Multiple Antennas in Wireless Communications

Transcription

1 Multiple Antennas in Wireless Communications Luca Sanguinetti Department of Information Engineering Pisa University April, 2009 Luca Sanguinetti (IET) MIMO April, / 75

2 Outline MIMO Capacity of MIMO systems channel known at the transmitter and receiver channel known at the receiver Luca Sanguinetti (IET) MIMO April, / 75

3 MIMO Basic MIMO channel model A MIMO channel is represented by a channel matrix Luca Sanguinetti (IET) MIMO April, / 75

4 MIMO An overview MIMO channels arise in many different scenarios wireline systems wireless systems Some examples Frequency-Selective Channels Multicarrier Channels Multi-Antenna Channels... Luca Sanguinetti (IET) MIMO April, / 75

5 MIMO Frequency-Selective Channels Luca Sanguinetti (IET) MIMO April, / 75

6 MIMO Multicarrier Channels - OFDM Luca Sanguinetti (IET) MIMO April, / 75

7 MIMO Multicarrier Channels - OFDM The received samples y take the form with H = y = Hx + n h(0) 0 0 h(l) h(1) h(0) h(l) h(l) h(0) h(l) h(l) h(0) Luca Sanguinetti (IET) MIMO April, / 75

8 MIMO Multi-Antenna Channels Luca Sanguinetti (IET) MIMO April, / 75

9 MIMO Multi-Antenna Channels Basic narrow-band system model y = Hx + n with H = H 1,1 H 1,2 H 1,M H 2,1 H 2,2 H 2, H N,1 H N,2 H N,M In broad-band channel matrix entries are frequency (delay) dependent Luca Sanguinetti (IET) MIMO April, / 75

10 MIMO Multi-Antenna Channels Advantages no additional bandwidth or time array gain diversity gain Luca Sanguinetti (IET) MIMO April, / 75

11 Capacity Shannon capacity - Definition Claude Shannon (April 30, 1916 February 24, 2001) Channel capacity was pioneered by Claude Shannon back in 1948 [Shannon, 1948] using a mathematical theory of communication the central and most famous success of information theory Luca Sanguinetti (IET) MIMO April, / 75

12 Capacity Shannon capacity - Definition The capacity of a channel is operationally defined as the maximum data rate at which reliable communication can be performed without any constraint on transmitter and receiver complexity Denote by C the channel capacity. Then, for any rate R < C there exist rate R channel codes with arbitrarily low error probability Luca Sanguinetti (IET) MIMO April, / 75

13 Capacity Shannon capacity - Definition Although theoretically possible to communicate at R < C code design is a very difficult problem! reasonable block length and encoding/decoding complexity Tremendous progress have been made for single-antenna Gaussian channels TURBO codes, LDPC codes,... Such solutions do not apply to multi-antenna Gaussian channels due to the spatial dimension Luca Sanguinetti (IET) MIMO April, / 75

14 Capacity of single antenna Gaussian channels Shannon capacity The capacity is mathematically defined as C = max I(x, y) f(x),e{ x 2 } P where f(x) denotes the input distribution while I(x, y) = x X y Y p(x, y) log p(x, y) p(x)p(y) is the mutual information Luca Sanguinetti (IET) MIMO April, / 75

15 Capacity of single antenna Gaussian channels Shannon capacity The capacity of a single antenna Gaussian channel is given by Luca Sanguinetti (IET) MIMO April, / 75

16 Capacity of single antenna Gaussian channels Shannon capacity The capacity of a single antenna Gaussian channel is given by C = log(1 + SNR) bit/s/hz Luca Sanguinetti (IET) MIMO April, / 75

17 Capacity of single antenna Gaussian channels Shannon capacity Observe that when SNR 0 C SNR log e the capacity increases linearly with received power When SNR 1 C log SNR the capacity increases logarithmically with received power Luca Sanguinetti (IET) MIMO April, / 75

18 Capacity of single antenna Gaussian channels Shannon capacity Luca Sanguinetti (IET) MIMO April, / 75

19 Capacity of single antenna Gaussian channels Shannon capacity Luca Sanguinetti (IET) MIMO April, / 75

20 MIMO flat fading model y = Hx + n (1) Luca Sanguinetti (IET) MIMO April, / 75

21 Operating conditions 1 H is deterministic known at the transmitter and receiver 2 H is random with a given probability distribution known at the transmitter and receiver (slow-fading) known at the receiver (fast-fading) Luca Sanguinetti (IET) MIMO April, / 75

