L interférence dans les réseaux non filaires

Transcription

1 L interférence dans les réseaux non filaires Du contrôle de puissance au codage et alignement Jean-Claude Belfiore Télécom ParisTech 7 mars 2013 Séminaire Comelec

2 Parts Part 1 Part 2 Part 3 Part 4 Part 5 Interference in Wireless Networks Resource Allocation Han & Kobayashi Interference Alignment The Compute-and-Forward tool 2 / 54

3 Part I Interference in Wireless Networks

4 Introduction Outline of current Part 1 Introduction 2 In cellular systems 3 In ad hoc networks 4 / 54

5 Introduction Next Frontier for Wireless Networks Cells in cellular wireless networks are becoming smaller and smaller as the density of users per space unit is becoming higher and higher. In wireless sensor networks, the density of sensors is becoming higher and higher as well. 5 / 54

6 Introduction Next Frontier for Wireless Networks Cells in cellular wireless networks are becoming smaller and smaller as the density of users per space unit is becoming higher and higher. In wireless sensor networks, the density of sensors is becoming higher and higher as well. Unwanted signals Each node receives a combination of its own signal and many unwanted ones. Wireless networks become more and more Interference Limited. 5 / 54

7 In cellular systems Outline of current Part 1 Introduction 2 In cellular systems 3 In ad hoc networks 6 / 54

8 In cellular systems Cellular network R Base Station T T T T Figure: One Cell : Many Users 7 / 54

9 In cellular systems Cellular network R Base Station T The Relay Channel Figure: Accessing to hidden Terminals 7 / 54

10 In cellular systems Cellular network Base Station T T The Multiple Access Channel Figure: Uplink 7 / 54

11 In cellular systems Cellular network Base Station T T The Broadcast Channel Figure: Downlink 7 / 54

12 In cellular systems Cellular network Base Station 2 Other system T Base Station 1 T The Interference Channel Figure: Many cells sharing the same Physical Resources 7 / 54

13 In ad hoc networks Outline of current Part 1 Introduction 2 In cellular systems 3 In ad hoc networks 8 / 54

14 In ad hoc networks Ad Hoc and Wireless Sensor Networks Interferences Ad Hoc and Wireless sensor networks can experience a high level of interference between nodes when the number of nodes per area unit is high and the physical resource is scarce. 9 / 54

15 In ad hoc networks Ad Hoc and Wireless Sensor Networks Interferences Ad Hoc and Wireless sensor networks can experience a high level of interference between nodes when the number of nodes per area unit is high and the physical resource is scarce. Properties of the Wireless Medium Main properties are Braodcast property. Superposition property. 9 / 54

16 Part II Resource Allocation

17 Solve the problem at the RRM level Outline of current Part 4 Solve the problem at the RRM level 5 Power Control 6 Subchannel allocation 11 / 54

18 Solve the problem at the RRM level At the Physical Layer Orthogonal Multiplexing Transmit signals from different cells/users at different Subbands (FDMA) Time Slots (TDMA) Interference problem is solved by avoiding Interference. But it is not enough / 54

19 Solve the problem at the RRM level At the Physical Layer Orthogonal Multiplexing Transmit signals from different cells/users at different Subbands (FDMA) Time Slots (TDMA) Interference problem is solved by avoiding Interference. But it is not enough... Interference as noise At receivers, the sum of all interfering signals is considered as noise. Definition of new parameters as P i hii 2 SINR i = N + 2 j i P j h ji for user i, where P j is the transmit power of user j, h ji is the attenuation from transmit cell j to receive cell i and N is the power of noise. 12 / 54

20 Solve the problem at the RRM level At the Physical Layer Orthogonal Multiplexing Transmit signals from different cells/users at different Subbands (FDMA) Time Slots (TDMA) Interference problem is solved by avoiding Interference. But it is not enough... Interference as noise At receivers, the sum of all interfering signals is considered as noise. Definition of new parameters as P i hii 2 SINR i = N + 2 j i P j h ji for user i, where P j is the transmit power of user j, h ji is the attenuation from transmit cell j to receive cell i and N is the power of noise. Interference has to be mitigated at the R adio R esource M anagement level. 12 / 54

21 Power Control Outline of current Part 4 Solve the problem at the RRM level 5 Power Control 6 Subchannel allocation 13 / 54

22 Power Control An Optimization Problem (Power Minimization) Consider many interfering cells (or pairs of users) sharing the same physical resource (time or frequency), 14 / 54

