Demixing Radio Waves in MIMO Spatial Multiplexing: Geometry-based Receivers Francisco A. T. B. N. Monteiro

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2 Demixing Radio Waves in MIMO Spatial Multiplexing: Geometry-based Receivers 1- Spatial multiplexing: the problem - Lattice geometry 3- Detection: linear, SIC, LR and SD 4- Novel enhanced SIC 5- Mapping onto lattices with trellises /56

3 Radio waves as symbols [Picture from the Institute of Sound and Vibration, University of Southampton, UK] 3 Demixing Radio Waves in MIMO Spatial Multiplexing 3/56

4 D orthogonal lattices in SISO 30 db 8 16-QAM SNR=30dB 8 64-QAM SNR=30dB Demixing Radio Waves in MIMO Spatial Multiplexing 4/56

5 D orthogonal lattices in SISO 0 db 8 16-QAM SNR=0dB 8 64-QAM SNR=0dB Demixing Radio Waves in MIMO Spatial Multiplexing 5/56

6 Multiple-input multiple-output 6 Demixing Radio Waves in MIMO Spatial Multiplexing 6/56

New doors open beyond Shannon s capacity 1993-1998: Paulraj et al.

Gerard Foschini deduced the theoretical capacity for MIMO and proves it

Gerard Foschini (Bell Labs, NJ) Multipath can be beneficial by opening

7 New doors open beyond Shannon s capacity : Paulraj et al. research on MIMO (Stanford, CA). Gerard Foschini deduced the theoretical capacity for MIMO and proves it experimentally (Bell Labs, NJ). Gerard Foschini (Bell Labs, NJ) Multipath can be beneficial by opening simultaneous channels. Multiple-input multiple-output (MIMO) is born. After 10 years of academic research standards: 80.11n and 4G (LTE-A, WiMax). 7 Demixing Radio Waves in MIMO Spatial Multiplexing 7/56

8 MIMO (lattice) detection problem ( ) xn T y N R y h h h x n NT y h h h 1 N x n T = + y h h h N x n N 1 N N N N N y = Hx + n 1 1 R R R R T T R xˆ ML { } y x = min H x 8 Demixing Radio Waves in MIMO Spatial Multiplexing 8/56

9 Lattices Lattices are discrete subgroups in a Euclidean space n n Λ = y R : y = hx = H x, x i i i i= 1 Z H h h 11 1n = = h a n1 nn h, h, h 1 n Dimension 9 Demixing Radio Waves in MIMO Spatial Multiplexing 9/56

10 MIMO Transmission (coded or uncoded) x 1 x x3 x N T 10 Demixing Radio Waves in MIMO Spatial Multiplexing 10/56

11 MIMO channel is a linear transformation Example: the hexagonal lattice (A ) H x 1 x x3 x N T dimension dimension 1 11 Demixing Radio Waves in MIMO Spatial Multiplexing 11/56

12 MIMO channel is a linear transformation x 1 y x dimension dimension y 3 3 x y dimension -4-4 dimension 1 dimension -4-4 dimension 1 y y y Example: 3 dimensions (3 antennas using PAM) N 1 1 T = + 1 N NR h h h h h h h h h N 1 N N N R R R T xˆ T ML = x x x NT min x n n n NR { } y Hx 1 Demixing Radio Waves in MIMO Spatial Multiplexing 1/56

13 CVP in lattices in 8 dimensions (or more) Closest vector problem (CVP)in a lattice (NP-hard problem) 4 Closest point to the red one? Dimension Dimension Dimension 1 13 Demixing Radio Waves in MIMO Spatial Multiplexing 13/56

14 Demixing Radio Waves in MIMO Spatial Multiplexing: Geometry-based Receivers 1- Spatial multiplexing: the problem - Lattice geometry 3- Detection: linear, SIC, LR and SD 4- Novel enhanced SIC 5- Mapping onto lattices with trellises 14 14/56

15 Every lattice has a dual Primal lattice Dual lattice { y : y H x, x } i Λ = = R Z { n Λ = z Z : z y Z, y Λ} D ( ) Vol Λ = det( H) -The n-dimensional primal lattice can be split into parallel layers. -Infinite number way of defining these layers. D { : ( 1 T ), 1 } n z R z H xx Z Λ = = Vol ( ) Λ = D 1 Vol ( Λ) Vectors in the dual lattice define parallel planes in the primal lattice. 15 Demixing Radio Waves in MIMO Spatial Multiplexing 15/56