22 Deterministic channel - Via singular value decomposition Computing the singular vector decomposition of H produces H = UΛV H U C N N and V C M M are unitary matrices Λ C N M is diagonal 1 with K non-negative elements K = rank (H) min (M, N) Define x = V H x ỹ = U H y ñ = U H n 1 We call this matrix diagonal even though it may not be square Luca Sanguinetti (IET) MIMO April, / 75

23 Deterministic channel - Via singular value decomposition Luca Sanguinetti (IET) MIMO April, / 75

24 Deterministic channel - Via singular value decomposition Then, we may write (1) as or, equivalently, ỹ = Λ x + ñ (2) ỹ k = λ k x k +ñ k k =1, 2,..., K (3) The MIMO channel has been decomposed into K independent parallel Gaussian channels. Luca Sanguinetti (IET) MIMO April, / 75

25 Deterministic channel - Via singular value decomposition Luca Sanguinetti (IET) MIMO April, / 75

26 Deterministic channel - Via singular value decomposition The capacity is thus given by C = K log (1 + P k λ k ) bit/s/hz (4) k=1 where P k is the power allocated to x k such that K P k P (5) k=1 Luca Sanguinetti (IET) MIMO April, / 75

27 Deterministic channel - Unknown at the transmitter If no channel state information is available at the transmitter and the capacity takes the form C = K k=1 P k = P K ( log 1+ Pλ ) k K (6) bit/s/hz (7) Luca Sanguinetti (IET) MIMO April, / 75

28 Analogy with OFDM Luca Sanguinetti (IET) MIMO April, / 75

29 Analogy with OFDM The received samples y take the form with H = y = Hx + n h(0) 0 0 h(l) h(1) h(0) h(l) h(l) h(0) h(l) h(l) h(0) Luca Sanguinetti (IET) MIMO April, / 75

30 Analogy with OFDM Observing that H in (8) is circulant The eigenvalue decomposition of circulant matrices is H = F H DF where F is the unitary discrete Fourier transform matrix and while D = diag {H(k); k =1, 2,..., N 1} L 1 H(k) = l=0 h(l)e j2πl(k 1)/N is the channel frequency response over the nth subcarrier. Luca Sanguinetti (IET) MIMO April, / 75

31 Analogy with OFDM Define Then x = F H x y = Fỹ n = Fñ ỹ(k) =H(k) x(k) + ñ(k) k =1, 2,..., N The original frequency selective channel has been decomposed into a set of N independent parallel channels. Luca Sanguinetti (IET) MIMO April, / 75

32 Analogy with OFDM Luca Sanguinetti (IET) MIMO April, / 75

33 Deterministic channel - Unknown at the transmitter Exercise. Take K = M = N and assume H = I K. Determine the channel capacity. Verify that the capacity tends to the limiting value log(e) as K. Luca Sanguinetti (IET) MIMO April, / 75

34 Deterministic channel - Unknown at the transmitter Exercise. Assume that the channel power is fixed and equal to K k=1 λ k = ρ. Verify that (7) is maximized when H is orthogonal. Luca Sanguinetti (IET) MIMO April, / 75

35 Deterministic channel - Known at the transmitter If channel state information is available at the transmitter, we obtain C = max { P k 0, P K k=1 P k P } K k=1 log (1+ P ) k λ k bit/s/hz (8) The solution is found to be ) P k = (µ 1λk + (9) where µ is chosen to satisfy (5) and (x) + = max(x, 0). Luca Sanguinetti (IET) MIMO April, / 75

36 Deterministic channel - Known at the transmitter Water-filling or water-pouring algorithm Initialization: 1 Set k =1 Power allocation repeat the following procedure until P k 0 k: ( 1 µ = P K k+1 1+ K k+1 i=1 ) 1 λ i 2 P i = µ 1 λ i i =1, 2,..., K k +1 3 if P K k+1 < 0 then let P K k+1 =0 4 k = k +1 Luca Sanguinetti (IET) MIMO April, / 75