23 Power Control An Optimization Problem (Power Minimization) Consider many interfering cells (or pairs of users) sharing the same physical resource (time or frequency), Fixed Rate Target Data rates R k are given = Target SINR, γ k are given. Optimization problem: min P i P i subject to SINR k γ k and P i P max 14 / 54

24 Power Control An Optimization Problem (Power Minimization) Consider many interfering cells (or pairs of users) sharing the same physical resource (time or frequency), Fixed Rate Target Data rates R k are given = Target SINR, γ k are given. Optimization problem: min P i P i subject to SINR k γ k and P i P max Problem with solution [Pischella & B., 08] This problem has solutions whenever hkk 2 k, 2 > γ k. (1) j i h jk When (1) is not satisfied for some cell, then we say that the network is interferencelimited. No more degree of freedom is available. 14 / 54

25 Power Control Rate Maximization Same assumptions as before. Now we want to maximize a function ϕ(r 1,...,R K ) of the user rates. 15 / 54

26 Power Control Rate Maximization Same assumptions as before. Now we want to maximize a function ϕ(r 1,...,R K ) of the user rates. Which function ϕ? A natural function can be the the weighted sum rate w k R k k where weights w k are proportional to users queue length. 15 / 54

27 Power Control Rate Maximization Same assumptions as before. Now we want to maximize a function ϕ(r 1,...,R K ) of the user rates. Which function ϕ? A natural function can be the the weighted sum rate w k R k k where weights w k are proportional to users queue length. Resolution [Pischella & B., 10] It is a nonconvex optimization problem that can only be solved when SINR is high enough. 15 / 54

28 Subchannel allocation Outline of current Part 4 Solve the problem at the RRM level 5 Power Control 6 Subchannel allocation 16 / 54

29 Subchannel allocation OFDMA Use OFDMA ccess for the orthogonality option. 17 / 54

30 Subchannel allocation OFDMA Use OFDMA ccess for the orthogonality option. Rate-Constrained Users Frequency allocation can be done optimizing a criterion fostering carriers with better SINR. Solution is water filling -type. 17 / 54

31 Subchannel allocation OFDMA Use OFDMA ccess for the orthogonality option. Rate-Constrained Users Frequency allocation can be done optimizing a criterion fostering carriers with better SINR. Solution is water filling -type. Rate Maximization Subcarriers are not chosen when SINR falls below some threshold. Graph coloring algorithms may also be used for subcarrier allocation. 17 / 54

32 Part III Han & Kobayashi

33 Beyond Interference as Noise Outline of current Part 7 Beyond Interference as Noise 8 Han and Kobayashi [Han & Kobayashi, 81] 9 The W curve 19 / 54

34 Beyond Interference as Noise What is possible in Interference-Limited Networks? T 1 h 11 R 1 T 2 h 22 R 2 Figure: Channel Model 20 / 54

35 Beyond Interference as Noise What is possible in Interference-Limited Networks? T 1 h 11 R 1 h 21 h 12 T 2 h 22 R 2 Figure: Channel Model 20 / 54

36 Beyond Interference as Noise What is possible in Interference-Limited Networks? T 1 h 11 R 1 h 21 h 12 Model of the 2-user Interference Channel. Additive Gaussian noise is present at each receiver. T 2 h 22 R 2 Figure: Channel Model 20 / 54

37 Beyond Interference as Noise What is possible in Interference-Limited Networks? T 1 h 11 R 1 h 21 h 12 Model of the 2-user Interference Channel. Additive Gaussian noise is present at each receiver. T 2 h 22 R 2 Figure: Channel Model Is it better to decode interference? When the level of interference becomes high enough, then it is better, for the receiver, to decode both the legitimate user and the interferers. For low level interference: still consider interference as noise. 20 / 54

38 Han and Kobayashi [Han & Kobayashi, 81] Outline of current Part 7 Beyond Interference as Noise 8 Han and Kobayashi [Han & Kobayashi, 81] 9 The W curve 21 / 54

39 Han and Kobayashi [Han & Kobayashi, 81] Han and Kobayashi Coding Scheme Transmitters Each transmitter splits the data into Private data. Common data. It transmits both flows using superposition coding. 22 / 54

40 Han and Kobayashi [Han & Kobayashi, 81] Han and Kobayashi Coding Scheme Transmitters Each transmitter splits the data into Private data. Common data. It transmits both flows using superposition coding. Receivers R i decodes all data coming from T i. It only decodes common data from T j, j i. Private data from T j are treated as noise. 22 / 54

