Multiple Antennas in Wireless Communications

Multiple Antennas in Wireless Communications Luca Sanguinetti Department of Information Engineering Pisa University luca.sanguinetti@iet.unipi.it April, 2009 Luca Sanguinetti (IET) MIMO April, 2009 1 / 93

Diversity in wireless communications Many ways to obtain diversity 1 Time diversity multiple time slots (coherence time) not effective over slow fading channels techniques: channel coding plus interleaving 2 Frequency diversity multiple frequency bands (coherence bandwidth) not effective over flat fading channels techniques: OFDM, RAKE, equalization 3 Spatial diversity multiple antennas (coherence distance) techniques:... Luca Sanguinetti (IET) MIMO April, 2009 2 / 93

Diversity in wireless communications Time diversity Basic concept Luca Sanguinetti (IET) MIMO April, 2009 3 / 93

Diversity in wireless communications Frequency diversity Basic concept Luca Sanguinetti (IET) MIMO April, 2009 4 / 93

Diversity in wireless communications Space diversity - Receive diversity Basic concept Techniques selection diversity MRC Luca Sanguinetti (IET) MIMO April, 2009 5 / 93

Diversity in wireless communications Many ways to obtain diversity A real channel exhibits time diversity: T frequency diversity: F space diversity: S The available diversity is then given by T F S Equality is reached only when there is full independence of fading across space, time and frequency [Sayed, 2002] Luca Sanguinetti (IET) MIMO April, 2009 6 / 93

Diversity in wireless communications Many ways to obtain diversity Frequency & Time diversity require expensive resources (bandwidth or time)! do not provide array gain! Receive diversity does not sacrifice bandwidth or time may provide array gain Luca Sanguinetti (IET) MIMO April, 2009 7 / 93

Spatial diversity Receive diversity Advantages large performance improvement GSM [Mouly, 1992] two receive antennas at the base station uplink transmissions are improved without adding any cost, size or power consumption to the mobile Can we get the same results in the downlink employing multiple transmit antennas? Answer: YES! Transmit diversity: since 1990s Luca Sanguinetti (IET) MIMO April, 2009 8 / 93

Spatial diversity Transmit diversity Basic concept Luca Sanguinetti (IET) MIMO April, 2009 9 / 93

Transmit diversity Techniques Delay diversity [Wittneben, 1991] [Winter, 1998] Luca Sanguinetti (IET) MIMO April, 2009 10 / 93

Transmit diversity Delay diversity Advantages convert spatial diversity in frequency diversity compliant with existing systems Disadvantages the achievable diversity gain depends on coding/interleaving units no array gain Luca Sanguinetti (IET) MIMO April, 2009 11 / 93

Transmit diversity Techniques Phase diversity [Hiroike, 1992] Luca Sanguinetti (IET) MIMO April, 2009 12 / 93

Transmit diversity Phase diversity Advantages convert spatial diversity in time diversity compliant with existing systems Disadvantages the achievable diversity gain depends on coding/interleaving units no array gain Luca Sanguinetti (IET) MIMO April, 2009 13 / 93

Transmit diversity Techniques Transmit MRC (T-MRC) [Lo, 1999] Luca Sanguinetti (IET) MIMO April, 2009 14 / 93

Transmit diversity Probability of error 10 0 QPSK, N = 1 10-1 Rayleigh Fading 10-2 BER 10-3 10-4 SISO T-MRC, M =2 10-5 -3 0 3 6 9 12 E b /N 0, db 15 18 21 24 Luca Sanguinetti (IET) MIMO April, 2009 15 / 93

Transmit diversity T-MRC Advantages diversity gain at the receiver using transmit array array gain at the receiver using transmit array no processing at the receiver low cost of implementation (shared among all users) appealing for cellular or wireless local are networks Disadvantage channel knowledge at the transmitter Luca Sanguinetti (IET) MIMO April, 2009 16 / 93

