Capacity Pre-log of Noncoherent SIMO Channels via Hironaka s Theorem

Size: px

Start display at page:

Download "Capacity Pre-log of Noncoherent SIMO Channels via Hironaka s Theorem"

Jemima Mason
5 years ago
Views:

1 Capacity Pre-log of Noncoherent SIMO Channels via Hironaka s Theorem Veniamin I. Morgenshtern 22. May 2012 Joint work with E. Riegler, W. Yang, G. Durisi, S. Lin, B. Sturmfels, and H. Bőlcskei

2 SISO Fading Channel TX x t h t RX y t 2/27

3 SISO Fading Channel TX x t h t RX y t y t = p snr h t x t + w t 2/27

4 SISO Fading Channel TX x t h t RX y t y t = p snr h t x t + w t h t.. channel gain (random, changes in time) w t CN(0, 1), i.i.d. across t 2/27

5 SISO Fading Channel TX x t h t RX y t y t = p snr h t x t + w t h t.. channel gain (random, changes in time) w t CN(0, 1), i.i.d. across t Coherent setting: h t is known at RX 2/27

6 SISO Fading Channel TX x t h t RX y t y t = p snr h t x t + w t h t.. channel gain (random, changes in time) w t CN(0, 1), i.i.d. across t Coherent setting: h t is known at RX Noncoherent setting: h t is not known at RX 2/27

7 SISO Correlated Block-Fading Channel TX x t independent across blocks correlated [h 1 h 2... h T ] RX y t y t = p snr h t x t + w t t =(1,...,T) 3/27

8 SISO Correlated Block-Fading Channel TX x t independent across blocks correlated [h 1 h 2... h T ] RX y t y t = p snr h t x t + w t t =(1,...,T) [h 1 h T ] T CN(0, PP H ) P 2 C T Q ; rank P = Q<T 3/27

9 SISO Correlated Block-Fading Channel TX x t independent across blocks correlated [h 1 h 2... h T ] RX y t y t = p snr h t x t + w t t =(1,...,T) [h 1 h T ] T CN(0, PP H ) P 2 C T Q ; rank P = Q<T Channel gains in whitened form: 2 3 h h 2 s = P , s q CN(0, 1), iid across q h T s Q 3/27

10 Capacity TX x t h t RX y t Fundamental problem: What is the ultimate limit on the rate of reliable communication over the channel capacity? 4/27

11 Capacity TX x t h t RX y t Fundamental problem: What is the ultimate limit on the rate of reliable communication over the channel capacity? Shannon coding theorem: " T X C(snr), (1/T )supi({x t }; {y t }); f x( ) E t=1 x t 2 # apple T 4/27

12 Capacity TX x t h t RX y t Fundamental problem: What is the ultimate limit on the rate of reliable communication over the channel capacity? Shannon coding theorem: " T X C(snr), (1/T )supi({x t }; {y t }); f x( ) E t=1 x t 2 # apple T Mutual information: I({x t }; {y t })=h({x t }) h({x t } {y t }) Di erential Entropy: h({y t }) 4/27

13 Capacity Pre-log AWGN Channel (h t =18t) [Shannon, 48]: C AWGN = log(1 + snr) 5/27

14 Capacity Pre-log AWGN Channel (h t =18t) [Shannon, 48]: C AWGN = log(1 + snr) Coherent setting with T =1[Telatar, 99]: C coh = E h log(1 + snr h 2 ) log(snr), snr!1 5/27

15 Capacity Pre-log AWGN Channel (h t =18t) [Shannon, 48]: C AWGN = log(1 + snr) Coherent setting with T =1[Telatar, 99]: C coh = E h log(1 + snr h 2 ) log(snr), snr!1 Pre-Log C(snr) = lim snr!1 log(snr) 5/27

16 The Noncoherent Penalty Noncoherent setting, T =1[Lapidoth & Mozer, 03]: C ncoh log log(snr) 6/27

17 The Noncoherent Penalty Noncoherent setting, T =1[Lapidoth & Mozer, 03]: C ncoh log log(snr) Noncoherent setting, rank P = 1[Marzetta & Hochwald, 99]: C ncoh (1 1/T ) log(snr) 6/27