41 Han and Kobayashi [Han & Kobayashi, 81] Han and Kobayashi Coding Scheme Transmitters Each transmitter splits the data into Private data. Common data. It transmits both flows using superposition coding. Receivers R i decodes all data coming from T i. It only decodes common data from T j, j i. Private data from T j are treated as noise. Achievable Rates By optimizing the ratio between R c (common) and R p (private), significantly higher rates are achievable. 22 / 54

42 Han and Kobayashi [Han & Kobayashi, 81] A toy example PAM Constellation 8 PAM Transmitted Constellation 1 bit private and 2 bits common Non legitimate receiver decodes Figure: Example of an 8 PAM constellation 23 / 54

43 Han and Kobayashi [Han & Kobayashi, 81] A toy example PAM Constellation 8 PAM Transmitted Constellation 1 bit private and 2 bits common Non legitimate receiver decodes Figure: Example of an 8 PAM constellation Hierarchical Modulation. 23 / 54

44 The W curve Outline of current Part 7 Beyond Interference as Noise 8 Han and Kobayashi [Han & Kobayashi, 81] 9 The W curve 24 / 54

45 The W curve Generalized Degrees of Freedom Generalized D.O.F. Define α = loginr logsnr. Use Han and Kobayashi. 25 / 54

46 The W curve Generalized Degrees of Freedom Generalized D.O.F. Define α = loginr logsnr. Use Han and Kobayashi. Then the generalized DOF are defined as C(SNR,α) D(α) = lim SNR logsnr C is the capacity of the channel. C logsnr in absence of interference. 25 / 54

47 The W curve Generalized Degrees of Freedom Generalized D.O.F. Define α = loginr logsnr. Use Han and Kobayashi. Then the generalized DOF are defined as C(SNR,α) D(α) = lim SNR logsnr C is the capacity of the channel. C logsnr in absence of interference. D(α) 1 2/3 1/2 Interference as noise Generalized Degrees of freedom (W curve) [Etkin, Tse & Wang, 08] D(α) = 1 α 0 α 1/2 priv. D(α) = α 1/2 α 2/3 priv. + com. D(α) = 1 α/2 2/3 α 1 priv. + com. D(α) = α/2 1 α 2 com. D(α) = 1 α 2 com. 1/2 2 /3 1 2 α No more degree of freedom 25 / 54

48 Part IV Interference Alignment

49 Alignment Principles Interference Alignment consists in reserving space (not linear in general) for all interfering signals. The remaining space will be used by the wanted signal and will be free of interference. 27 / 54

50 Alignment Principles Interference Alignment consists in reserving space (not linear in general) for all interfering signals. The remaining space will be used by the wanted signal and will be free of interference. Which spaces Several types of alignment have been studied among which, 1 Linear over R or C. Needs space, time or frequency diversity. 2 Linear over Q. Does not need diversity. 3 Arithmetic (coding over residue rings). Does not need diversity. 27 / 54

51 Alignment Principles Interference Alignment consists in reserving space (not linear in general) for all interfering signals. The remaining space will be used by the wanted signal and will be free of interference. Which spaces Several types of alignment have been studied among which, 1 Linear over R or C. Needs space, time or frequency diversity. 2 Linear over Q. Does not need diversity. 3 Arithmetic (coding over residue rings). Does not need diversity. Main Constraint Needs channels knowledge at all transmitting sides. 27 / 54

52 Linear Interference Alignment [Cadambe & Jafar, 09] Outline of current Part 10 Linear Interference Alignment [Cadambe & Jafar, 09] 11 Integer Interference Alignment [Jafarian & Vishwanath, 11] 12 What is really alignment? 28 / 54

53 Linear Interference Alignment [Cadambe & Jafar, 09] Principle An example Suppose we have 3 receivers for 5 unknowns, y 1 = 3x 1 + 2x 2 + 3x 3 + x 4 + 5x 5 y 2 = 2x 1 + 4x 2 + x 3 3x 4 + 5x 5 y 3 = 4x 1 + 3x 2 + 5x 3 + 2x 4 + 8x 5 where receiver 1 only wants x / 54

54 Linear Interference Alignment [Cadambe & Jafar, 09] Principle An example Suppose we have 3 receivers for 5 unknowns, y 1 = 3x 1 + 2x 2 + 3x 3 + x 4 + 5x 5 y 2 = 2x 1 + 4x 2 + x 3 3x 4 + 5x 5 y 3 = 4x 1 + 3x 2 + 5x 3 + 2x 4 + 8x 5 Remark 1 In fact, the interfering beams span a 2 D space leaving one dimension free from interference for the wanted signal, H 4 = H 3 H 2 and H 5 = H 3 + H 2. where receiver 1 only wants x / 54