Transmit diversity Channel acquisition at the transmitter Open loop systems operate in TDD mode exploit channel reciprocity not suitable for fast time-varying channels Luca Sanguinetti (IET) MIMO April, 2009 17 / 93

Transmit diversity Channel acquisition at the transmitter Closed loop systems operate in TDD or FDD mode require an error-free feedback channel Luca Sanguinetti (IET) MIMO April, 2009 18 / 93

Transmit diversity Channel unknown at the transmitter Any other idea to achieve diversity without channel knowledge? how about using two time intervals in the first interval only the first antenna is used in the second interval only the second antenna is used y(1) = h 1 x + n(1) y(2) = h 2 x + n(2) The above equation is of the same form of MRC the error rate is the same However, the date rate is halved Luca Sanguinetti (IET) MIMO April, 2009 19 / 93

Transmit diversity Channel unknown at the transmitter Transmit diversity is easy to achieve if a sacrifice in rate is acceptable The point is: How can we maximize the rate and at the same time minimizing the error rate? Luca Sanguinetti (IET) MIMO April, 2009 20 / 93

Transmit diversity Alamouti code The Alamouti code was proposed in [Alamouti, 1998] two transmit antennas multiple receive antennas Adopted in wireless standards WCDMA CDMA2000 Luca Sanguinetti (IET) MIMO April, 2009 21 / 93

Alamouti code Block diagram Assume N =1 Encoding rule [ x1 x (x 1,x 2 ) X = 2 x 2 x 1 ] = 1 2 [ s1 s 2 s 2 s 1 ] (1) The scaling factor 1 2 is used to meet the power constraint Luca Sanguinetti (IET) MIMO April, 2009 22 / 93

Alamouti code Signal model The received samples at time 1 and 2 are given by y(1) = h 1 x 1 + h 2 x 2 + n(1) (2) y(2) = h 1 x 2 + h 2x 1 + n(2) (3) In matrix form y = hx + n (4) with y = [ y(1) y(2) ] h = [ h 1 h 2 ] n = [ n(1) n(2) ] Luca Sanguinetti (IET) MIMO April, 2009 23 / 93

Alamouti code ML detection Maximum-likelihood (ML) detection (ŝ 1, ŝ 2 ) = arg min X Exhaustive search over Q 4 alternatives! y h X 2 Luca Sanguinetti (IET) MIMO April, 2009 24 / 93

Alamouti code ML detection Substituting (1) into (4) yields y h X 2 z 1 + z 2 (5) with z 1 = x 1 2 ( h 1 2 + h 2 2) w 1 (6) z 2 = x 2 2 ( h 1 2 + h 2 2) w 2 (7) and w 1 = h 1 x 1 y(1) + h 1 x 1 y (1) + h 2 x 1y(2) + h 2 x 1 y (2) (8) w 2 = h 2 x 2y(1) + h 2 x 2 y (1) + h 1 x 2 y(2) + h 1 x 2y (2) (9) Luca Sanguinetti (IET) MIMO April, 2009 25 / 93

Alamouti code ML detection From (5) (9) it follows that the ML function has been decomposed into two parts one of which z 1 = x 1 2 ( h 1 2 + h 2 2) w 1 is only a function of x 1 while the other one z 2 = x 2 2 ( h 1 2 + h 2 2) w 2 is only a function of x 2 Luca Sanguinetti (IET) MIMO April, 2009 26 / 93

Alamouti code ML detection ML detection is performed separately using linear processing with ŝ 1 = arg max { z 1 } ŝ 2 = arg max { z 2 } x 1 x 2 z 1 = x 1 2 ( h 1 2 + h 2 2) w 1 ( z 2 = x 2 2 h 1 2 + h 2 2) w 2 and w 1 = h 1 x 1 y(1) + h 1 x 1 y (1) + h 2 x 1y(2) + h 2 x 1 y (2) w 2 = h 2 x 2 y(1) + h 2 x 2 y (1) + h 1 x 2y(2) + h 1 x 2 y (2) Luca Sanguinetti (IET) MIMO April, 2009 27 / 93

