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, / 46

2 Outline Fundamentals of system design in MIMO systems linear receivers non-linear receivers Luca Sanguinetti (IET) MIMO April, / 46

3 Multiplexing in wireless comunications Basic concept Luca Sanguinetti (IET) MIMO April, / 46

4 Multiplexing in wireless communications Many ways to obtain multiplexing Time division multiplexing multiple time slots TDM TDMA Frequency division multiplexing multiple frequency bands OFDM OFDMA Code division multiplexing multiple spreading codes CDM CDMA Space division multiplexing multiple antennas SDM SDMA Luca Sanguinetti (IET) MIMO April, / 46

5 Multiplexing in wireless communications Many ways to obtain multiplexing Time & Frequency & Code division multiplexing require expensive resources (bandwidth or time)! Luca Sanguinetti (IET) MIMO April, / 46

6 Multiplexing in wireless communications Many ways to obtain multiplexing Spatial division multiplexing no extra bandwidth or time required x= x 1 x 2 M x M H y = y 1 y 2 M y N Luca Sanguinetti (IET) MIMO April, / 46

7 Multiplexing in wireless communications Spatial multiplexing as a space-time code with rate = M Space-time codes no channel information at the transmitter maximum rate = M T 1 Symbol Information mapper bits Space-time encoder M T X = ML detector The error probability is upper bound by ( K Pr(error) k=1 λ k ) 1 K ρ M KN Luca Sanguinetti (IET) MIMO April, / 46

8 Multiplexing in wireless communications Spatial multiplexing as a space-time code with rate = M Consider the simplest spatial multiplexing scheme symbols are demultiplexed and transmitted over antennas no temporal coding M 1 Symbol Information mapper bits DEMUX X = ML detector Luca Sanguinetti (IET) MIMO April, / 46

9 Multiplexing in wireless communications Spatial multiplexing as a space-time code with rate = M Consider the simplest spatial multiplexing scheme symbols are demultiplexed and transmitted over antennas no temporal coding M 1 Symbol Information mapper bits DEMUX X = ML detector The error probability is upper bound by Pr(error) 1 [ ρ ] N λ N M Luca Sanguinetti (IET) MIMO April, / 46

10 Multiplexing in wireless communications Spatial multiplexing as a space-time code with rate = M Spatial multiplexing with no temporal coding space-time code with rate = M diversity gain N Luca Sanguinetti (IET) MIMO April, / 46

11 Spatial division multiplexing Horizontal encoding (HE) Information bits DEMUX Coding - Interleaving Coding - Interleaving 1 M 1 Receiver N 1 M Luca Sanguinetti (IET) MIMO April, / 46

12 Spatial division multiplexing Vertical encoding (VE) Information bits Coding - Interleaving DEMUX 1 1 Receiver 1 M N M Luca Sanguinetti (IET) MIMO April, / 46

13 Spatial division multiplexing Diversity gain HE: any given symbol is transmitted by a single antenna diversity gain N array gain N coding gain depends on the temporal code simple receiver design! VE: any given symbol is transmitted by all antennas full diversity gain MN! array gain N coding gain depends on the temporal code much more complexity in receiver design Various combinations are possible [Foschini, 1996] Luca Sanguinetti (IET) MIMO April, / 46

14 Signal model Assume HE, T = 1 and N M Received samples (no temporal coding) or, equivalently, y = Hx + n (1) with h i C N y = h i x i + M k=1,k i h k x k }{{} multi stream interference +n (2) The problem is now to face with multi-stream interference Luca Sanguinetti (IET) MIMO April, / 46

15 Interference mitigation How to mitigate multi-stream interference? intersymbol interference mitigation multiuser detection [Verdú, 1998] Non-linear receivers maximum-likelihood detector successive interference cancellation Linear receivers zero-forcing equalizer minimum mean square-error equalizer Luca Sanguinetti (IET) MIMO April, / 46

16 ML detection ML performs the following non-linear optimization ˆx = arg min y H x 2 (3) x which represents a constrained least-squares problem Advantages minimum error rate maximum diversity gain Disadvantages high complexity exhaustive search over Q M alternatives! Luca Sanguinetti (IET) MIMO April, / 46

17 Probability of error analysis Rayleigh fading 10-2 M = 1, N = 1 Error rate M = 1, N = ML - M = 2, N = SNR, db Luca Sanguinetti (IET) MIMO April, / 46

18 Probability of error analysis 10 0 Rayleigh fading 10-1 ML 10-2 Error rate M = 2, N = 3 M = 3, N = SNR, db Luca Sanguinetti (IET) MIMO April, / 46

19 ML detection - Sphere decoding ML can be posed as integer least-squares problem sphere decoding [Kannan, 1983] and [Fincke, 1985] Main idea searching only over lattice points Hx lying within an hypersphere of radius R around y Lattice points R Hx Received signal Luca Sanguinetti (IET) MIMO April, / 46