55 Linear Interference Alignment [Cadambe & Jafar, 09] Principle An example Suppose we have 3 receivers for 5 unknowns, y 1 = 3x 1 + 2x 2 + 3x 3 + x 4 + 5x 5 y 2 = 2x 1 + 4x 2 + x 3 3x 4 + 5x 5 y 3 = 4x 1 + 3x 2 + 5x 3 + 2x 4 + 8x 5 Remark 1 In fact, the interfering beams span a 2 D space leaving one dimension free from interference for the wanted signal, H 4 = H 3 H 2 and H 5 = H 3 + H 2. where receiver 1 only wants x 1. Remark 2 Vector u = [ ] is orthogonal to all interfering vectors. 29 / 54

56 Linear Interference Alignment [Cadambe & Jafar, 09] Principle An example Suppose we have 3 receivers for 5 unknowns, y 1 = 3x 1 + 2x 2 + 3x 3 + x 4 + 5x 5 y 2 = 2x 1 + 4x 2 + x 3 3x 4 + 5x 5 y 3 = 4x 1 + 3x 2 + 5x 3 + 2x 4 + 8x 5 where receiver 1 only wants x 1. Remark 2 Vector u = [ ] is orthogonal to all interfering vectors. Remark 1 In fact, the interfering beams span a 2 D space leaving one dimension free from interference for the wanted signal, H 4 = H 3 H 2 and H 5 = H 3 + H 2. Recovering x 1 Projecting the received vector along u gives, 9x 1 = 17y 1 y 2 10y 3 recovering x 1 from the 3 observed values y i. 29 / 54

57 Linear Interference Alignment [Cadambe & Jafar, 09] Principle An example Suppose we have 3 receivers for 5 unknowns, y 1 = 3x 1 + 2x 2 + 3x 3 + x 4 + 5x 5 y 2 = 2x 1 + 4x 2 + x 3 3x 4 + 5x 5 y 3 = 4x 1 + 3x 2 + 5x 3 + 2x 4 + 8x 5 where receiver 1 only wants x 1. Remark 2 Vector u = [ ] is orthogonal to all interfering vectors. Remark 1 In fact, the interfering beams span a 2 D space leaving one dimension free from interference for the wanted signal, H 4 = H 3 H 2 and H 5 = H 3 + H 2. Recovering x 1 Projecting the received vector along u gives, 9x 1 = 17y 1 y 2 10y 3 recovering x 1 from the 3 observed values y i. Linear Interference alignment allows many interfering users to communicate simultaneously over a small number of signalling dimensions by putting the interfering signals in a space of small dimension so that the desired signal can be projected into the null space of the interference. 29 / 54

58 Linear Interference Alignment [Cadambe & Jafar, 09] Feasibility Feasibility of interference alignment depends on the structure and probabilistic behavior of the channel matrix. 30 / 54

59 Linear Interference Alignment [Cadambe & Jafar, 09] Feasibility Feasibility of interference alignment depends on the structure and probabilistic behavior of the channel matrix. MIMO case For instance, for the symmetric K user interference channel where all transmitters have n t antennas and all receivers, n r antennas, interference alignment is feasible with probability 1 when d n t + n r K + 1 where d is the number of flows of each pair. 30 / 54

60 Linear Interference Alignment [Cadambe & Jafar, 09] Feasibility Feasibility of interference alignment depends on the structure and probabilistic behavior of the channel matrix. MIMO case For instance, for the symmetric K user interference channel where all transmitters have n t antennas and all receivers, n r antennas, interference alignment is feasible with probability 1 when d n t + n r K + 1 where d is the number of flows of each pair. Symbol extension Beamforming across multiple channel uses to increase space dimension. 30 / 54

61 Integer Interference Alignment [Jafarian & Vishwanath, 11] Outline of current Part 10 Linear Interference Alignment [Cadambe & Jafar, 09] 11 Integer Interference Alignment [Jafarian & Vishwanath, 11] 12 What is really alignment? 31 / 54

62 Integer Interference Alignment [Jafarian & Vishwanath, 11] Alignment on Ideals This type of alignment will be studied on an example. We assume 3 pairs of transmitters/receivers. PAM constellations are used (transmission of integer symbols), i.e. x i Z. 32 / 54