18 The Noncoherent Penalty Noncoherent setting, T =1[Lapidoth & Mozer, 03]: C ncoh log log(snr) Noncoherent setting, rank P = 1[Marzetta & Hochwald, 99]: C ncoh (1 1/T ) log(snr) Noncoherent setting, rank P = Q [Liang & Veeravalli, 04]: C ncoh (1 Q/T ) log(snr) 6/27

19 The Noncoherent Penalty Noncoherent setting, T =1[Lapidoth & Mozer, 03]: C ncoh log log(snr) Noncoherent setting, rank P = 1[Marzetta & Hochwald, 99]: C ncoh (1 1/T ) log(snr) Noncoherent setting, rank P = Q [Liang & Veeravalli, 04]: C ncoh (1 Q/T ) log(snr) TX x t independent across blocks correlated [h 1 h 2... h T ] RX y t 6/27

20 The Noncoherent Penalty Noncoherent setting, T =1[Lapidoth & Mozer, 03]: C ncoh log log(snr) Noncoherent setting, rank P = 1[Marzetta & Hochwald, 99]: C ncoh (1 1/T ) log(snr) Noncoherent setting, rank P = Q [Liang & Veeravalli, 04]: C ncoh (1 Q/T ) log(snr) TX x t independent across blocks correlated [h 1 h 2... h T ] RX y t Number of unknown channel parameters per block is Q = rank P: [h 1 h 2...h T ] T = P[s 1 s 2...s Q ] T 6/27

21 Eliminating the Noncoherent Penalty by Adding Receive Antennas (Only!) 7/27

22 The Noncoherent Penalty in the SIMO Case Q Q M y mt = p snr h mt x t + w mt (m =1,...M, t=1,...,t) Total number of unknown channel parameters is MQ 8/27

23 The Noncoherent Penalty in the SIMO Case Q Q M y mt = p snr h mt x t + w mt (m =1,...M, t=1,...,t) Total number of unknown channel parameters is MQ Is the pre-log then given by (1 MQ/T )? 8/27

24 The Noncoherent Penalty in the SIMO Case Q Q M y mt = p snr h mt x t + w mt (m =1,...M, t=1,...,t) Total number of unknown channel parameters is MQ Is the pre-log then given by (1 MQ/T )? No! We can use only one receive antenna to get a pre-log of (1 Q/T ) 8/27

25 The Noncoherent Penalty in the SIMO Case Q Q M y mt = p snr h mt x t + w mt (m =1,...M, t=1,...,t) Total number of unknown channel parameters is MQ Is the pre-log then given by (1 MQ/T )? No! We can use only one receive antenna to get a pre-log of (1 Q/T ) We can actually do better! 8/27

26 Main Result [VM et. al., 2012] Under technical conditions on P, SIMO =min[1 1/T, R(1 Q/T )] 9/27

27 Main Result [VM et. al., 2012] Under technical conditions on P, SIMO =min[1 1/T, R(1 Q/T )] 1 1/T. 2(1 Q/T ) 1 Q/T T 1 T Q... R 9/27

28 Main Result [VM et. al., 2012] Under technical conditions on P, SIMO =min[1 1/T, R(1 Q/T )] 1 1/T. 2(1 Q/T ) 1 Q/T T 1 T Q... R Multiple antennas at the receiver can recover degrees of freedom 9/27

29 Implications M =2,T 2Q 1 SIMO =1 1/T! > 1 Q/T {z} SISO 10 / 27

30 Implications M =2,T 2Q 1 SIMO =1 1/T! > 1 Q/T {z} SISO M =2,T!1,Q T/2 (i.e., Q/T =1/2) SISO =1 Q/T 1/2 SIMO =1 1/T 1 10 / 27

31 Implications M =2,T 2Q 1 SIMO =1 1/T! > 1 Q/T {z} SISO M =2,T!1,Q T/2 (i.e., Q/T =1/2) SISO =1 Q/T 1/2 SIMO =1 1/T 1 The channel identification penalty vanishes if a second receive antenna is used 10 / 27

32 Linear Algebra: Guessing Pre-Log 11 / 27

33 Guessing the Pre-Log Noiseless I/O relation: ŷ mt = h mt x t 12 / 27

34 Guessing the Pre-Log Noiseless I/O relation: ŷ mt = h mt x t Rule of thumb: = number of RVs x t that can be identified uniquely from {ŷ mt } T 12 / 27

