Sub-modularity and Antenna Selection in MIMO systems

Transcription

1 Sub-modularity and Antenna Selection in MIMO systems Rahul Vaze Harish Ganapathy

2 Point-to-Point MIMO Channel 1 1 Tx H Rx N t N r

3 Point-to-Point MIMO Channel 1 1 Tx H Rx N t N r Antenna Selection Transmit Side Receive Side

4 Point-to-Point MIMO Channel 1 1 Tx H Rx N t N r Antenna Selection Transmit Side Receive Side Metrics Mutual Information Reliability

5 Point-to-Point MIMO Channel 1 1 Tx H Rx N t N r Antenna Selection Transmit Side Receive Side Metrics Mutual Information Reliability Advantages Simplified Circuitry Fewer Tx/Rx Chains

6 Relay Selection Source F G Destination

7 Relay Selection Source F G Destination Protocols Amplify-forward Decode-forward

8 Relay Selection Source F G Destination Protocols Amplify-forward Decode-forward Metrics Mutual Information Reliability

9 Relay Selection Source F G Destination Protocols Amplify-forward Decode-forward Metrics Mutual Information Reliability Implementation Centralized Distributed

10 Prior Work

11 P2P Prior Work

12 Prior Work P2P Transmi Receiv

13 Prior Work P2P Transmi Receiv CSIT No CSIT

14 Prior Work P2P Relay Transmi Receiv CSIT No CSIT

15 Prior Work P2P Relay Transmi Receiv AF DF CSIT No CSIT

16 Prior Work P2P Relay Transmi Receiv AF DF CSIT No CSIT Distributed Centralize

17 Prior Work P2P Relay Transmi Receiv AF DF CSIT No CSIT Distributed Centralize CSIT NoCSI

18 Prior Work P2P Relay Transmi Receiv AF DF CSIT No CSIT Distributed Centralize CSIT NoCSI Lot of work assuming Genie-Aided Antenna Selection No provably good simple algorithm for Antenna Selection

19 Implementation Most analytical work assumes brute-force (exponential complexity)

20 Implementation Most analytical work assumes brute-force (exponential complexity) - If subset size is then number of L computations Prohibitive for large antennas, e.g. Massive MIMO Nt L

21 Implementation Most analytical work assumes brute-force (exponential complexity) - If subset size is then number of L computations Prohibitive for large antennas, e.g. Massive MIMO Nt L Lots of greedy/heuristic algorithms No theoretical guarantees

22 Objective for Point-to-Point MIMO Channel 1 1 Tx H Rx N t N r

23 Objective for Point-to-Point MIMO Channel 1 1 Tx H Rx N t N r Find the size L receive antenna subset that maximizes the mutual information max R L {1,2,...,N r }, R L =L log det I + PNt H RL H RL

24 Objective for Relay Selection Source f i g j Destination

25 Objective for Relay Selection Beamforming weight Source f i w i g j Destination

26 Objective for Relay Selection Source Beamforming weight f i w i g j Destination Find the size L relay antennas subset that maximizes the mutual information max max T L {1,2,...,N} w log 1+ w w w ( + I)w

27 Objective for Relay Selection Source Beamforming weight f i w i g j Destination Find the size L relay antennas subset that maximizes the mutual information max max T L {1,2,...,N} w log 1+ w w w ( + I)w = apple gt1 f t1 t 1,..., g t L f tl t L w =[w t1,...,w tl ] T, T,

28 Our Contribution

29 Our Contribution P2P Greedy Algorithm with linear complexity achieves at least (1 1/e) fraction of the optimal solution

30 Our Contribution P2P Greedy Algorithm with linear complexity achieves at least (1 1/e) fraction of the optimal solution Implication: Genie-aided analysis holds

31 Our Contribution P2P Greedy Algorithm with linear complexity achieves at least (1 1/e) fraction of the optimal solution Implication: Genie-aided analysis holds Relay Greedy Algorithm with linear complexity achieves the optimal solution

32 Some Priliminaries

33 Let f :2 U! R Some Priliminaries

34 Some Priliminaries Let f :2 U! R Then f is called monotone if f(s [{a}) f(s). S U, a 2 U

35 Some Priliminaries Let f :2 U! R Then f is called monotone if f(s [{a}) Then f is called sub-modular if f(s). S U, a 2 U

36 Some Priliminaries Let f :2 U! R Then f is called monotone if f(s [{a}) Then f is called sub-modular if f(s). S U, a 2 U f(s [{a}) f(s) f(t [{a}) f(t ), S T.