16 Vectors in the Dual Lattice define hyperplanes (/3) Dimension = λ (,1) ( D) min Dimension 1 Dimension ( D) ( D) h Dimension 1 h (a) Selection of (,1) in the dual lattice. 16 Demixing Radio Waves in MIMO Spatial Multiplexing 16/56

17 Vectors in the Dual Lattice define hyperplanes (3/3) Dimension Dimension 1 1 ( 1,4) Dimension ( 1,4) ( D) ( D) h Dimension 1 h (b) Selection of ( 1,4) in the dual lattice. 17 Demixing Radio Waves in MIMO Spatial Multiplexing 17/56

18 Demixing Radio Waves in MIMO Spatial Multiplexing: Geometry-based Receivers 1- Spatial multiplexing: the problem - Lattice geometry 3- Detection: linear, SIC, LR and SD 4- Novel enhanced SIC 5- Mapping onto lattices with trellises 18 18/56

19 The problem with inversion receiver (ZF) y W = H ZF 1 + = ( ) 1 H (or pseudo-inverse H H H H H ˆx dimension 0 dimension dimension Z dimension 1 Z 19 Demixing Radio Waves in MIMO Spatial Multiplexing 19/56

20 Linear receivers y=hx+n Zero Forcing (ZF) 1 slice( ) ˆx= H y y W = H ZF (or pseudo-inverse 1 ( H ) + 1 H H = H H H ˆx Minimum Mean Squared Error (MMSE) E Wy x y 1 W = H H H 1 H + I H Slicing ˆx MMSE SNR N T 0 Demixing Radio Waves in MIMO Spatial Multiplexing 0/56

21 The geometry of successive interference cancelation H= QR R r r r = 0 r r r 33 1 Demixing Radio Waves in MIMO Spatial Multiplexing 1/56

22 V-BLAST: vertical Bell Labs space-time [1999] or SIC: successive interference cancelation or Babai s algorithm or the nearest plane algorithm [1986] j span( H ) Proj ( ) h H j j h j span( H j ) ( n 1)-dimensional h j h j+1 Demixing Radio Waves in MIMO Spatial Multiplexing /56

23 Sphere decoding y Hx ξ y H = QR ˆx [Figure by Dr. Wai Ho Mow, Univ. of hong Kong] m m y rxj ( ) ξ ij i i= 1 j= i 3 Demixing Radio Waves in MIMO Spatial Multiplexing 3/56

24 Tree decoding or... sphere decoding 4 Demixing Radio Waves in MIMO Spatial Multiplexing 4/56

25 Most used techniques Linear Zero Forcing (ZF) Minimum Mean Squared Error (MMSE) V-BLAST (OSIC with ZF criterion) Non- Linear V-BLAST (OSIC with MMSE criterion) Lattice Reduction Aided (with ZF criterion) Lattice Reduction Aided (with MMSE criterion) Sphere decoder (with different enumerations) Maximum Likelihood (ML) 5 Demixing Radio Waves in MIMO Spatial Multiplexing 5/56

26 Demixing Radio Waves in MIMO Spatial Multiplexing: Geometry-based Receivers 1- Spatial multiplexing: the problem - Lattice geometry 3- Detection: linear, SIC, LR and SD 4- Novel enhanced SIC 5- Mapping onto lattices with trellises 6 6/56

27 Dual-Lattice-Aided detection: 1 st step: Find Successive Minima in the Dual Lattice 10 Dimension h ( D) ( D) h1 1 st :Use the L densest families of hyperplanes; Dimension 1 7 Demixing Radio Waves in MIMO Spatial Multiplexing 7/56

28 Dual-Lattice-Aided detection: nd step: project onto densest hyperplanes Dimension st :Use the L densest families of hyperplanes; nd : Project onto those hyperplanes Dimension 1 8 Demixing Radio Waves in MIMO Spatial Multiplexing 8/56

29 LDA Results 10 0 x 64-QAM SER ZF MMSE OSIC-ZF LLL-ZF LLL-OSIC-ZF DLA (L=16, ν max =) ML (SD) SNR [db] 9 Demixing Radio Waves in MIMO Spatial Multiplexing 9/56

