ nas, 2018] Define the F q T − → F q T functions: ◮ σ ( a ) = a q ◮ For all i ∈ N , define N i ( a ) = σ i − 1 ( a ) σ i − 2 ( a ) · · · σ ( a ) a ◮ For all i ∈ N and a ∈ F q T , define D i a ( b ) = σ i ( b ) N i ( a ) Let q > L , T ≥ n t , x be a primitive element of F q T and let sub-codeword generators for C be  β 1 β 2 · · · β n t  D x i − 1 ( β 1 ) D x i − 1 ( β 2 ) · · · D x i − 1 ( β n t )   D 2 D 2 D 2   x i − 1 ( β 1 ) x i − 1 ( β 2 ) · · · x i − 1 ( β n t ) G i =   . . .  ...  . . .   . . .   D k − 1 D k − 1 D k − 1 x i − 1 ( β 1 ) x i − 1 ( β 2 ) · · · x i − 1 ( β n t ) for i = 1 , 2 , . . . , L . Theorem ([Mart´ ınez-Pe˜ nas, 2018]) C is MSRD.

Rate–Diversity Optimal Multiblock Space–Time Codes Let: ◮ T ≥ n t ◮ k = Ln t − d + 1 ◮ q = | Π | 2 > L with φ the corresponding Gaussian integer map ∈ F k × Ln t ◮ G = � � · · · G 1 G 2 G L be a linearized q T Reed–Solomon code generator � � �� ˜ � ˜ ˜ � u ∈ F k � X = · · · φ ( M B ( u G 1 ) ⊺ ) φ ( M B ( u G 2 ) ⊺ ) φ ( M B ( u G L ) ⊺ ) � q T Corollary X is a rate–diversity optimal L-block n t × T code achieving transmit diversity gain d.

◮ Define bit rate R b = 1 LT log 2 |X| ◮ There is no tradeoff between R b and d : R b = R · log 2 |A| ◮ There exist d = Ln t codes with arbitrarily large |A| ◮ Multiblock diversity–multiplexing optimal codes [Lu, 2008], [Yang and Belfiore, 2007] ◮ Codes from cyclic division algebras [Sethuraman et al., 2003] (repeat each codeword L times)

R b /n t = 2 and d = ⌈ (1 − ε ) Ln t ⌉ 10 7 ε = 0 10 6 5 0 0 . = 10 5 ε 1 0 . 10 4 = ε |A| 10 3 ε = 0 . 25 10 2 10 1 7 5 = 0 ε . 10 0 0 5 10 15 20 25 30 Ln t

n r = n t = T = L = 2 and R b ≈ 4 Reference code (Mult. CDA): 2-block diversity–multiplexing optimal code Mult. CDA, d = 4 due to [Lu, 2008], [Yang and Belfiore, 2007] 0 . 5 10 0 Mult. CDA, d = 4, R b = 4 . 00 Sum-Rank, d = 3, R b = 4 . 09 0 10 − 1 Codeword Error Rate − 0 . 5 10 − 2 0 0 . 5 − 0 . 5 Sum-Rank, d = 3 10 − 3 0 . 5 10 − 4 0 10 − 5 0 5 10 15 20 25 − 0 . 5 SNR (dB) − 0 . 5 0 0 . 5

R b = 4, n t = 2, and d = ⌈ (1 − ε ) Ln t ⌉ 10 7 L + 1 0 10 6 = ε 10 5 1 0 Mult. CDA . = 10 4 ε |A| 10 3 ε = 0 . 25 Sum-Rank 10 2 ε = 0 . 5 10 1 10 0 0 1 2 3 4 5 6 8 10 12 14 16 L

Concluding Remarks ◮ Linearized Reed–Solomon codes [Mart´ ınez-Pe˜ nas, 2018] translate to minimal-delay rate–diversity optimal multiblock space–time codes ◮ Such codes can perform well with small constellations compared to full diversity alternatives ◮ Future work: ◮ Further performance analysis, coding gain analysis ◮ Algebraic soft-decision decoding