37 Some Priliminaries Let f :2 U! R Then f is called monotone if f(s [{a}) Then f is called sub-modular if f(s). S U, a 2 U f(s [{a}) f(s) f(t [{a}) f(t ), S T. Diminishing Returns Property: Value of adding an element to smaller set is more than that of the bigger set

38 Some Priliminaries Let f :2 U! R Then f is called monotone if f(s [{a}) Then f is called sub-modular if f(s). S U, a 2 U f(s [{a}) f(s) f(t [{a}) f(t ), S T. Diminishing Returns Property: Value of adding an element to smaller set is more than that of the bigger set Then f is called modular if

39 Some Priliminaries Let f :2 U! R Then f is called monotone if f(s [{a}) Then f is called sub-modular if f(s). S U, a 2 U f(s [{a}) f(s) f(t [{a}) f(t ), S T. Diminishing Returns Property: Value of adding an element to smaller set is more than that of the bigger set Then f is called modular if f(s [{a}) f(s) =f(t [{a}) f(t ), S T.

40 Some Priliminaries Let f :2 U! R Then f is called monotone if f(s [{a}) Then f is called sub-modular if f(s). S U, a 2 U f(s [{a}) f(s) f(t [{a}) f(t ), S T. Diminishing Returns Property: Value of adding an element to smaller set is more than that of the bigger set Then f is called modular if f(s [{a}) f(s) =f(t [{a}) f(t ), S T. Non-Diminishing Returns Property: Value of adding an element to smaller set is equal to the bigger set

41 Why Sub-Modular Functions? Greedy Method : At each step add an element that maximizes the incremental gain.

42 Why Sub-Modular Functions? Greedy Method : At each step add an element that maximizes the incremental gain. Theorem (Nemhauser et. al. 1978): If f is monotone and sub-modular, then the greedy method achieves at least (1 1/e) fraction of the optimal solution.

43 Why Sub-Modular Functions? Greedy Method : At each step add an element that maximizes the incremental gain. Theorem (Nemhauser et. al. 1978): If f is monotone and sub-modular, then the greedy method achieves at least (1 1/e) fraction of the optimal solution. Theorem (Rado1968, Edmonds1971): If f is monotone and modular, then the greedy method achieves the optimal solution.

44 Receive Antenna Selection is Sub-Modular 1 Tx S Rx N t N r

45 Receive Antenna Selection is Sub-Modular 1 Tx S Rx N t N r

46 Receive Antenna Selection is Sub-Modular 1 a Tx S Rx N t N r

47 Receive Antenna Selection is Sub-Modular 1 h a Tx H S S Rx N t N r

48 Receive Antenna Selection is Sub-Modular 1 h a Tx H S S Rx N t N r f(s [{a}) f(s) = log det I S +1 + P N t apple HS h [H S h ] log det I S + P H S H S, N t

49 Receive Antenna Selection is Sub-Modular 1 h a Tx H S S Rx N t N r f(s [{a}) f(s) = log det = log det I S +1 + P apple HS N t h I Nt + P N t [H S h ] [H S h ] apple HS h log det log det I S + P H S H S, N t I Nt + P H S N H S t

50 Receive Antenna Selection is Sub-Modular 1 h a Tx H S S Rx N t N r f(s [{a}) f(s) = log det = log det I S +1 + P apple HS N t h I Nt + P N t [H S h ] [H S h ] apple HS h log det log det I S + P H S H S, N t I Nt + P H S N H S t f(s [{a}) f(s) Mutual Information between A and C with Gaussian Signalling A S h H S C N t B

51 Receive Antenna Selection is Sub-Modular Tx 1 a S Rx N t

52 Receive Antenna Selection is Sub-Modular Tx 1 a S Rx S T N t T

53 Receive Antenna Selection is Sub-Modular Tx 1 a h H T N t S T Rx S T

54 Receive Antenna Selection is Sub-Modular Tx N t f(t [{a}) f(t ) is Mutual Information between X and W 1 a h H T S T Rx S T X h W N t S H S Y H T\S T \S Z

55 Receive Antenna Selection is Sub-Modular X h Tx N t f(t [{a}) f(t ) is Mutual Information between X and W 1 a W N t h H T S T Rx f(s [{a}) Mutual Information between A and C A S T h f(s) C N t Y S H S H T\S B S H S T \S Z

56 Receive Antenna Selection is Sub-Modular X Y H S h Tx H T\S N t f(t [{a}) f(t ) is Mutual Information between X and W S 1 a W N t apple h H T S T Physically Degraded Rx S T f(s [{a}) f(s) Mutual Information between A and C A B S h H S C N t Z T \S

57 Result

58 Greedy Algorithm : Result

59 Result Greedy Algorithm : Start with R L =

60 Result Greedy Algorithm : Start with R L = At step i, R L = R L {i }, i = arg Choose the best among the rest max log det I + P H RL {i}h i {1,2,...,N r },i /R L N R L {i} t,

61 Result Greedy Algorithm : Start with R L = At step i, R L = R L {i }, i = arg max log det I + P H RL {i}h i {1,2,...,N r },i /R L N R L {i} t Repeat until R L = L. Choose the best among the rest,

62 Result Greedy Algorithm : Start with R L = At step i, R L = R L {i }, i = arg max log det I + P H RL {i}h i {1,2,...,N r },i /R L N R L {i} t Repeat until R L = L. Choose the best among the rest, Theorem: Greedy Algorithm for receive antenna selection with linear complexity achieves at least (1 1/e) fraction of the optimal solution.

