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A NACHRICHTENTECHNIK LRA Decision-Feeback Equalization Structure | - PowerPoint PPT Presentation

Oct 15, 2023 •455 likes •629 views

Introduction Equalization Structure of the Signals | Maximum-Likelihood Detection | Linear Equalization Lattice-Reduction-Aided Equalization LRA Scheme | IF Scheme Robert F.H. Fischer Sebastian Stern Factorization Task Criteria | Constraints

  1. Introduction Equalization Structure of the Signals | Maximum-Likelihood Detection | Linear Equalization Lattice-Reduction-Aided Equalization LRA Scheme | IF Scheme Robert F.H. Fischer Sebastian Stern Factorization Task Criteria | Constraints Lattices and Lattice Problems Shortest Independent Vector Problem | Lattice Basis Reduction Institut für Nachrichtentechnik, Universität Ulm Numerical Results A NACHRICHTENTECHNIK LRA Decision-Feeback Equalization Structure | Sorting | Algorithm Numerical Results Summary Fischer: Lattice Reduction and Factorization for Equalization 1 multipoint-to-point transmission, MIMO multiple-access channel K non-cooperating single-antenna users abstract high-level view of digital communications central base station with N R receive antennas – a point x drawn from some signal constellation A is transmitted > joint processing/decoding at the receiver side possible (a point can represent log 2 |A| bits of information) = n – the channel adds (interference and) noise n x y x ˆ n q 1 c 1 x 1 – the received symbols is y = x + n M A = {− 1 , +1 } ENC y H – at the receiver, decisions have to be taken q K c K x K M ENC y = Hx + n since we can use quadrature modulation (modulation of amplitude and phase) , all signals are complex-valued F p C channel coding done over the finite field F p for reducing the error rate, channel coding is employed ( q k and c k taken from F p ) in block codes (codelength η ) not all A η combinations are used but only mapping M of finite-field symbols c k to complex-valued points x k those which can be distinguished reliably taken from some signal constellation A a trade-off between transmission rate (bit rate) and error rate is possible Fischer: Lattice Reduction and Factorization for Equalization 2 Fischer: Lattice Reduction and Factorization for Equalization 3

  2. How to perform equalization / decoding? y = Hx + n r 1 ˆ q 1 x 1 ˆ M − 1 DEC ENC − 1 y done symbol-by-symbol (independently over the time steps) in the uncoded case ? r K q K ˆ x K ˆ M − 1 DEC ENC − 1 C C F p linear equalization joint equalization / decoding typically much to complex according to zero-forcing (ZF) or minimum mean-squared error (MMSE) criterion > separate equalization / decoding = decision-feedback equalization (DFE) aka successive interference cancellation, (V-)BLAST channel decoding – individual (per user) lattice-reduction-aided (LRA) / integer-forcing (IF) schemes – over a temporal block (code word) low-complexity, high-performance schemes maximum-likelihood detection (MLD) / lattice decoding low-complexity equalization strategy (as for the uncoded case) optimum procedure, highest complexity – over the users – per time step Fischer: Lattice Reduction and Factorization for Equalization 4 Fischer: Lattice Reduction and Factorization for Equalization 5 Construction n signal point lattice q c x r x ˆ ˆ q M − 1 M ENC DEC ENC − 1 Λ a Λ s Λ s Λ s F p F p C C C F p typically: Λ a = Z or Λ a = G = Z + j Z R V ( Λ s ) R V ( Λ s ) „ shaping “ lattice encoding ENC over F p Λ s Λ s A mapping M to signal point in C and its Voronoi region R V ( Λ s ) 0101 1001 (typically a sublattice of Λ a : Λ s ⊂ Λ a ) 0001 0100 1000 1101 C 0000 0011 0111 1100 signal constellation lattice decoding (in signal space) 0010 0110 1011 1111 Λ a w.r.t. to Λ c A = Λ a ∩ R V ( Λ s ) 1010 1110 demapping M − 1 to ˆ c ∈ F p encoder inverse ENC − 1 lattice code do everything in N dimensions C = Λ a ∩ R V ( Λ s ) demapping modulo Λ s , i.e., mod M − 1 Fischer: Lattice Reduction and Factorization for Equalization 6 Fischer: Lattice Reduction and Factorization for Equalization 7