20 ML detection - sphere decoding How to choose R? Too large, too many points! Too small, no points! How to determine the lattice points lying inside the given sphere? Answer: Fincke and Pohst algorithm [Fincke, 1985] Advantage reduced complexity - polynomial! Disadvantage still too complex for practical applications! Luca Sanguinetti (IET) MIMO April, / 46

21 Decorellating detector (DD) DD is based on zero-forcing criterion complete elimination of multi-stream interference This is achieved as follows ˆx = ( H H H ) 1 H H y Using (1) yields ˆx = x + ( H H H ) 1 H H n Luca Sanguinetti (IET) MIMO April, / 46

22 DD Each symbol can be now decoded independently with SNR i = ρ [ M (H H H) 1] i,i (4) where ρ = E{ x i 2 }/σ 2 (σ 2 denotes noise variance) Significant reduction of complexity is achieved compared to MLD Luca Sanguinetti (IET) MIMO April, / 46

23 DD - Block diagram Decorrelator for stream #1 ˆx 1 y Decorrelator for stream #2 Decorrelator for stream #M ˆx 2 ˆx M Luca Sanguinetti (IET) MIMO April, / 46

24 DD - Analogy with CDM Consider a CDM system multiple streams are transmitted using orthogonal spreading codes c 1 s 1 Front End s M c M Luca Sanguinetti (IET) MIMO April, / 46

25 DD - Analogy with CDM Received samples (AWGN channel) M y = c i x i + n = Cx + n i=1 Decision statistic with DD ˆx = x + ( C H C ) 1 C H n Multiple symbols are decoupled exploiting code orthogonality In MIMO systems multiple symbols are separated by means of spatial signatures Luca Sanguinetti (IET) MIMO April, / 46

26 DD In the presence of ill-conditioned matrix (correlated channel) [ (H H H ) 1 ] i,i SNR i = ρ [ M (H H H) 1] 0 i,i Large degradation of the system performance! The rank of H plays a crucial rule Luca Sanguinetti (IET) MIMO April, / 46

27 DD How to deal with ill-conditioned channels? Reducing the number of multiple streams Employing antenna selection algorithms Stream #1 Stream #K Antenna selection 1 M Channel information Some channel information at the transmitter is required! Luca Sanguinetti (IET) MIMO April, / 46

28 DD SNR i are distributed as (Rayleigh fading channel) f(z) = ( ) M M ρ(n M)! e ρ z M N M ρ z u(z) Chi-square random variable with k = 2(N M + 1) Luca Sanguinetti (IET) MIMO April, / 46

29 DD SNR i are distributed as (Rayleigh fading channel) f(z) = ( ) M M ρ(n M)! e ρ z M N M ρ z u(z) Chi-square random variable with k = 2(N M + 1) Then, we get E {SNR i } = N M + 1 ρ (5) M Pr(error) ρ (N M+1) (6) Luca Sanguinetti (IET) MIMO April, / 46

30 DD From (5) and (6) it follows that Array gain = N M + 1 Diversity gain = lim ρ log (P e ) log (ρ) = N M + 1 The maximum diversity gain should be N!! The reason is that: M 1 degrees of freedom are used to remove interference N M + 1 are employed to improve system performance Luca Sanguinetti (IET) MIMO April, / 46

31 DD- Error rate analysis 10 0 Rayleigh fading, DD M = 1, N = 1 Error rate M = 1, N = M = 2, N = 2 M = 3, N = SNR, db Luca Sanguinetti (IET) MIMO April, / 46

32 DD- Error rate analysis 10 0 Rayleigh fading, DD M = 1, N = 1 Error rate 10-3 M = 1, N = M = 2, N = 3 M = 3, N = SNR, db Luca Sanguinetti (IET) MIMO April, / 46

33 DD How about capacity? { M } C = E log (1 + SNR i ) i=1 Using (4) yields M ρ C = E log 1 + [ i=1 M (H H H) 1] i,i (7) Luca Sanguinetti (IET) MIMO April, / 46

34 DD For ρ 1 we get ( ρ ) C = M log + O(1) (8) M with { } O(1) = ME χ 2 2(M N+1) (9) Spatial multiplexing gain is achieved! The term O(1) however introduces some degradations! Luca Sanguinetti (IET) MIMO April, / 46

35 DD - Capacity Analysis 70 M = N = bit/s/hz Capacity DD SNR, db Luca Sanguinetti (IET) MIMO April, / 46

36 DD & MRC DD eliminates multi-stream interference at the expense of noise enhancement MRC mitigates thermal noise with no interference is the optimal strategy Luca Sanguinetti (IET) MIMO April, / 46

37 DD & MRC DD eliminates multi-stream interference at the expense of noise enhancement MRC mitigates thermal noise with no interference is the optimal strategy M = N = C/R (8,8) Matched filter DD Luca Sanguinetti (IET) MIMO April, / 46 SNR, db