Alamouti code ML detection If x 1 2 = x 2 2 =1 or, equivalently, z 1 x 1 h 1y(1) h 2 y (2) 2 z 2 x 2 h 2y(1) + h 1 y (2) 2 z 1 x 1 r 1 2 (10) z 2 x 2 r 2 2 (11) with r = [ r1 r 2 ] [ h = 1 h 2 h 2 h 1 ][ y(1) y (2) ] [ = H y(1) y (2) ] (12) Luca Sanguinetti (IET) MIMO April, 2009 28 / 93

Alamouti code ML detection Observe that [ y(1) y (2) and ] [ h1 h = 2 h 2 h 1 ][ x1 r = H H [ y(1) y (2) x 2 ] [ n(1) + n (2) ] = Hx + n ] = c x + H H n (13) H is orthogonal c = H H H = ( h 1 2 + h 2 2) I 2 Luca Sanguinetti (IET) MIMO April, 2009 29 / 93

Alamouti code ML detection Simple detection Luca Sanguinetti (IET) MIMO April, 2009 30 / 93

Alamouti code Diversity gain analysis From (13) we get r 1 = r 2 = ( h 1 2 + h 2 2) x 1 + h 1 n(1) + h 2 n(2) ( h 1 2 + h 2 2) x 2 + h 2 n(1) h 1 n(2) Compute the SNRs (recall that x i = s i / 2) SNR 1 = ρ h 2 2 SNR 2 = ρ h 2 2 { where ρ = E s 2}/ σ 2 (σ 2 denotes the noise variance) Luca Sanguinetti (IET) MIMO April, 2009 31 / 93

Alamouti code Diversity gain analysis How about the distribution of SNRs? Chi-square of order 2M = 4, i.e., χ 2 2k with k =2 At this stage, recall the line of reasoning employed for MRC Does Alamouti code achieve the maximum diversity gain? Luca Sanguinetti (IET) MIMO April, 2009 32 / 93

Alamouti code Array gain analysis How about the average SNRs? { E {SNR 1 } =E h 2} ρ 2 { E {SNR 2 } =E h 2} ρ 2 Recall that E { χ 2 2k} =2kγ where γ =1/2 is the variance of the real and imaginary parts of channel gains (Rayleigh fading model) Does Alamouti code achieve any array gain? Answer: NO! Luca Sanguinetti (IET) MIMO April, 2009 33 / 93

Alamouti code Comparison with MRC 10 0 QPSK 10-1 Rayleigh Fading 10-2 BER 10-3 10-4 Alamouti MRC 10-5 -3 0 3 6 9 12 E b /N 0, db 15 18 21 24 Luca Sanguinetti (IET) MIMO April, 2009 34 / 93

Alamouti code Uncoherent detection Coherent detection requires channel information at the receiver When channel information is not available at the receiver [ ] [ ][ ] [ ] y(1) x1 x y = = 2 h1 n(1) y(2) x 2 x + = Xh + n 1 h 2 n(2) Assume that the transmission is preceded by a known X ĥ = 1 x 1 2 + x 2 2 XH y = h + ñ The channel estimate is used to detect the next X The latter is in turn used to update channel estimate A loss of 3 db is achieved Luca Sanguinetti (IET) MIMO April, 2009 35 / 93

Alamouti code Comparison with transmit MRC If channel information is available at the transmitter with h =[h 1,h 2,..., h M ] and y = hgs + n (14) g = hh h 2 (15) Luca Sanguinetti (IET) MIMO April, 2009 36 / 93

Alamouti code Comparison with transmit MRC The received SNR is given by SNR = h 2 ρ T-MRC achieves the maximum diversity gain Luca Sanguinetti (IET) MIMO April, 2009 37 / 93

Alamouti code Comparison with transmit MRC The received SNR is given by SNR = h 2 ρ T-MRC achieves the maximum diversity gain In case of Rayleigh fading E {SNR} =E { h 2} ρ = Mρ T-MRC achieves the maximum array gain Luca Sanguinetti (IET) MIMO April, 2009 37 / 93