35 Coherent SISO Channel T =3 ŷ 1 = h 1 x 1 ŷ 2 = h 2 x 2 ŷ 3 = h 3 x 3 13 / 27

36 Coherent SISO Channel T =3 ŷ 1 = h 1 x 1 ŷ 2 = h 2 x 2 ŷ 3 = h 3 x 3 3 linear equations in 3 unknowns 13 / 27

37 Coherent SISO Channel T =3 ŷ 1 = h 1 x 1 ŷ 2 = h 2 x 2 ŷ 3 = h 3 x 3 3 linear equations in 3 unknowns ) unique solution 13 / 27

38 Coherent SISO Channel T =3 ŷ 1 = h 1 x 1 ŷ 2 = h 2 x 2 ŷ 3 = h 3 x 3 3 linear equations in 3 unknowns ) unique solution ) SISO = 3/3 =1[Telatar, 99] 13 / 27

39 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 14 / 27

40 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 Quadratic equations! 14 / 27

41 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 Quadratic equations! Define z i =1/x i 14 / 27

42 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 z 1 ŷ 1 = s 1 z 2 ŷ 2 = s 2 z 3 ŷ 3 =(s 1 + s 2 ) Quadratic equations! Define z i =1/x i 3 linear equations in 5 unknowns: z 1, z 2, z 3, s 1, s 2 14 / 27

43 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 z 1 ŷ 1 = s 1 z 2 ŷ 2 = s 2 z 3 ŷ 3 =(s 1 + s 2 ) Quadratic equations! Define z i =1/x i 3 linear equations in 5 unknowns: z 1, z 2, z 3, s 1, s 2 ) infinitely many solutions 14 / 27

44 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 z 1 ŷ 1 = s 1 z 2 ŷ 2 = s 2 z 3 ŷ 3 =(s 1 + s 2 ) Quadratic equations! Define z i =1/x i 3 linear equations in 5 unknowns: z 1, z 2, z 3, s 1, s 2 ) infinitely many solutions Transmit 2 pilot-symbols to eliminate ambiguity 14 / 27

45 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 1ŷ 1 = s 1 1ŷ 2 = s 2 z 3 ŷ 3 =(s 1 + s 2 ) Quadratic equations! Define z i =1/x i 3 linear equations in 5 unknowns: z 1, z 2, z 3, s 1, s 2 ) infinitely many solutions Transmit 2 pilot-symbols to eliminate ambiguity 3 linear equations in 3 unknowns: z 3, s 1, s 2 14 / 27

46 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 1ŷ 1 = s 1 1ŷ 2 = s 2 z 3 ŷ 3 =(s 1 + s 2 ) Quadratic equations! Define z i =1/x i 3 linear equations in 5 unknowns: z 1, z 2, z 3, s 1, s 2 ) infinitely many solutions Transmit 2 pilot-symbols to eliminate ambiguity 3 linear equations in 3 unknowns: z 3, s 1, s 2 ) unique solution 14 / 27

47 Noncoherent SISO Channel T =3,Q= h h 2 5 = h {z } P apple s1 s 2 ŷ 1 = s 1 x 1 ŷ 2 = s 2 x 2 ŷ 3 =(s 1 + s 2 )x 3 1ŷ 1 = s 1 1ŷ 2 = s 2 z 3 ŷ 3 =(s 1 + s 2 ) Quadratic equations! Define z i =1/x i 3 linear equations in 5 unknowns: z 1, z 2, z 3, s 1, s 2 ) infinitely many solutions Transmit 2 pilot-symbols to eliminate ambiguity 3 linear equations in 3 unknowns: z 3, s 1, s 2 ) unique solution ) SISO =(3 2)/3 =1/3 =1 Q/T [Liang & Veeravalli, 04] 14 / 27

48 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 15 / 27

49 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 Quadratic equations! 15 / 27

50 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 Quadratic equations! Define z i =1/x i 15 / 27

51 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 z 1 ŷ 11 = s 11 z 2 ŷ 12 = s 12 z 3 ŷ 13 =(s 11 + s 12 ) z 1 ŷ 21 = s 21 z 2 ŷ 22 = s 22 z 3 ŷ 23 =(s 21 + s 22 ) Quadratic equations! Define z i =1/x i 15 / 27