63 Integer Interference Alignment [Jafarian & Vishwanath, 11] Alignment on Ideals This type of alignment will be studied on an example. We assume 3 pairs of transmitters/receivers. PAM constellations are used (transmission of integer symbols), i.e. x i Z. Decoded signals are supposed to be: y 1 = x 1 + 4x 2 + 3x 3 y 2 = 2x 1 + x 2 + 3x 3 (2) y 3 = 6x 1 + 2x 2 + x 3 32 / 54

64 Integer Interference Alignment [Jafarian & Vishwanath, 11] Alignment on Ideals This type of alignment will be studied on an example. We assume 3 pairs of transmitters/receivers. PAM constellations are used (transmission of integer symbols), i.e. x i Z. Decoded signals are supposed to be: y 1 = x 1 + 4x 2 + 3x 3 y 2 = 2x 1 + x 2 + 3x 3 (2) y 3 = 6x 1 + 2x 2 + x 3 Now, precoding is done at the transmitters such that x 1 = u 1 ; x 2 = 3u 2 ; x 3 = 2u 3 where u i carries information. 32 / 54

65 Integer Interference Alignment [Jafarian & Vishwanath, 11] Alignment on Ideals This type of alignment will be studied on an example. We assume 3 pairs of transmitters/receivers. PAM constellations are used (transmission of integer symbols), i.e. x i Z. Decoded signals are supposed to be: y 1 = x 1 + 4x 2 + 3x 3 y 2 = 2x 1 + x 2 + 3x 3 (2) y 3 = 6x 1 + 2x 2 + x 3 Now, precoding is done at the transmitters such that x 1 = u 1 ; x 2 = 3u 2 ; x 3 = 2u 3 where u i carries information. Equation (2) becomes r 1 = y 1 = u u 2 + 6u 3 r 2 = y 2 = 2u 1 + 3u 2 + 6u 3. r 3 = 1 2 y 3 = 3u 1 + 3u 2 + u 3 32 / 54

66 Integer Interference Alignment [Jafarian & Vishwanath, 11] Alignment on Ideals This type of alignment will be studied on an example. We assume 3 pairs of transmitters/receivers. PAM constellations are used (transmission of integer symbols), i.e. x i Z. Decoded signals are supposed to be: y 1 = x 1 + 4x 2 + 3x 3 y 2 = 2x 1 + x 2 + 3x 3 (2) y 3 = 6x 1 + 2x 2 + x 3 Now, precoding is done at the transmitters such that x 1 = u 1 ; x 2 = 3u 2 ; x 3 = 2u 3 where u i carries information. Equation (2) becomes r 1 = y 1 = u u 2 + 6u 3 r 2 = y 2 = 2u 1 + 3u 2 + 6u 3. r 3 = 1 2 y 3 = 3u 1 + 3u 2 + u 3 If u 1 Z 6, u 2 Z 2 and u 3 Z 3, then, u 1 = r 1 mod 6, u 2 = r 2 mod 2 and u 3 = r 3 mod / 54

67 Integer Interference Alignment [Jafarian & Vishwanath, 11] Alignment on Ideals This type of alignment will be studied on an example. We assume 3 pairs of transmitters/receivers. PAM constellations are used (transmission of integer symbols), i.e. x i Z. Decoded signals are supposed to be: y 1 = x 1 + 4x 2 + 3x 3 y 2 = 2x 1 + x 2 + 3x 3 (2) y 3 = 6x 1 + 2x 2 + x 3 Alignment Interference is aligned on ideals (6Z, 2Z and 3Z). Residue Symbols are received free of interference. Now, precoding is done at the transmitters such that x 1 = u 1 ; x 2 = 3u 2 ; x 3 = 2u 3 where u i carries information. Equation (2) becomes r 1 = y 1 = u u 2 + 6u 3 r 2 = y 2 = 2u 1 + 3u 2 + 6u 3. r 3 = 1 2 y 3 = 3u 1 + 3u 2 + u 3 If u 1 Z 6, u 2 Z 2 and u 3 Z 3, then, u 1 = r 1 mod 6, u 2 = r 2 mod 2 and u 3 = r 3 mod / 54

68 What is really alignment? Outline of current Part 10 Linear Interference Alignment [Cadambe & Jafar, 09] 11 Integer Interference Alignment [Jafarian & Vishwanath, 11] 12 What is really alignment? 33 / 54

69 What is really alignment? Framework Linear or integer Techniques of alignment all rely in fact on the same idea. Linear: Received signal (noiseless) belongs to a ring of multivariate polynomials (variables are channel gains). Interferences are put in an ideal. Integer: After approximation, channel gains are integer-valued. Received signal (noiseless) belongs to the ring of integers. Interferences are put in an ideal as well. 34 / 54