Alamouti code Comparison with transmit MRC 10 0 QPSK 10-1 Rayleigh Fading 10-2 BER 10-3 10-4 Alamouti T-MRC 10-5 -3 0 3 6 9 12 E b /N 0, db 15 18 21 24 Luca Sanguinetti (IET) MIMO April, 2009 38 / 93

Alamouti code Rate How about rate? Luca Sanguinetti (IET) MIMO April, 2009 39 / 93

Alamouti code Rate How about rate? Denote by N s the no. of transmitted symbols Since N s = 2 and T =2 Rate = N s T Rate =1 It is a full rate code Luca Sanguinetti (IET) MIMO April, 2009 39 / 93

Alamouti code Multiple receive antennas What happens when multiple receive antennas are used? It easily turns out that r 1 = x 1 r 2 = x 2 N i=1 N i=1 ( h i,1 2 + h i,2 2) + ( h i,1 2 + h i,2 2) + N h i,1 n i (1) + h i,2 n i (2) i=1 N h i,2n i (1) h i,1n i (2) where h i,k denote the channel gain between the kth transmit and ith receive antennas i=1 ML detection is still performed with linear processing Luca Sanguinetti (IET) MIMO April, 2009 40 / 93

Alamouti code Multiple receive antennas Luca Sanguinetti (IET) MIMO April, 2009 41 / 93

Alamouti code Multiple receive antennas How about diversity gain and array gain? SNR 1 = SNR 2 = ρ 2 2 N h i,k 2 k=1 i=1 SNRs have a Chi-square distribution with order 4N Full diversity gain is achieved, i.e., MN with M =2 Full receive array gain is achieved, i.e., N Luca Sanguinetti (IET) MIMO April, 2009 42 / 93

Alamouti code Summary Advantages simple decoding full diversity gain full rate no channel knowledge is required at transmitter Luca Sanguinetti (IET) MIMO April, 2009 43 / 93

Codes Design criteria A good code follows a design criterion for Gaussian channels: Euclidean distance for binary channels: Hamming distance Luca Sanguinetti (IET) MIMO April, 2009 44 / 93

Codes Design criteria A good code follows a design criterion for Gaussian channels: Euclidean distance for binary channels: Hamming distance What is the Alamouti criterion? [Tarokh, 1998] [Tarokh, 1999] Luca Sanguinetti (IET) MIMO April, 2009 44 / 93

Space-time coding Block diagram Luca Sanguinetti (IET) MIMO April, 2009 45 / 93

Space-time coding Signal model Assume a quasi-static channel 1 Y = HX + N (16) H = h 1,1 h 1,2 h 1,M h 2,1 h 2,2 h 2,M..... h N,1 h N,2 h N,M Maximum-likelihood detection (ŝ 1, ŝ 2,..., ŝ Ns ) = arg min X Exhaustive search over Q MT alternatives. 1 Constant over T time intervals Y H X 2 (17) Luca Sanguinetti (IET) MIMO April, 2009 46 / 93

Space-time coding Probability of error analysis Let X X with X = {X (1), X (2) X,..., X } and X (i) = X (i) 1 X (i) 2 (1) X(i) 1 (2) X(i) 1 (T ) (1) X(i) 2 (2) X(i) 2 (T )..... X (i) M (1) X(i) M (2) X(i) M (T ) Luca Sanguinetti (IET) MIMO April, 2009 47 / 93

Space-time coding Probability of error analysis Assume that X (1) is transmitted MLD operates as follows ˆX = arg min X X Y (1) H X 2 An error occurs if ˆX = X (i) with i 1 The error probability is upper bound by ( ) Pr error X (1) is sent X i=1 where Pr ( X (1) X (i)) is the pairwise probability ( Pr X (1) X (i)) (18) Luca Sanguinetti (IET) MIMO April, 2009 48 / 93