52 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 z 1 ŷ 11 = s 11 z 2 ŷ 12 = s 12 z 3 ŷ 13 =(s 11 + s 12 ) z 1 ŷ 21 = s 21 z 2 ŷ 22 = s 22 z 3 ŷ 23 =(s 21 + s 22 ) Quadratic equations! Define z i =1/x i 6 linear equations in 7 unknowns: z 1, z 2, z 3, s 11, s 12, s 21, s / 27

53 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 z 1 ŷ 11 = s 11 z 2 ŷ 12 = s 12 z 3 ŷ 13 =(s 11 + s 12 ) z 1 ŷ 21 = s 21 z 2 ŷ 22 = s 22 z 3 ŷ 23 =(s 21 + s 22 ) Quadratic equations! Define z i =1/x i 6 linear equations in 7 unknowns: z 1, z 2, z 3, s 11, s 12, s 21, s 22 ) infinitely many solutions 15 / 27

54 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 1ŷ 11 = s 11 z 2 ŷ 12 = s 12 z 3 ŷ 13 =(s 11 + s 12 ) 1ŷ 21 = s 21 z 2 ŷ 22 = s 22 z 3 ŷ 23 =(s 21 + s 22 ) Quadratic equations! Define z i =1/x i 6 linear equations in 7 unknowns: z 1, z 2, z 3, s 11, s 12, s 21, s 22 ) infinitely many solutions Transmit 1 pilot-symbol to eliminate ambiguity 15 / 27

55 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 1ŷ 11 = s 11 z 2 ŷ 12 = s 12 z 3 ŷ 13 =(s 11 + s 12 ) 1ŷ 21 = s 21 z 2 ŷ 22 = s 22 z 3 ŷ 23 =(s 21 + s 22 ) Quadratic equations! Define z i =1/x i 6 linear equations in 7 unknowns: z 1, z 2, z 3, s 11, s 12, s 21, s 22 ) infinitely many solutions Transmit 1 pilot-symbol to eliminate ambiguity 6 linear equations in 6 unknowns: z 2, z 3, s 11, s 12, s 21, s / 27

56 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 1ŷ 11 = s 11 z 2 ŷ 12 = s 12 z 3 ŷ 13 =(s 11 + s 12 ) 1ŷ 21 = s 21 z 2 ŷ 22 = s 22 z 3 ŷ 23 =(s 21 + s 22 ) Quadratic equations! Define z i =1/x i 6 linear equations in 7 unknowns: z 1, z 2, z 3, s 11, s 12, s 21, s 22 ) infinitely many solutions Transmit 1 pilot-symbol to eliminate ambiguity 6 linear equations in 6 unknowns: z 2, z 3, s 11, s 12, s 21, s 22 ) unique solution 15 / 27

57 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 1ŷ 11 = s 11 z 2 ŷ 12 = s 12 z 3 ŷ 13 =(s 11 + s 12 ) 1ŷ 21 = s 21 z 2 ŷ 22 = s 22 z 3 ŷ 23 =(s 21 + s 22 ) Quadratic equations! Define z i =1/x i 6 linear equations in 7 unknowns: z 1, z 2, z 3, s 11, s 12, s 21, s 22 ) infinitely many solutions Transmit 1 pilot-symbol to eliminate ambiguity 6 linear equations in 6 unknowns: z 2, z 3, s 11, s 12, s 21, s 22 ) unique solution ) SIMO =(3 1)/3 =2/3 =1 1/T 15 / 27

58 Noncoherent SIMO Channel (T =3,Q=2,M=2) ŷ 11 = s 11 x 1 ŷ 12 = s 12 x 2 ŷ 13 =(s 11 + s 12 )x 3 ŷ 21 = s 21 x 1 ŷ 22 = s 22 x 2 ŷ 23 =(s 21 + s 22 )x 3 1ŷ 11 = s 11 z 2 ŷ 12 = s 12 z 3 ŷ 13 =(s 11 + s 12 ) 1ŷ 21 = s 21 z 2 ŷ 22 = s 22 z 3 ŷ 23 =(s 21 + s 22 ) Quadratic equations! Define z i =1/x i 6 linear equations in 7 unknowns: z 1, z 2, z 3, s 11, s 12, s 21, s 22 ) infinitely many solutions Transmit 1 pilot-symbol to eliminate ambiguity 6 linear equations in 6 unknowns: z 2, z 3, s 11, s 12, s 21, s 22 ) unique solution ) SIMO =(3 1)/3 =2/3 =1 1/T > 1/3 = SISO 15 / 27