38 Minimum mean square-error detector (MMSED) Based on the minimization of the following cost function J = E { x ˆx 2} Multi-stream interference and thermal noise are jointly mitigated Using the orthogonality principle ˆx = ( H H H + M ρ I M) 1 H H y (10) Luca Sanguinetti (IET) MIMO April, / 46

39 MMSED Substituting (1) into (10) yields ˆx = x + η where η is the disturbance term with covariance matrix Then C η = σ 2 ( H H H + ρ M I M ) 1 SNR i = 1 [ ( ) ] 1 H H H + M ρ I M i,i 1 Luca Sanguinetti (IET) MIMO April, / 46

40 MMSED For ρ 0 Then, we get ˆx = ( H H H + M ) 1 ρ I M H H y ρ M HH y E {SNR i } = N M ρ For ρ 0, it coincides with MRC Luca Sanguinetti (IET) MIMO April, / 46

41 MMSED For ρ ˆx = ( H H H + M ρ I M) 1 H H y ( H H H ) 1 H H y Then, we get E {SNR i } = N M + 1 M ρ For ρ, it coincides with DD Luca Sanguinetti (IET) MIMO April, / 46

42 MMSED - Capacity Analysis M = N = C/R (8,8) Matched filter DD MMSED SNR, db Luca Sanguinetti (IET) MIMO April, / 46

43 Bell Labs Layered Space-Time (BLAST) BLAST is a multistage detector based on successive interference cancellation the single data streams are successively decoded and subtracted MMSED for stream #1 Decode stream #1 ˆx 1 y Subtract stream #1 MMSED for stream #2 Decode stream #2 ˆx 2 Subtract stream #1, 2,, M-2 MMSED for stream #M-1 Decode stream #M-1 ˆx M1 Subtract stream #1, 2,, M-1 MMSED for stream #M Decode stream #M-1 ˆx M Luca Sanguinetti (IET) MIMO April, / 46

44 BLAST Rewrite (1) into the following form with h i C M y = M 1 i=1 h i x i + n Decision statistic at the jth iteration ˆx (j) i = wz (j) i with w is the ith row of MMSE/DD matrix z (j) i i 1 = y k=1 h kˆx (j) k Luca Sanguinetti (IET) MIMO April, / 46

45 BLAST + DD Assume ρ 1: SNR i is Chi-square distributed with 2(N M + i) degrees of freedom Then, we get and E {SNR i } = N M + i M ρ log (P e (i)) lim = N M + i ρ log (ρ) Luca Sanguinetti (IET) MIMO April, / 46

46 DD For ρ 1 we get with ( ρ ) C = M log + O(1) M O(1) = M k=1 { } E χ 2 2(M N+k) Spatial multiplexing gain is achieved! The term O(1) is now better than (9)! Luca Sanguinetti (IET) MIMO April, / 46

47 BLAST Small improvement with respect to linear receivers the BER is dominated by the first stream decoded the capacity is dominated by the first stream decoded Drawbacks to address no reliability strategy error propagation phenomena Any idea to overcome the above problems? Luca Sanguinetti (IET) MIMO April, / 46

48 BLAST Small improvement with respect to linear receivers the BER is dominated by the first stream decoded the capacity is dominated by the first stream decoded Drawbacks to address no reliability strategy error propagation phenomena Any idea to overcome the above problems? Ordering decode and successively subtract according to some metric This leads to a slight increase of complexity but it reduces the probability of error it increases the capacity Luca Sanguinetti (IET) MIMO April, / 46

49 BLAST + MMSED M = N = C/R (8,8) Matched filter DD MMSED BLAST + MMSED SNR, db Luca Sanguinetti (IET) MIMO April, / 46

50 References [Fincke, 1985] Fincke, U and Pohst, M, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis Mathematics of Computation, vol 44, no 170, pp , April 1985 [Foschini, 1998] G J Foschini, M J Gans, On limits of wireless communications in a fading environment when using multiple antennas Wireless Personal Communications, vol 6, pp , Oct 1998 [Gesbert, 2003] Gesbert, D and Shafi, M and Da shan Shiu and Smith, P J and Naguib, A, From theory to practice: an overview of MIMO space-time coded wireless systems IEEE Journal on Selected Areas in Communications, vol 21, no 3, pp , April 2003 [Kannan, 1983] Kannan, R, Improved algorithms on integer programming and related lattice problems InProc of ACM Symposium on Theory of Computation, Boston, MA, pp , April 1983 [Paulraj, 2004] Paulraj, A J, Gore, D A, Nabar, R U, Bolcskei, H, An overview of MIMO communications A key to gigabit wireless In Proceedings of the IEEE, vol 92, no2, Feb 2004 [Verdú, 1998] Verdú, S, Multiuser Detection Cambridge, UK: Cambridge University Press Luca Sanguinetti (IET) MIMO April, / 46