Space-time coding Probability of error analysis Assume the worst case is Pr ( X (1) X (2)), then (18) becomes ( ) ( Pr error X (1) is sent Pr X (1) X (2)) (19) Define with ( A X (1), X (2)) ( = D X (1), X (2)) ( D H X (1), X (2)) (20) ( D X (1), X (2)) = X (1) X (2) Compute the EVD of A A = UΛU with Λ = diag{λ 1,λ 2,..., λ K } and K denotes the rank of A Luca Sanguinetti (IET) MIMO April, 2009 49 / 93

Space-time coding Probability of error analysis It can be shown that in the case of Rayleigh fading ( Pr X (1) X (2)) { )} =E Pr (X (1) X (2) H ( K k=1 λ k ) 1 K ρ M KN From (19) we get ( ) Pr error X (1) is sent ( K k=1 λ k ) 1 K ρ M KN Luca Sanguinetti (IET) MIMO April, 2009 50 / 93

Space-time coding Array gain & Diversity gain 10 0 10-1 Average SER 10-2 10-3 Diversity gain Coding gain 10-4 10-5 -3 0 3 6 9 12 ρ, db 15 18 21 24 Luca Sanguinetti (IET) MIMO April, 2009 51 / 93

Space-time coding Probability of error analysis Define the coding gain as G c = ( K k=1 λ k ) 1 K (21) and compute the diversity gain as ( ( log Pr error X (1) is sent )) G d = lim = KN (22) ρ log (ρ) Then ( ) ( ρ Pr error X (1) is sent c) M G Gd (23) Luca Sanguinetti (IET) MIMO April, 2009 52 / 93

Space-time coding Design criteria Rank or diversity criterion To achieve the maximum diversity gain MN, A ( X (i), X (j)) must be full rank j i. If j i the minimum rank of A ( X (i), X (j)) is K, then a diversity gain of KN is achieved Determinant criterion To optimize the coding gain, the minimum determinant of A ( X (i), X (j)) j i must be maximized Luca Sanguinetti (IET) MIMO April, 2009 53 / 93

Space-time coding Summary Characteristics: map data symbols onto codewords (in any form) (s 1,s 2,..., s Ns ) X channel not known at the transmitter Aims: maximize coding and diversity gain Approaches: space-time block codes [Tarokh, 1999] space-time trellis codes [Tarokh, 1998] Luca Sanguinetti (IET) MIMO April, 2009 54 / 93

Space-time block codes Definition operates on a block of data at a time the output only depends on the current input bits Other codes: for example convolutional codes the output depends on the current and previous inputs Is the Alamouti code a Space-Time Block Code (STBC)? Luca Sanguinetti (IET) MIMO April, 2009 55 / 93

Space-time block codes Alamouti code Does Alamouti code satisfy the rank criterion? Answer: YES! Luca Sanguinetti (IET) MIMO April, 2009 56 / 93

Space-time block codes Alamouti code Does Alamouti code satisfy the rank criterion? Answer: YES! Take two codewords [ X (j) x1 x = ] 2 x 2 x 1 X (i) = [ x 1 x 2 x 2 x 1 ] Using the identities rank (C) = rank ( BB H) = rank (B), we get ( ( rank A X (i), X (j))) ( = rank D (X (i), X (j))) A ( X (i), X (j)) is full-rank if and only if D ( X (i), X (j)) is full-rank Luca Sanguinetti (IET) MIMO April, 2009 56 / 93

Space-time block codes Alamouti code Observe that ( ( det D X (i), X (j))) ( = det X (i) X (j)) = x 1 x 1 2 + x 2 x 2 2 is zero only if x 1 = x 1 and x 2 = x 2 D ( X (i), X (j)) is always full rank when X (i) X (j) That is why Alamouti code achieves full spatial diversity, i.e., 2N Luca Sanguinetti (IET) MIMO April, 2009 57 / 93

Space-time block codes Alamouti code Does Alamouti code provide some coding gain? Answer: NO! Luca Sanguinetti (IET) MIMO April, 2009 58 / 93

