Augmented Lattice Reduction for MIMO decoding

Transcription

1 Augmented Lattice Reduction for MIMO decoding LAURA LUZZI joint work with G. Rekaya-Ben Othman and J.-C. Belfiore at Télécom-ParisTech NANYANG TECHNOLOGICAL UNIVERSITY SEPTEMBER 15, 2010 Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 1

2 Motivation Multiple antenna systems allow to improve data rates and reliability. In order to increase data rates, both the number of antennas and the size of the signal set can be increased. This entails a high decoding complexity which is a real challenge for practical implementation. Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 2

3 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 3

4 MIMO systems bits modulation coding X H Y decoding demodulation channel multiplexing gain - send independent data on each antenna - improve the rate diversity gain - send independent copies of the same data - improve reliability Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 4

5 System model: spatial multiplexing y n 1 = H n m x m 1 + w n 1 received signal channel codeword noise m transmit antennas, n receive antennas H, w are random matrices with i.i.d. complex Gaussian entries x i S Z[i] is the signal transmitted by antenna i Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 5

6 System model: Space-Time Coding Y n t = H n m X m t + W n t received signal channel codeword noise m transmit antennas, n receive antennas, t frame length codewords are represented by matrices or space-time blocks the matrix element x i,j C represents the signal sent by antenna i at time j Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 6

7 Diversity order of a coding scheme d = lim SNR log(p e ) log(snr), where P e is the error probability, and SNR is the signal-to-noise ratio. Maximum diversity order For spatial multiplexing: d max = n (receive diversity) For space-time coding: d max = mn (transmit and receive diversity) Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 7

8 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 8

9 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 9

10 Lattice point representation and decoding Spatial multiplexing case y = Hx + w The received vector is the translated version of a point in the lattice generated by H. Coded case Y = HX + W Equivalent lattice formulation after vectorizing matrices: y = H l Φs + w H l linear map corresponding to left multiplication by H Φ generator matrix of the code s vector of information signals Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 10

11 Decoding Optimal decoding amounts to solving the Closest Vector Problem (CVP) in the lattice generated by H: ˆx = argmin y Hx 2 x S ML solution ML decoders Sphere Decoder, Schnorr-Euchner algorithm... optimal performance but exponential complexity Suboptimal decoders zero forcing (ZF), successive interference cancellation (SIC)... polynomial complexity, but they don t attain maximal diversity Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 11

12 Channel preprocessing Example: ZF decoding y = Hx + w ˆx ZF = H 1 y = x + H 1 w if H is orthogonal, ZF decoding is optimal if H is ill-conditioned, the noise H 1 w is amplified Solution: channel preprocessing by lattice reduction improves the performance of suboptimal decoders Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 12

13 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 13

14 Lattice reduction Two matrices H and H generate the same lattice if H = HT with T unimodular. Lattice reduction: find a lattice basis formed by short and nearly orthogonal vectors the most popular lattice reduction algorithm is the LLL algorithm, thanks to its polynomial complexity Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 14

15 The LLL algorithm Gram-Schmidt Orthogonalization (GSO) h 1 h 1 for i = 2,..., m do for j = 1,..., i 1 do µ i,j hi,h j h j 2 end h i h i i 1 j=1 µ i,jh j end LLL-reduced basis H is LLL-reduced if its GSO satisfies the following properties: Size reduction: µ k,l 1 2, 1 l < k m, Lovasz condition: h k + µ k,k 1h 2 k h 2 k 1, 1 < k m Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 15

16 The LLL algorithm Compute GSO k 2 while k m do Size reduction RED(k,k-1) if Lovasz(k,k-1) is not satisfied then swap h k and h k 1 update GSO k max(k 1, 2) end else for l = k 2,..., 1 do Size reduction RED(k,l) end k k + 1 end end Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 16

17 Preprocessing using LLL reduction m transmit antennas, n receive antennas y n 1 = H n m x m 1 + w n 1 received signal channel codeword noise H red = HU LLL-reduced form Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 17

18 Preprocessing using LLL reduction m transmit antennas, n receive antennas y n 1 = H n m x m 1 + w n 1 received signal channel codeword noise H red = HU LLL-reduced form LLL-ZF decoder compute the pseudoinverse H red ) ˆx LLL ZF = U ( H red y Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 17

19 Preprocessing using LLL reduction m transmit antennas, n receive antennas y n 1 = H n m x m 1 + w n 1 received signal channel codeword noise H red = HU LLL-reduced form LLL-ZF decoder compute the pseudoinverse H red ) ˆx LLL ZF = U ( H red y LLL-SIC decoder QR decomposition of H red ỹ = Q H y = Rx + Q H w recursively compute x m = ỹm, r mm ỹi m x i = j=i+1 r ij x j, i = m 1,..., 1 r ii ˆx LLL SIC = U x Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 17

20 Performance of LLL-aided decoding Proposition [Taherzadeh, Mobasher, Khandani 2007] LLL-ZF and LLL-SIC decoders attain the maximal receive diversity n. the average complexity of LLL-aided decoding is polynomial, which makes it an attractive technique however, LLL becomes less effective for high-dimensional lattices as a consequence, the performance gap with respect to ML decoding increases greatly when the number of antennas is large Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 18