70 What is really alignment? Framework Linear or integer Techniques of alignment all rely in fact on the same idea. Linear: Received signal (noiseless) belongs to a ring of multivariate polynomials (variables are channel gains). Interferences are put in an ideal. Integer: After approximation, channel gains are integer-valued. Received signal (noiseless) belongs to the ring of integers. Interferences are put in an ideal as well. Coding Let R denote the ring of all possible (noiseless) received signal. Put all interferers in an ideal J of R and encode the desired signal in the residue ring R/J. 34 / 54

71 Part V The Compute-and-Forward tool

72 Lattices Outline of current Part 13 Lattices 14 Compute-and-Forward [Nazer & Gastpar 09] 15 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] 16 Is practical alignment feasible? 36 / 54

73 Lattices From Modulation and Coding / 54

74 Lattices From Modulation and Coding... What a lattice element could be F 2 F 2 Binary Encoder Data Modulator Labeling Lattice element? in the signal space Figure: Encoder and Modulator 37 / 54

75 Lattices From Modulation and Coding... What a lattice element could be F 2 F 2 Binary Encoder Data Modulator Labeling Lattice element? in the signal space Figure: Encoder and Modulator Requirements Encoder must be linear Modulation should be PAM, QAM or HEX Labeling (modulator) between binary codewords and modulated symbols has to respect some criteria 37 / 54

76 Lattices An example: the D 4 lattice (partition) QAM Partition à la Ungerboeck A subset B subset Figure: Labeling of subsets A and B 38 / 54

77 Lattices An example: the D 4 lattice (coding) Encoder Binary data Labeling (QAM1,QAM2) (A, A) (B, B) Binary data (uncoded) Figure: D 4 encoder 39 / 54

78 Lattices An example: the D 4 lattice (coding) Encoder Binary data Labeling (QAM1,QAM2) (A, A) (B, B) Binary data (uncoded) Figure: D 4 encoder The binary code is the (2,1) repetition code (linear) Modulation is QAM, labeling is the Ungerboeck labeling 39 / 54

79 Lattices An example: the D 4 lattice (coding) Encoder Binary data Labeling (QAM1,QAM2) (A, A) (B, B) Binary data (uncoded) Figure: D 4 encoder The binary code is the (2,1) repetition code (linear) Modulation is QAM, labeling is the Ungerboeck labeling One of the simplest examples of Construction A D 4 = (1 + ı)z[ı] 2 + (2,1) F2 39 / 54

80 Lattices Definition Lattice points An element v of Λ can be written as : v = a 1 v 1 + a 2 v a n v n, a 1,a 2,...,a n Z where (v 1,v 2,...,v n) is a basis of R n. The lattice Λ can be defined as : { } n Λ = a i v i a i Z i=1 40 / 54

81 Lattices Lattices : Generator matrix The set of vectors v 1,v 2,...,v n is a lattice basis. Definition Matrix M whose columns are vectors v 1,v 2,...,v n is a generator matrix of the lattice denoted Λ M. 41 / 54

82 Lattices Lattices : Generator matrix The set of vectors v 1,v 2,...,v n is a lattice basis. Definition Matrix M whose columns are vectors v 1,v 2,...,v n is a generator matrix of the lattice denoted Λ M. Each vector x = (x 1,x 2,...,x n) in Λ M, can be written as, x = M z where z = (z 1,z 2,...,z n) Z n. Lattice Λ M may be seen as the result of a linear transform applied to lattice Z n (cubic lattice). 41 / 54

83 Lattices Lattices : Geometric properties The generator matrix M describes the lattice Λ M, but it is not unique. All matrices M T where T has integer entries and dett = ±1 are generator matrices of Λ M. T is called a unimodular matrix. G = M M is the Gram matrix of the lattice. 42 / 54

84 Lattices Lattices : Geometric properties The generator matrix M describes the lattice Λ M, but it is not unique. All matrices M T where T has integer entries and dett = ±1 are generator matrices of Λ M. T is called a unimodular matrix. G = M M is the Gram matrix of the lattice. Definitions The fundamental parallelotope of Λ M is the region, P = { x R n x = a 1 v 1 + a 2 v a n v n, 0 a i < 1, i = 1...n } The fundamental volume is the volume of the fundamental parallelotope. It is denoted Vol ( Λ M ). The fundamental volume of the lattice is vol ( Λ M ) = det(m), which is det(g) either. 42 / 54