37 Deterministic channel - Known at the transmitter Luca Sanguinetti (IET) MIMO April, / 75

38 Deterministic channel - Known at the transmitter Question. What is the near optimal power allocation strategy for large values of P? Luca Sanguinetti (IET) MIMO April, / 75

39 Deterministic channel - Known at the transmitter Question. What is the near optimal power allocation strategy for large values of P? P k = P K for k =1, 2,..., K. Luca Sanguinetti (IET) MIMO April, / 75

40 Deterministic channel - Known at the transmitter Question. What is the near optimal power allocation strategy for small values of P? Luca Sanguinetti (IET) MIMO April, / 75

41 Deterministic channel - Known at the transmitter Question. What is the near optimal power allocation strategy for small values of P? P k = P for k = max λ i. i Luca Sanguinetti (IET) MIMO April, / 75

42 Deterministic channel - Known at the transmitter For large values of P with C = K k=1 For small values of P ( log 1+ P ) K λ k K log P + O(1) (10) O(1) = K log k=1 ( ) λk K ( ) C Ploge max λ i i (11) Luca Sanguinetti (IET) MIMO April, / 75

43 Deterministic channel - Known at the transmitter Assume P 1 and use Jensens inequality 1 K K k=1 ( ) ( ( K )) P log K λ P 1 k log K K λ k k=1 Define the channel power as K N M λ k = tr{h H H} = H n,m 2 k=1 n=1 m=1 Luca Sanguinetti (IET) MIMO April, / 75

44 Deterministic channel - Known at the transmitter Assume P 1 and use Jensens inequality 1 K K k=1 ( ) ( ( K )) P log K λ P 1 k log K K λ k k=1 Define the channel power as K N M λ k = tr{h H H} = H n,m 2 k=1 n=1 m=1 Well-conditioned channel matrices facilitate communication at high SNR Luca Sanguinetti (IET) MIMO April, / 75

45 Capacity of SIMO channels Deterministic channel Assume M = 1 (single transmit antenna), i.e., H = h C N 1 Then single eigenmode λ 1 = h H h = h 2 1 Channel unknown at the transmitter C = log (1+P h 2) 2 Channel known at the transmitter C = log (1+P h 2) Assuming that h 2 = N we get C = log (1 + PN) Luca Sanguinetti (IET) MIMO April, / 75

46 Capacity of MISO channels Deterministic channel Assume N = 1 (single receive antenna), i.e., H = h C 1 M Then single eigenmode λ 1 = h H h = h 2 1 Channel unknown at the transmitter C = log (1+ PM ) h 2 2 Channel known at the transmitter C = log (1+P h 2) Assuming that h 2 = M we get C = log (1 + PM) Luca Sanguinetti (IET) MIMO April, / 75

47 Deterministic channel - Known at the transmitter Summarizing (assume full rank channel matrix) 1 SIMO 2 MISO 3 MIMO C log (PN) C log (PM) C K log (P ) = min(m, N) log (P ) Luca Sanguinetti (IET) MIMO April, / 75

48 Spatial-Multiplexing gain Using multiple transmit and receive antennas rate is increased by a factor min(m, N) no additional power consumption is required Such a gain is known as multiplexing gain or better spatial multiplexing gain Luca Sanguinetti (IET) MIMO April, / 75

49 Spatial-Multiplexing gain Luca Sanguinetti (IET) MIMO April, / 75

50 MIMO Tradeoff between gains on MIMO channels Array and diversity gains basic concept: coherent combination of multiple signals multiple receive antennas: they may be simultaneously achieved multiple transmit antennas: array gain requires channel knowledge diversity gain does not Array and spatial-multiplexing gains maximum array gain: maximum singular value should be used [Andersen, 2000] maximum spatial-multiplexing gain: water-filling over singular values [Telatar, 1999] Luca Sanguinetti (IET) MIMO April, / 75

51 MIMO Tradeoff between gains on MIMO channels Diversity and spatial-multiplexing gains traditionally the design has been focused on either maximum diversity gain [Tarokh, 1998] maximum spatial-multiplexing gain [Telatar, 1999] There exists a fundamental trade-off [Zheng, 2003] spatial-multiplexing gain is related to data rate diversity gain is related to error rate diversity-multiplexing tradeoff is essentially tradeoff between error rate and data rate Luca Sanguinetti (IET) MIMO April, / 75