Space-time block codes Alamouti code Does Alamouti code provide some coding gain? Answer: NO! Recall that SNR 1 = SNR 2 = ρ 2 2 N h i,k 2 k=1 i=1 Then E{SNR 1 } =E{SNR 2 } = ρn Only receive array gain is achieved, i.e., N The code does not produce any additional coding gain Luca Sanguinetti (IET) MIMO April, 2009 58 / 93

Space-time block codes Alamouti code Why simple ML detection? Look at the structure of X X [ x1 x X = 2 x 2 x 1 ] Its rows are orthogonal to each other ( X H X = x 1 2 + x 2 2) I 2 Such codes are called Orthogonal Space-Time Block Codes (O-STBCs) [Tarokh, 1999] Luca Sanguinetti (IET) MIMO April, 2009 59 / 93

Space-time block codes Codes with more than two antennas Alamouti code: O-STBC Advantages full diversity full rate low complexity Disadvantage no coding gain only two transmit antennas, i.e., M =2 Any O-STBC with such advantages for M>2? Luca Sanguinetti (IET) MIMO April, 2009 60 / 93

Space-time block codes Codes with more than two antennas Consider a code operating according to the following encoding rule X X is a linear combination of (s 1,s 2,... s Ns ) and ( s 1,s 2,... s N s ) Satisfying ( X H X = K s 1 2 + s 2 2 +...+ s Ns 2) I M (24) where K is a constant and I M is the identity matrix of order M Luca Sanguinetti (IET) MIMO April, 2009 61 / 93

Space-time block codes Codes with more than two antennas Using (24) we get ( ( det A X (i), X (j))) = which is zero if and only if x k = x k k M x k x k k=1 This means that O-STBCs provide full diversity, i.e, MN 2 Luca Sanguinetti (IET) MIMO April, 2009 62 / 93

Space-time block codes Codes with more than two antennas As a direct consequence of (24) reduces to (ŝ 1, ŝ 2,..., ŝ Ns ) = arg min X ŝ k = arg min s k f k ( s k ) Y H X 2 ML detection with O-STBCs is performed with linear effort separately for each s k O-STBCs are characterized by low-complexity Luca Sanguinetti (IET) MIMO April, 2009 63 / 93

Space-time block codes Codes with more than two antennas As a consequence of the above result SNR = N M i=1 k=1 h i,k 2 ρ M SNRs have a Chi-square distribution of order 2MN How about diversity gain? Full diversity gain MN is achieved How about coding gain? It is N only the receive array gain is achieved (as Alamouti code) Luca Sanguinetti (IET) MIMO April, 2009 64 / 93

Space-time block codes Codes with more than two antennas O-STBCs with M>2 [Tarokh, 1999] Do they achieve full diversity? Answer: YES! Do they provide coding gain? Answer: NO! (as Alamouti code) Do they have simple linear processing? Answer: YES! Do they provide full rate? Answer: NO! (Alamouti code is full rate) Luca Sanguinetti (IET) MIMO April, 2009 65 / 93

Space-time block codes Codes with more than two antennas O-STBCs with M>2 [Tarokh, 1999] Luca Sanguinetti (IET) MIMO April, 2009 66 / 93

Space-time block codes Real constellation Example 1 Rate 1 O-STBC for M =4 X = x 1 x 2 x 3 x 4 x 2 x 1 x 4 x 3 x 3 x 4 x 1 x 2 x 4 x 3 x 2 x 1 Luca Sanguinetti (IET) MIMO April, 2009 67 / 93

Space-time block codes Complex constellation Example 2 Rate 1/2 O-STBC for M =4 X = x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 2 x 1 x 4 x 3 x 2 x 1 x 4 x 3 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 4 x 3 x 2 x 1 x 4 x 3 x 2 x 1 Example 3 Rate 1/2 O-STBC for M =3 X = x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 2 x 1 x 4 x 3 x 2 x 1 x 4 x 3 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 Luca Sanguinetti (IET) MIMO April, 2009 68 / 93