85 Lattices Lattices : Geometric properties The generator matrix M describes the lattice Λ M, but it is not unique. All matrices M T where T has integer entries and dett = ±1 are generator matrices of Λ M. T is called a unimodular matrix. G = M M is the Gram matrix of the lattice. Definitions The fundamental parallelotope of Λ M is the region, P = { x R n x = a 1 v 1 + a 2 v a n v n, 0 a i < 1, i = 1...n } The fundamental volume is the volume of the fundamental parallelotope. It is denoted Vol ( Λ M ). The fundamental volume of the lattice is vol ( Λ M ) = det(m), which is det(g) either. A bad basis is a basis with long vectors (large orthogonality defect). A good basis (or reduced basis) is a basis with short vectors (small orthogonality defect). 42 / 54

86 Lattices Z 2 lattice v 2 v 1 Z 2 lattice (v 1, v 2 ) Lattice Point Lattice Basis Fundamental Parallelotope Voronoi region 43 / 54

87 Lattices Z 2 lattice v 2 Properties Generator matrix is M = [ ] v 1 A QAM constellation is a finite part of Z 2. Z 2 lattice (v 1, v 2 ) Lattice Point Lattice Basis Fundamental Parallelotope Voronoi region 43 / 54

88 Lattices A 2 lattice v2 v1 The A2 lattice (v1, v2) Lattice point Lattice basis Fundamental parallelotope Voronoi region 44 / 54

89 Lattices A 2 lattice v2 v1 Properties Generator matrix is M = [ ] The A2 lattice An HEX constellation is a finite part of A 2, the hexagonal lattice. (v1, v2) Lattice point Lattice basis Fundamental parallelotope Voronoi region 44 / 54

90 Compute-and-Forward [Nazer & Gastpar 09] Outline of current Part 13 Lattices 14 Compute-and-Forward [Nazer & Gastpar 09] 15 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] 16 Is practical alignment feasible? 45 / 54

91 Compute-and-Forward [Nazer & Gastpar 09] Principles z h1x1 h j x j Relay k j =1 a j x j hkxk 46 / 54

92 Compute-and-Forward [Nazer & Gastpar 09] Principles Relay The Relay wants to reliably decode the result of computation λ = k j=1 a j x j. If x j are lattice points of some integer lattice, then λ is also a lattice point for some lattice. z h1x1 h j x j Relay k j =1 a j x j hkxk 46 / 54

93 Compute-and-Forward [Nazer & Gastpar 09] Principles Relay The Relay wants to reliably decode the result of computation λ = k j=1 a j x j. If x j are lattice points of some integer lattice, then λ is also a lattice point for some lattice. z h1x1 h j x j Relay k j =1 a j x j hkxk Received signal Received signal is y = k j=1 h j x j + z where x j Λ are lattice points, h j R and z iid Gaussian noise. Note that a j Z. 46 / 54

94 Compute-and-Forward [Nazer & Gastpar 09] Principles Relay The Relay wants to reliably decode the result of computation λ = k j=1 a j x j. If x j are lattice points of some integer lattice, then λ is also a lattice point for some lattice. z h1x1 h j x j Relay k j =1 a j x j hkxk Received signal Received signal is y = k j=1 h j x j + z where x j Λ are lattice points, h j R and z iid Gaussian noise. Note that a j Z. Goal Compute λ reliably. 46 / 54

95 Compute-and-Forward [Nazer & Gastpar 09] Computation Rate Computation Rate The computation rate defined in is ( R comp (h,a) = log 2 ( a 2 SNR h a ) 2 ) SNR h 2 and is achievable by using lattice codes for x i. 47 / 54

96 Compute-and-Forward [Nazer & Gastpar 09] Computation Rate Computation Rate The computation rate defined in is ( R comp (h,a) = log 2 ( a 2 SNR h a ) 2 ) SNR h 2 and is achievable by using lattice codes for x i. Maximization Maximization of R comp requires to choose a opt Z k ] as [Feng et al. 11], ( a opt = argmin a 0 a I SNR 1 + SNR h 2 H ) a = argmin a 0 a.q.a where H = [h i h j ]. Minimization of the symmetric form Q (Lattice Shortest Vector problem. Λ has Gram matrix Q). 47 / 54

97 Compute-and-Forward [Nazer & Gastpar 09] Computation Rate Computation Rate The computation rate defined in is ( R comp (h,a) = log 2 ( a 2 SNR h a ) 2 ) SNR h 2 and is achievable by using lattice codes for x i. Maximization Maximization of R comp requires to choose a opt Z k ] as [Feng et al. 11], ( a opt = argmin a 0 a I SNR 1 + SNR h 2 H ) a = argmin a 0 a.q.a where H = [h i h j ]. Minimization of the symmetric form Q (Lattice Shortest Vector problem. Λ has Gram matrix Q). Successive Minima Find the k successive minima of Λ. Reduce Λ. 47 / 54