30 LDA Results x3 64-QAM 10-1 SER ZF MMSE OSIC-ZF LLL-ZF LLL-OSIC-ZF DLA (L=4, ν max =1) DLA (L=4, ν max =) DLA (L=4, ν max =3) DLA (L=4, ν max =4) 4dB ML (SD) SNR [db] 30 Demixing Radio Waves in MIMO Spatial Multiplexing 30/56

31 Results x4 64-QAM SER ZF MMSE OSIC-ZF LLL-ZF LLL-OSIC-ZF DLA (L=3, ν max =) DLA (L=3, ν max =4) ML (SD) 10dB SNR [db] 31 Demixing Radio Waves in MIMO Spatial Multiplexing 31/56

32 Dual-lattice aided detection The dual lattice has a clear geometrical interpretation; Pre-processing: sphere-decoder to find L successive minima (i.e., L densest hyperplanes); Only linear O(n 3 ) operations per received vector (projections); Dual-Lattice-Aided outperforms V-BLAST (SIC). 3 Demixing Radio Waves in MIMO Spatial Multiplexing 3/56

33 The closest vector problem in multi-dimensional wireless channels 1- Spatial multiplexing: the problem - Lattice geometry 3- Detection: linear, SIC, LR and SD 4- Novel enhanced SIC 5- Mapping onto lattices with trellises 33 33/56

34 The notion of coverage Improves coverage of Voronoi cell by increasing the inradius of the decision region Zero-forcing SIC ZF with Lattice Reduction (Figure co-authored with Dr. Karen Su, University of Cambridge ) 34 Demixing Radio Waves in MIMO Spatial Multiplexing 34/56

35 Equivalent basis: reduced and not reduced H eq = Q H M Dimension Low orthogonally defect (i.e., almost orthogonal vectors) Short vectors 35 Demixing Radio Waves in MIMO Spatial Multiplexing 35/56

36 Linear receivers: two blocks Zero Forcing (ZF) y=hx+n 1 slice( ) ˆx= H y y W = H ZF (or pseudo-inverse 1 ( H ) + 1 H H = H H H ˆx Minimum Mean Squared Error (MMSE) E Wy x y 1 W = H H H 1 H + I H Slicing ˆx MMSE SNR N T 36 Demixing Radio Waves in MIMO Spatial Multiplexing 36/56

37 The same two blocks in both Successive interference cancellation and sphere decoding y H = QR ˆx 37 Demixing Radio Waves in MIMO Spatial Multiplexing 37/56

38 A more general interpretation Λ Focusing Λ F Specific detection algorithm ˆx ZF is a constant focusing onto the integer lattice P 1 FZF N E =Z FZF P M Focusing onto one of the easy lattices F P 1 P F M Interested in easy lattices that have a trellis representation 38 Demixing Radio Waves in MIMO Spatial Multiplexing 38/56

39 Step 1- Create a synthetic lattice nearby Step - linear focusing n Z Synthetic lattices in L R Received random lattice Λ L R ˆx 39 Demixing Radio Waves in MIMO Spatial Multiplexing 39/56

40 Step 1- Create a synthetic lattice nearby Step - linear focusing n Z Synthetic lattices in L R Received random lattice Λ L R ˆx 40 Demixing Radio Waves in MIMO Spatial Multiplexing 40/56

41 Step 1- Create a synthetic lattice nearby Step - linear focusing n Z Synthetic lattices in L R Received random lattice Λ L R ˆx 41 Demixing Radio Waves in MIMO Spatial Multiplexing 41/56

42 Step 1- Create a synthetic lattice nearby Step - linear focusing n Z Synthetic lattices in L R Received random lattice Λ L R ˆx 4 Demixing Radio Waves in MIMO Spatial Multiplexing 4/56

43 Lattices diverge... but with a good match close to origin! 4 3 dimension 1 0 D Slice In a 8D random lattice #cosets=15, dimension 1 43 Demixing Radio Waves in MIMO Spatial Multiplexing

44 Focusing and detection in the L R set Near optimum detection; With full diversity; Can outperform LLL Lattice Reduction Aided Small trellises up to 4 4 antennas with 64-QAM. 44 Demixing Radio Waves in MIMO Spatial Multiplexing 44/56

45 D lattices in L R 10 8 Z Words A Words B 6 rz 1 rz dimension rz+ c 1 1 rz+ c dimension 1 Checkerboard lattice 0 1 H = 1 r r 1 = = c c 11 1 = = Demixing Radio Waves in MIMO Spatial Multiplexing 45/56