59 Equations for General P (T =3,Q=2,M=2) 2 p 11 p p 21 p ŷ 12 0 p 31 p ŷ p 11 p p 21 p 22 ŷ p 31 p 32 {z 0 ŷ 23 } B s 11 s 12 s 21 s 22 z 2 z = Solution is unique i B is full-rank ŷ ŷ / 27

60 Information Theory 17 / 27

61 Vector Notations (T =3,Q =2,M =2) y 11 y 12 y 13 y 21 y 22 y 23 {z } y s 11 x 1 = p s 12 x 2 snr (s 11 + s 12 )x s 21 x s 22 x (s 21 + s 22 )x 3 {z } ŷ w 11 w 12 w 13 w 21 w 22 w {z } w P = s =[s 11 s 12 s 21 s 22 ] T x =[x 1 x 2 x 3 ] T 18 / 27

62 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. 19 / 27

63 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) =h(y) h(y x) 19 / 27

64 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) =h(y) h(y x) y is Gaussian conditioned on x ) h(y x) 4 log(snr) 19 / 27

65 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) =h(y) h(y x) y is Gaussian conditioned on x ) h(y x) 4 log(snr) h(y) = h p snrŷ + w 19 / 27

66 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) =h(y) h(y x) y is Gaussian conditioned on x ) h(y x) 4 log(snr) h(y) = h p snrŷ + w h p snrŷ + w w 19 / 27

67 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) =h(y) h(y x) y is Gaussian conditioned on x ) h(y x) 4 log(snr) h(y) = h p snrŷ + w h p snrŷ + w w = h p snrŷ 19 / 27

68 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) =h(y) h(y x) y is Gaussian conditioned on x ) h(y x) 4 log(snr) h(y) = h p snrŷ + w h p snrŷ + w w = h p snrŷ = 6 log(snr)+h(ŷ) {z} finite? 19 / 27

69 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) = h(y) h(y x) 6 log(snr) 4 log(snr) + c y is Gaussian conditioned on x ) h(y x) 4 log(snr) h(y) = h p snrŷ + w h p snrŷ + w w = h p snrŷ = 6 log(snr)+h(ŷ) {z} finite? 19 / 27

70 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) = h(y) h(y x) 6 log(snr) 4 log(snr) + c y is Gaussian conditioned on x ) h(y x) 4 log(snr) h(y) = h p snrŷ + w h p snrŷ + w w = h p snrŷ = 6 log(snr)+h(ŷ) {z} finite? SIMO (6 4)/3 =2/3 19 / 27

71 The Lower Bound (T =3,Q=2,M=2) Transmit x i CN(0, 1), i.i.d. I(x; y) = h(y) h(y x) 6 log(snr) 4 log(snr) + c y is Gaussian conditioned on x ) h(y x) 4 log(snr) h(y) = h p snrŷ + w h p snrŷ + w w = h p snrŷ = 6 log(snr)+h(ŷ) {z} finite? SIMO (6 4)/3 =2/3 > 1/3 = SISO 19 / 27

72 Is h(ŷ) Finite? Change of Variables 2 3 s 11 x 1 s 12 x 2 ŷ = (s 11 + s 12 )x 3 6 s 21 x s 22 x 2 5 (s 21 + s 22 )x 3 20 / 27

73 Is h(ŷ) Finite? Change of Variables 2 3 s 11 x 1 s 12 x 2 ŷ = (s 11 + s 12 )x 3 6 s 21 x s 22 x 2 5 (s 21 + s 22 )x 3 h(ŷ) h(ŷ x 1 ) (pilot-symbol in the noiseless case) 20 / 27

74 Is h(ŷ) Finite? Change of Variables 2 3 s 11 x 1 s 12 x 2 ŷ = (s 11 + s 12 )x 3 6 s 21 x s 22 x 2 5 (s 21 + s 22 )x 3 h(ŷ) h(ŷ x 1 ) (pilot-symbol in the noiseless case) For fixed x 1, the function ŷ = ŷ(s,x 2,x 3 ) is a bijection 20 / 27