98 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] Outline of current Part 13 Lattices 14 Compute-and-Forward [Nazer & Gastpar 09] 15 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] 16 Is practical alignment feasible? 48 / 54

99 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] Use Compute-and-Forward Each user transmits 2 data flows: private + common. Each receiver sees k = 2 or k = 3 data flows to decode. 49 / 54

100 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] Use Compute-and-Forward Each user transmits 2 data flows: private + common. Each receiver sees k = 2 or k = 3 data flows to decode. Find the k successive minima related to Gram matrix Q. 2 or 3 computation rates R 1, R 2 and maybe R 3 with R 1 > R 2 > R 3. R 2 is always achievable. 49 / 54

101 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] Use Compute-and-Forward Each user transmits 2 data flows: private + common. Each receiver sees k = 2 or k = 3 data flows to decode. Rates with common data Find the k successive minima related to Gram matrix Q. 2 or 3 computation rates R 1, R 2 and maybe R 3 with R 1 > R 2 > R 3. R 2 is always achievable Computation Rate for SNR = 40db First Equation Second Equation Sum Normalized Computation Rate h Figure: Sum Rate with Gauss Reduction 49 / 54

102 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] Achievable Rates Symmetric interference channel: Received Signals are y 1 = x 1 + gx 2 + z 1 and y 2 = gx 1 + x 2 + z 2 ; 50 / 54

103 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] Achievable Rates Symmetric interference channel: Received Signals are y 1 = x 1 + gx 2 + z 1 and y 2 = gx 1 + x 2 + z 2 ; Medium SNR Upper Bound And Achievable Rate On The Capacity (SNR = 20db) 4 The Upper Bound Achievable Rate 3.5 Sum Rate[Bits/Channel Use] g Figure: SNR = 20 db ; Achievable Sum-Rate 50 / 54

104 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] Achievable Rates Symmetric interference channel: Received Signals are y 1 = x 1 + gx 2 + z 1 and y 2 = gx 1 + x 2 + z 2 ; Medium SNR High SNR Upper Bound And Achievable Rate On The Capacity (SNR = 20db) Upper Bound And Achievable Rate On The Capacity (SNR = 50db) The Upper Bound Achievable Rate 9 8 The Upper Bound Achievable Rate 7 Sum Rate[Bits/Channel Use] Sum Rate[Bits/Channel Use] g g Figure: SNR = 20 db ; Achievable Sum-Rate Figure: SNR = 50 db ; Achievable Sum-Rate 50 / 54

105 Is practical alignment feasible? Outline of current Part 13 Lattices 14 Compute-and-Forward [Nazer & Gastpar 09] 15 1 interferer: (Almost) Achieving Capacity? [Nazer, Ordentlich & Erez, 12] 16 Is practical alignment feasible? 51 / 54

106 Is practical alignment feasible? From theory to practice Linear alignment Problems to overcome: Perfect knowledge of all channel gains at all transmitters. Based on Zero Forcing (becomes efficient when SNR ). No idea of its behavior at finite SNR. 52 / 54

107 Is practical alignment feasible? From theory to practice Linear alignment Problems to overcome: Perfect knowledge of all channel gains at all transmitters. Based on Zero Forcing (becomes efficient when SNR ). No idea of its behavior at finite SNR. Integer alignment (or more generally lattice alignment) Problems to overcome: Approximation of channel gains by integers generates additional noise. No design criterion for lattice codes right now. 52 / 54

108 Is practical alignment feasible? From theory to practice Linear alignment Problems to overcome: Perfect knowledge of all channel gains at all transmitters. Based on Zero Forcing (becomes efficient when SNR ). No idea of its behavior at finite SNR. Integer alignment (or more generally lattice alignment) Problems to overcome: Approximation of channel gains by integers generates additional noise. No design criterion for lattice codes right now. Fractal behavior Some values of channel gains lead to performances much worse than very close other ones. 52 / 54

109 Conclusion Open Problems On the Coding+Alignment side Find Lattice Codes adapted to the interference channel and find a practical way to align interferers. 53 / 54

110 Conclusion Open Problems On the Coding+Alignment side Find Lattice Codes adapted to the interference channel and find a practical way to align interferers. Other points Asynchronous Codes? / 54

111 Merci!!