52 Capacity of single antenna Gaussian channels Shannon capacity The capacity is mathematically defined as C = max I(x, y) (12) f(x),e{ x 2 } P where f(x) denotes the input distribution while I(x, y) = x X y Y p(x, y) log p(x, y) p(x)p(y) (13) is the mutual information Luca Sanguinetti (IET) MIMO April, / 75

53 Capacity of single antenna Gaussian channels Shannon capacity Recall that the entropy is given H(y) = y Y p(y) log p(y) while the conditional entropy takes the form H(y x) = p(x, y) log p(y x) x X y Y Luca Sanguinetti (IET) MIMO April, / 75

54 Deterministic channel - Alternative derivation The mutual information can be written as I(x, y) =H(y) H(y x) Since x and n are independent H(y x) = H(n) I(x, y) =H(y) H(n) Maximizing I(x, y) reduces to maximizing H(y) Luca Sanguinetti (IET) MIMO April, / 75

55 Deterministic channel - Alternative derivation For this purpose, we recall that Then x is zero-mean with covariance E{xx H } = Q x is independent from n y is zero-mean with covariance E{yy H } = I N + HQH H Luca Sanguinetti (IET) MIMO April, / 75

56 Deterministic channel - Alternative derivation Lemma (1) If z C n with zero mean and satisfying E{zz H } = A. Then the entropy of z satisfies H(z) log det(πea) with equality if and only if z is circularly symmetric complex Gaussian with E{zz H } = A From the above Lemma, it follows that H(y) is maximized when y is circularly symmetric complex Gaussian When does it happen? Luca Sanguinetti (IET) MIMO April, / 75

57 Deterministic channel - Alternative derivation Lemma (2) If x C M is circularly symmetric complex Gaussian then Hx is circularly symmetric complex Gaussian for any H C N M. Lemma (3) If x and n are independent circularly symmetric complex Gaussians, then y = Hx + n is circularly symmetric complex Gaussian. Then, the answer to the question When does it happen? is: when x is circularly symmetric complex Gaussian Luca Sanguinetti (IET) MIMO April, / 75

58 Deterministic channel - Alternative derivation Collecting all the above facts together, we get I(x, y) =H(y) H(n) log det ( I N + H H QH ) Using det ( I N + H H QH ) = det ( I M + HH H Q ) yields I(x, y) log det ( I M + HH H Q ) Computing the SVD, i.e., HH H = UΛΛ H U H, produces I(x, y) det ( I N + Λ H QΛ H) Luca Sanguinetti (IET) MIMO April, / 75

59 Deterministic channel - Alternative derivation We are now left with C = max det ( I N + Λ H QΛ H) Q, tr{q} P Using det (A) K [A] k,k we eventually obtain k=1 C = max { P k 0, P K k=1 P k P } log (1+ P ) k λ k (14) with P k =[Q] k,k The result in (14) is equivalent to (4) obtained via SVD! Luca Sanguinetti (IET) MIMO April, / 75

60 Random channel Assume that H is a random matrix varying independently at each channel use independent of x and n Assume H 1 known at the transmitter and receiver (slow-fading) 2 known at the receiver (fast-fading) Luca Sanguinetti (IET) MIMO April, / 75

61 Random channel - Preliminaries 10% Outage capacity Ergodic capacity Luca Sanguinetti (IET) MIMO April, / 75

62 Random channel - Known at the transmitter When channel information is available at the transmitter C =E H { ( max log det I + HQ H H)} Q 0,tr(Q) P The solution is found to be water-filling for each H Note that the channel capacity in (??) is valid for any fading distribution Luca Sanguinetti (IET) MIMO April, / 75

63 Random channel - Unknown at the transmitter When no channel information is available at the transmitter with C = max I(x; y, H) (15) f(x) and I(x; y, H) =I(x; H) +I(x; y H) =I(x; y H) (16) I(x; y H) = E H { I(x; y H = H) } (17) Luca Sanguinetti (IET) MIMO April, / 75

64 Random channel Substituting (16) and (17) into (15) produces C = max f(x) E H { I(x; y H = H) } (18) Theorem (1) The capacity of the channel is achieved when x is circularly symmetric complex Gaussian with zero-mean and covariance Q = P M I M. The capacity is then given by { log det (I N + PM )} H H H (19) C =E H Luca Sanguinetti (IET) MIMO April, / 75