46 n-d lattices in L R [ In D: Hexagonal lattice (A) ] In 4D: Schläfli lattice (D 4 ) In 8D: Gosset lattice (E 8 ) Z / Z Z / 6Z Z / 6Z 6 3 Z / Z Z / Z Z / Z Z / Z Z / Z Z / Z Z / Z Z / Z Z / Z 46 Demixing Radio Waves in MIMO Spatial Multiplexing 46/56

47 Lattices in L R Λ Λ=Λ + R Λ R Primal Lattice Λ = rz rz R 1 n dimension 0 Λ ΛR are the cosets representatives induced by the quotient group numb. of cosets= Λ = Λ R Vol Vol ( ΛR ) ( Λ ɶ ) ΛΛR dimension H = Z Demixing Radio Waves in MIMO Spatial Multiplexing 47/56

48 Number of cosets n=4 real dimensions ( configurations), limiting to 100 cosets pdf Number of cosets * 48 Demixing Radio Waves in MIMO Spatial Multiplexing 48/56

49 Number of cosets n=6 real dimensions (3 3 configurations), limiting to 00 cosets pdf Number of cosets * 49 Demixing Radio Waves in MIMO Spatial Multiplexing 49/56

50 Number of cosets n=8 real dimensions (4 4 configurations), limiting to 500 cosets pdf * Number of cosets 50 Demixing Radio Waves in MIMO Spatial Multiplexing 50/56

51 Number of cosets n=1 real dimensions (6 6 configurations), limiting to 10,000 cosets 8 x pdf Number of cosets 51 Demixing Radio Waves in MIMO Spatial Multiplexing 51/56

52 Performance 10 0 x 64-QAM SER 10-3 ZF. MMSE OSIC-ZF 10-4 LLL-ZF LLL-OSIC-ZF Focusing (E[Φ]=46) ML (SD) SNR [db] 5 Demixing Radio Waves in MIMO Spatial Multiplexing 5/56

53 Performance x3 64-QAM SER ZF. MMSE OSIC-ZF LLL-ZF LLL-OSIC-ZF Focusing (E[Φ]=106) ML (SD) SNR [db] 53 Demixing Radio Waves in MIMO Spatial Multiplexing 53/56

54 Performance x4 64-QAM SER ZF MMSE OSIC-ZF LLL-ZF LLL-OSIC-ZF Focusing (E[Φ]=38) Focusing (E[Φ]=506) ML (SD) SNR [db] 54 Demixing Radio Waves in MIMO Spatial Multiplexing 54/56

55 Message to take home 55 Demixing Radio Waves in MIMO Spatial Multiplexing 55/56

56 Lattice are everywhere* Current growing/hot topics: Physical Layer Network Coding [Gastpar, Nazer, Proc. of the IEEE, 011]; Lattice-based cryptography for physical layer security. R. Zamir, ``Lattices are Everywhere'', talk at the Information Theory and Applications Workshop (ITA09), University of California at San Diego, February U. Erez, S. Litsyn and R. Zamir, "Lattices which are good for (almost) everything, IEEE Transactions on Information Theory, pp Oct Demixing Radio Waves in MIMO Spatial Multiplexing 56/56

57 Lattices in Cryptography Demixing Radio Waves in MIMO Spatial Multiplexing 57/56

58 State of the art in 01 LTE-Advanced: 15 b/s/hz requiring 8x8 antennas: its efficient detection is still an open problem. [IEEE Comms Mag Feb. 01] 58 Demixing Radio Waves in MIMO Spatial Multiplexing 58/56

59 Work done in these two groups Prof. Ian J. Wassell Prof. Frank R. Kschischang, IEEE Fellow Department of Engineering & The Computer Laboratory 59 Demixing Radio Waves in MIMO Spatial Multiplexing 59/56

60 THANK YOU 60 Demixing Radio Waves in MIMO Spatial Multiplexing

Improved Multiple Feedback Successive Interference Cancellation Algorithm for Near-Optimal MIMO Detection

Improved Multiple Feedback Successive Interference Cancellation Algorithm for Near-Optimal MIMO Detection Manish Mandloi, Mohammed Azahar Hussain and Vimal Bhatia Discipline of Electrical Engineering,