63 Simulation Result Performance of Greedy Receive Antenna Selection with N t =4, and L =4 Greedy antenna selection Optimal selection Spectral Efficiency (bps/hz) Number of receive antennas N r

64 Simulation Result Performance of Greedy Receive Antenna Selection with N t =4, and L =4 Greedy antenna selection Optimal selection Spectral Efficiency (bps/hz) Number of receive antennas N r Massive MIMO

65 Simulation Result Performance of Greedy Receive Antenna Selection with N t =4, and L =4 Greedy antenna selection Optimal selection Spectral Efficiency (bps/hz) Number of receive antennas N r Massive MIMO Performance Better than promised

66 Relay Selection Beamforming weight Source f i w i g j Destination

67 Relay Selection Source Beamforming weight f i w i g j Destination Find the L relay antennas subset that maximizes the mutual information max max T L {1,2,...,N} w log 1+ w w w ( + I)w

68 Relay Antenna Selection is Modular

69 Relay Antenna Selection is Modular max max T L {1,2,...,N} w log 1+ w w w ( + I)w = apple gt1 f t1 t 1,..., g t L f tl t L w =[w t1,...,w tl ] T, T, = g t1 0 0 t g tl t L

70 Relay Antenna Selection is Modular max max T L {1,2,...,N} w log 1+ w w w ( + I)w = apple gt1 f t1 t 1,..., g t L f tl t L w =[w t1,...,w tl ] T, T, max x x T Ax x T Bx = g t1 0 0 t g tl t L

71 Relay Antenna Selection is Modular max max T L {1,2,...,N} w log 1+ w w w ( + I)w = apple gt1 f t1 t 1,..., g t L f tl t L w =[w t1,...,w tl ] T, T, Rayleigh-Ritz Theorem max x x T Ax x T Bx = g t1 0 0 t g tl t L max T L {1,2,...,N} i TL g i 2 f i 2 f i 2 + g i 2 +1

72 Relay Antenna Selection is Modular max max T L {1,2,...,N} w log 1+ w w w ( + I)w = apple gt1 f t1 t 1,..., g t L f tl t L w =[w t1,...,w tl ] T, T, Rayleigh-Ritz Theorem max x x T Ax x T Bx = g t1 0 0 t g tl t L max T L {1,2,...,N} i TL g i 2 f i 2 f i 2 + g i 2 +1 Relay Selection Problem is Modular

73 Result

74 Result Theorem: Greedy Algorithm for relay antenna selection with linear complexity achieves the optimal solution.

75 Simulation Result Comparison of achievable rate with greedy and brute force relay antenna selection Spectral Efficiency (bps/hz) Greedy antenna selection Optimal selection Number of active antennas

76 Some Counter-Examples

77 Transmit Antenna Selection Tx h 1 Rx

78 Transmit Antenna Selection Tx h 1 Rx C {1} = log(1 + P h 1 2 )

79 Transmit Antenna Selection Tx h 1 h 2 Rx C {1} = log(1 + P h 1 2 )

80 Transmit Antenna Selection Tx h 1 h 2 Rx C {1} = log(1 + P h 1 2 ) C {1,2} = log 1+ P 2 ( h h 2 2 )

81 Transmit Antenna Selection Tx h 1 h 2 Rx C {1} = log(1 + P h 1 2 ) C {1,2} = log 1+ P 2 ( h h 2 2 ) Transmit Antenna Selection is NOT Monotone

82 Receive Antenna Selection with CSIT 1 1 Tx H Rx N t N r Channel

83 Receive Antenna Selection with CSIT 1 1 Tx H Rx N t N r Channel

84 Receive Antenna Selection with CSIT 1 1 Tx H Rx N t N r Channel With Watefilling Receive Antenna Selection is Monotone but NOT Sub-modular

85 Relay Antenna Selection with MIMO Source-Destination Source F G Destination

86 Relay Antenna Selection with MIMO Source-Destination Source F G Destination With MIMO S-D Relay Antenna Selection is NOT Sub-modular

87 Conclusions Theoretical Bounds on Greedy Algorithms Relay case - Greedy is Optimal Most other antenna selection problems are not sub-modular