65 Rayleigh fading model The direct computation of (19) is reported in [Telatar, 1999] while further results are illustrated in [Shin, 2003] Luca Sanguinetti (IET) MIMO April, / 75

66 Rayleigh fading model The direct computation of (19) is reported in [Telatar, 1999] while further results are illustrated in [Shin, 2003] It is found that m 1 m! C = log(e) (n 1)! [( n 1 m 1 l m l=0 µ=0 )( l+µ+n m p=0 n m 1 µ ( 1) l+µ (l+µ+n m)! l!µ! e M/P F p+1 (M/P ) ) ( n 1 m 2 l )( n m µ )] where m = min(m, N) and n = max(m, N) while F i (x) = 1 e xy y i dy Luca Sanguinetti (IET) MIMO April, / 75

67 Simulation results Luca Sanguinetti (IET) MIMO April, / 75

68 Rayleigh fading model Consider some special cases for gaining intuition increase SNR increase transmit antennas increase receive antennas Luca Sanguinetti (IET) MIMO April, / 75

69 Scaling laws If N and M are fixed and P increases C min (M, N) log (P )+O(1) (20) The ergodic capacity has a multiplexing gain of min (M, N) Each 3 db of SNR leads to an increase of min (M, N) bit/s/hz in spectral efficiency This is achieved without channel knowledge at the transmitter Luca Sanguinetti (IET) MIMO April, / 75

70 Scaling laws Luca Sanguinetti (IET) MIMO April, / 75

71 Scaling laws If N and P are fixed and M C = N log (1 + P ) The capacity is bounded in M and converges to the above result This is due to the fact the same amount of power P is divided between more and more antennas Luca Sanguinetti (IET) MIMO April, / 75

72 Scaling laws If M and P are fixed and N C = M log (1+ PM ) N (21) The capacity increases approximately as log (N) Adding more receive antennas increases the amount of power adding transmit antennas does not Luca Sanguinetti (IET) MIMO April, / 75

73 Scaling laws If M = N and P is fixed C K min (M, N) (22) where K is a constant depending on the ratio of M and N The capacity grows linearly with increases min (M, N) Luca Sanguinetti (IET) MIMO April, / 75

74 Scaling laws Luca Sanguinetti (IET) MIMO April, / 75

75 Random channel - Spatially correlated Assume that H is random with correlated entries H = T 1/2 WR 1/2 In such a case the capacity is known only for some special cases. A lower bound can be computed for x N (0, P/MI M ) C E W { ( log det I N + P M )} WT WR (23) Luca Sanguinetti (IET) MIMO April, / 75

76 Random channel - Spatially correlated Assume M = N and P C K log P M + log det (TR) (24) Since we have that log det (TR) 0 M spatial correlation degrades system performance the linear growth with respect to K is preserved The above result can be extended to the case M N. Luca Sanguinetti (IET) MIMO April, / 75

77 Random channel - Spatially correlated Loss due to the correlation Luca Sanguinetti (IET) MIMO April, / 75

78 Random channel - Rice model Assume that H is random with a line-of-sight component Assume for example H = r 1+r H + H = H A = H = H B = 1 1+r W [ [ ] ] Luca Sanguinetti (IET) MIMO April, / 75

79 Random channel - Rice model H A H B Luca Sanguinetti (IET) MIMO April, / 75

80 References [Andersen, 2000] Andersen, J. B., Array gain and capacity for known random channels with multiple element arrays at both ends. IEEE Journal on Selected Areas in Communications, vol. 18, no. 11, pp , Nov [Cover, 1991] Cover, T. M., and Thomas, J. A., Elements of Information Theory, Wiley, New York, [Shannon, 1948] Shannon, C. E., A mathematical theory of communications. The Bell System Technical Journal, vol. 27, pp , July Oct [Shin, 2003] Shin, H. and Lee, J., Capacity of multiple-antenna fading channels: spatial fading, correlation, double scattering and keyhole. IEEE Transactions on Information Theory, vol. 49, no. 10, pp , Oct [Telatar, 1999] Telatar, I. E.,Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications, vol. 10, no. 6, pp , Nov [Zheng, 2003] Zheng, L. and Tse, D. N. C.,Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels. IEEE Transactions on Information Theory, vol. 49, no. 5, pp , May Luca Sanguinetti (IET) MIMO April, / 75