21 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 19

22 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 20

23 Augmented lattice reduction - principle new technique which combines preprocessing and detection: lattice reduction is used directly to decode MIMO decoding amounts to solving the closest vector problem (CVP) in the lattice generated by the channel matrix reduce the CVP to the SVP (shortest vector problem) use LLL reduction to solve the SVP in polynomial time Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 21

24 Embedding method to reduce the CVP to the SVP Follow Kannan s approach (1987): embed the m-dimensional lattice generated by H into a suitable (m + 1)-dimensional lattice H = H y 0 t augmented matrix v = Hx y = w = H x t t 1 Strategy: if w and t are very small, v is the shortest vector in the augmented lattice. By finding v, we recover x. Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 22

25 Using LLL reduction to find the shortest lattice vector LLL-reduce H: Hred = HŨ Problem: in general, there s no guarantee that the shortest vector belongs to the reduced basis. However, the first column of H red satisfies h red 1 2 m 2 d H v shortest vector in L is called α-unique if u L, u α v u, v linearly dependent Exponential gap technique v is 2 m 2 -unique ±v is the first column of H red Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 23

26 LLL reduction finds the shortest lattice vector a(h) min 1 i m h i d H 2 m 1 2 Lemma a(h red ) d H Let t = a(hred) 2, and suppose that w d m+1 H 2. m+1 Then v = Hx y is a 2 m 2 -unique shortest vector in L( H). t Sketch of the proof: Suppose u = Hx qy such that u 2 m 2 v, and u, v linearly independent qt u q t q 2 m 2 v t H(x qx) Hx qy + q y Hx 2 m 2 v + 2 m 2 v w t ( ) 2 m 2 w 2 + t w < d t H contradiction. Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 24

27 Augmented lattice reduction- Decoding Choose t = a(h red) 2 m+1 the augmented lattice has an exponential gap LLL-reduce H: Hred = HŨ If w d H, v = H x is the first column of H 2 m+1 red 1 find the transmitted message x on the first column of Ũ : ˆx = 1 ũ m+1,1 (ũ 1,1,..., ũ m,1 ) T Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 25

28 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 26

29 Performance analysis Diversity gain d = lim SNR log(p e ) log(snr) Proposition If t = a(h red) 2 m+1, then augmented lattice reduction achieves the maximum receive diversity n. { P e (H) P w > Sketch of the proof. d } H 2 m+1 C(log(SNR))n+1 SNR n Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 27

30 6 6 MIMO system, spatial multiplexing 10 0 MIMO 6x6, spatial multiplexing, 16 QAM SER ML MMSE GDFE+LLL+ZF MMSE GDFE+LLL SIC MMSE GDFE+ALR MMSE GDFE+ILR SNR Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 28

31 8 8 MIMO system, spatial multiplexing 10 0 MIMO 8x8, spatial multiplexing, 16 QAM SER ML MMSE GDFE+LLL ZF MMSE GDFE+LLL SIC MMSE GDFE+ALR SNR Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 29

32 Coded case: 2 2 system, Golden Code 10 0 Golden Code 16 QAM ML MMSE GDFE + ALR MMSE GDFE + LLL + SIC 10 1 FER SNR Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 30

33 Comparison with Kim and Park s Method [N. Kim, H. Park, Improved lattice reduction aided detections for MIMO systems, Vehicular Technology Conference 2006] Same form of the augmented matrix H No exponential gap technique: - the parameter t is big (t > max r ii, where H red = QR) - the solution is found on the last column of Ũ no guarantee that LLL reduction will find the right vector. In fact, one can prove that the performance is the same as LLL-SIC. Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 31

34 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 32

35 Complexity analysis (for n = m) average number of iterations of the LLL algorithm: E[K(H)] O ( n 2 log n ) [Jalden et al. 2008] each iteration requires O(n 2 ) operations, which can be reduced to O(n) for LLL-SIC [Ling, Howgrave-Graham 2007] the total complexity of LLL-SIC is bounded by O(n 3 log n) Complexity bound for augmented lattice reduction E[K( H)] O(n 3 ) the total complexity is bounded by O(n 4 ). Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 33

36 Complexity: numerical simulations LLL ZF LLL SIC (using effective LLL) ALR ALR using effective LLL Sphere Decoder Average number of flops, SNR= N Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 34

37 Outline 1 MIMO systems 2 Decoding Lattice decoding Lattice reduction-aided decoding 3 Augmented lattice reduction Method Performance Complexity 4 Open problems Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 35

38 Open problems explain why the performance of augmented lattice reduction is better than LLL-SIC find more realistic bounds on the complexity for high-dimensional space-time codes, the gap between ML decoding and augmented lattice reduction is still huge. How can we bridge this gap? Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 36

39 Thank you for listening!! Laura Luzzi Augmented Lattice Reduction Nanyang Technological University 37