  3. (real-valued example K = 2 , Λ c = Z , |A| = 5 ) K -dim. lattice spanned by basis vectors b 1 , b 2 , . . . , b K — basis matrix � b 1 b 2 · · · b K � B = x Hx y = Hx + n real-valued lattice � z 1 � � � � K = B Z K Λ = λ = k =1 z k b k = B | z k ∈ Z def . . . z K for x ⊂ G K = ( Z + j Z ) K the noise-free receive vectors z = Hx are taken from the complex-valued lattice Λ = H G K spanned by the columns h k of the channel matrix � h 1 h 2 · · · h K � H = Fischer: Lattice Reduction and Factorization for Equalization 8 Fischer: Lattice Reduction and Factorization for Equalization 9 f X ( x ) : probability density function ML criterion simple strategy — filtering followed by individual decision/decoding � � � 2 n � y − Hx x = argmax x ∈A K f Y ( y | x ) = argmin ˆ x 1 x 1 ˆ x ∈A K y r H F LE x K x K ˆ this equalization strategy / scheme can be optimized either according to the zero-forcing (ZF) or minimum mean-squared error (MMSE) criterion zero-forcing criterion: ( I : identity matrix; ( · ) + : (left) pseudoinverse) � H H H � − 1 H H ! H + F LE · H = I > F LE , ZF = def = = = σ 2 n /σ 2 minimum mean-squared error criterion: ( ζ def x ) error signal e = F LE y − x ; error covariance matrix Φ ee = E { ee H } lattice decoding — high complexity per time step � Φ ee � � H H H + ζ I � − 1 H H ! e ffi cient implementation via the Sphere Decoder trace → min > F LE , MMSE = [AEVZ’02] = for combination with channel decoding generation of soft output required Fischer: Lattice Reduction and Factorization for Equalization 10 Fischer: Lattice Reduction and Factorization for Equalization 11

  4. � � � H H H � − 1 H H = f 1 ZF solution — F LE , ZF = r = F LE , ZF y = x + F LE , ZF n . . ; . f K – noise variance ( n i.i.d. components with variance σ 2 n ) σ 2 n k = σ 2 n · � f k � 2 – noise enhancement E k = σ 2 n k /σ 2 n = � f k � 2 � H H H + ζ I � − 1 H H (biased) MMSE solution — F LE , MMSE = � � � H � � H H H ) − 1 H H = f 1 or with H = we have F LE , MMSE = √ ζ I . . . f K – error covariance matrix � H H H + ζ I � − 1 = � H H H � − 1 Φ ee /σ 2 n = of equalizing the signal � H H H ) − 1 H H H � H H H ) − 1 = � H H H ) − 1 ) > noise enhancement – noise enhancement ( F LE , MMSE F H the noise is fi ltered, too = LE , MMSE = � Φ ee /σ 2 � k,k = � f k � 2 individual threshold decision per dimension not optimum E k = n Fischer: Lattice Reduction and Factorization for Equalization 12 Fischer: Lattice Reduction and Factorization for Equalization 13 � h 1 h 2 � � h 1 h 2 � H = H = � c 1 c 2 � C = Z ∈ Z 2 × 2 = HZ , | det( Z ) | =1 Fischer: Lattice Reduction and Factorization for Equalization 14 Fischer: Lattice Reduction and Factorization for Equalization 14

  5. q 1 ˆ x 1 ˆ M − 1 DEC ENC − 1 r ′ y r Z − 1 F LE ,C q K ˆ ˆ x K M − 1 DEC ENC − 1 C F p [YW’02], [WF’03] H n q 1 c 1 x 1 x 1 ¯ ˆ q 1 ˆ M − 1 M ENC DEC ENC − 1 y r x ˆ Z − 1 Z C F LE ,C ˆ q K c K x K x K ¯ q K ˆ M − 1 M ENC DEC ENC − 1 F p C C C F p � h 1 h 2 � H = � c 1 c 2 � C = Z ∈ Z 2 × 2 = HZ , | det( Z ) | =1 Fischer: Lattice Reduction and Factorization for Equalization 14 Fischer: Lattice Reduction and Factorization for Equalization 15 [NG’11] [NG’11] ˆ ˆ q 1 c 1 x 1 r 1 q 1 ¯ q 1 ¯ ˆ ˆ x 1 ¯ ¯ x 1 mod M − 1 mod M − 1 M ENC DEC ENC − 1 DEC ENC − 1 q ˆ y r q ˆ Z − 1 Z − 1 F LE F F ˆ ˆ q K c K x K r K q K ¯ ¯ q K ˆ ˆ x K ¯ ¯ x K mod M − 1 mod M − 1 M ENC DEC ENC − 1 DEC ENC − 1 F p C C F p C F p the receiver decodes an integer linear combination of the codewords the receiver decodes an integer linear combination of the codewords resolution of linear combinations at some central unit resolution of linear combinations at some central unit only fi nite- fi eld symbols are communicated — processing over F p only fi nite- fi eld symbols are communicated — processing over F p if a joint/central receiver is present, some preprocessing can be done prior to channel decoding — integer-forcing receiver [ZNEG’14] Fischer: Lattice Reduction and Factorization for Equalization 16 Fischer: Lattice Reduction and Factorization for Equalization 16

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