75 Is h(ŷ) Finite? Change of Variables 2 3 s 11 x 1 s 12 x 2 ŷ = (s 11 + s 12 )x 3 6 s 21 x s 22 x 2 5 (s 21 + s 22 )x 3 h(ŷ) h(ŷ x 1 ) (pilot-symbol in the noiseless case) For fixed x 1, the function ŷ = ŷ(s,x 2,x 3 ) is a bijection Change of Variables Lemma: h(ŷ x 1 )=h(s,x 2,x 3 x 1 )+E s,x 2,x 3 ) 20 / 27

76 Is h(ŷ) Finite? Change of Variables 2 3 s 11 x 1 s 12 x 2 ŷ = (s 11 + s 12 )x 3 6 s 21 x s 22 x 2 5 (s 21 + s 22 )x 3 h(ŷ) h(ŷ x 1 ) (pilot-symbol in the noiseless case) For fixed x 1, the function ŷ = ŷ(s,x 2,x 3 ) is a bijection Change of Variables Lemma: h(ŷ x 1 )=h(s,x 2,x 3 x 1 ) +2 E {z } s,x log 2,x 3 ) {z } finite? 20 / 27

77 Resolution of Singularities How can we show that E s,x 2,x 3 ) > 1? 21 / 27

78 Resolution of Singularities How can we show that E s,x 2,x 3 ) > 2,x 3 ) = J 1(x)J 2 (s)j 3 (x) 21 / 27

79 Resolution of Singularities How can we show that E s,x 2,x 3 ) > 2,x 3 ) = J 1(x)J 2 (s)j 3 (x) J 1 (x) and J 3 (x) are diagonal matrices 21 / 27

80 Resolution of Singularities How can we show that E s,x 2,x 3 ) > 2,x 3 ) = J 1(x)J 2 (s)j 3 (x) J 1 (x) and J 3 (x) are diagonal matrices det J 2 (s) is a homogeneous polynomial: det J 2 ( s) = D det J 2 (s), 8 2 C 21 / 27

81 Resolution of Singularities (Cont d) Polar coordinates: s! (r, ) Z exp( ksk 2 ) log det J 2 (s) ds C RQ Z apple exp( ksk 2 ) log det J 2 (s/ksk 2 ) ds + O(1) C Z RQ 1 Z apple exp( r 2 )r 2D 1 dr log f( ) d + O(1) 0 where =[0, ] 2D 2 [0, 2 ] is a compact set f is a real analytic function 22 / 27

82 Resolution of Singularities (Cont d) Hironaka s Theorem implies: If f 6 0 is a real analytic function, then Z log f( ) d < 1. R Analytic f(x) Normal crossing f(g(u)) = S u k 1 1 u k 2 k 2... u d d W R d g Proper analytic Manifold U 23 / 27

83 The Technical Condition on P: notjustrank f 6 0 i 2 3 p 11 p p 21 p p 21 0 p 31 p p p 11 p p 21 p 22 p p 31 p 32 0 p 32 is full-rank 24 / 27

84 The Technical Condition on P: notjustrank f 6 0 i 2 3 p 11 p p 21 p p 21 0 p 31 p p p 11 p p 21 p 22 p p 31 p 32 0 p 32 is full-rank For T =3, Q =2, M =2equivalent to: Every two rows of P are linearly independent 24 / 27

85 The Technical Condition on P: notjustrank f 6 0 i 2 3 p 11 p p 21 p p 21 0 p 31 p p p 11 p p 21 p 22 p p 31 p 32 0 p 32 is full-rank For T =3, Q =2, M =2equivalent to: Every two rows of P are linearly independent For example, P = satisfies this condition / 27

86 Connection to Linear Algebra 2 3 p 11 p p 21 p p 21 0 p 31 p p p 11 p is full-rank p 21 p 22 p p 31 p 32 0 p p 11 p p 21 p ŷ 12 0 i p 31 p ŷ p 11 p = B is full-rank a.e p 21 p 22 ŷ p 31 p 32 0 ŷ / 27

87 Open problems MIMO Stationary Channel Model 26 / 27

88 Thank you 27 / 27

Capacity Pre-Log of SIMO Correlated Block-Fading Channels

Capacity Pre-Log of SIMO Correlated Block-Fading Channels Wei Yang, Giuseppe Durisi, Veniamin I. Morgenshtern, Erwin Riegler 3 Chalmers University of Technology, 496 Gothenburg, Sweden ETH Zurich, 809