Space-time block codes Complex constellation Example 4 Rate 3/4 O-STBC for M =4 x 1 x 2 x 2 x 1 X = 2 x 3 x 3 x 2 1 x 1 +x 2 x 2 2 x 3 x 3 2 2 x 3 2 x 3 2 x 2 x 2 +x 1 x 1 2 2 x 3 x 3 x 1 x 1 +x 2 +x 2 2 2 x 1+x 1 +x 2 x 2 2 Luca Sanguinetti (IET) MIMO April, 2009 69 / 93

Space-time block codes Complex constellation Example 5 Rate 3/4 O-STBC for M =4 X = x 1 x 2 x 3 0 x 2 x 1 0 x 3 x 3 0 x 1 x 2 0 x 3 x 2 x 1 Luca Sanguinetti (IET) MIMO April, 2009 70 / 93

Space-time block codes Probability of error 10 0 QPSK 10-1 Rayleigh Fading 10-2 BER 10-3 10-4 Alamouti Tarokh Tarokh 10-5 -3 0 3 6 9 12 15 18 21 24 db Luca Sanguinetti (IET) MIMO April, 2009 71 / 93

Space-time block codes Probability of error 10 0 2 bit/s/hz 10-1 Rayleigh Fading 10-2 BER 10-3 10-4 Alamouti Tarokh Tarokh 10-5 -3 0 3 6 9 12 15 18 21 24 db Luca Sanguinetti (IET) MIMO April, 2009 72 / 93

Space-time block codes O-STBCs Advantages ML detection with linear effort full diversity Disadvantages No coding gain The maximum rate with complex constellation and M>2 is upper bounded by 3/4 Any alternative? Luca Sanguinetti (IET) MIMO April, 2009 73 / 93

Space-time block codes QO-STBCs Quasi O-STBCs proposed in [Jafarkhani, 2001], [Tirkkonen, 2000] and [Papadias, 2003] Four transmit antennas Easily constructed from two Alamouti codewords [ x1 x A = 2 x 2 x 1 ] [ x3 x B = 4 x 4 x 3 ] Luca Sanguinetti (IET) MIMO April, 2009 74 / 93

Space-time block codes QO-STBCs Example 1. Jafarkhani codeword [ ] A B X J = B A = x 1 x 2 x 3 x 4 x 2 x 1 x 4 x 3 x 3 x 4 x 1 x 2 x 4 x 3 x 2 x 1 Example 2. Tirkkonen codeword [Tirkkonen, 2000] [ ] A B X T = = B A x 1 x 2 x 3 x 4 x 2 x 1 x 4 x 3 x 3 x 4 x 1 x 2 x 4 x 3 x 2 x 2 Luca Sanguinetti (IET) MIMO April, 2009 75 / 93

Space-time block codes QO-STBCs Advantage Full rate with M =4 Disadvantages? How about diversity? How about complexity? Luca Sanguinetti (IET) MIMO April, 2009 76 / 93

Space-time block codes QO-STBCs Exercise. Show that rank(a(x (i) decoding operates according to J, X(j) J )) = 2 and prove that ML ˆX J = arg min X J f 1,4 ( x 1, x 4 )+f 2,3 ( x 2, x 3 ) Luca Sanguinetti (IET) MIMO April, 2009 77 / 93

Space-time block codes Probability of error 10 0 Rayleigh Fading Channel 2 bit/s/hz, N = 1 10-1 10-2 BER 10-3 10-4 Alamouti Tarokh QO-STBC 10-5 -3 0 3 6 9 12 15 18 21 24 db Luca Sanguinetti (IET) MIMO April, 2009 78 / 93

Space-time block codes QO-STBCs How about diversity? Rank criterion is not satisfied Only partial spatial diversity is achieved How about complexity? Rows not orthogonal to each other Higher complexity compared to O-STBCs complex symbol pair-wise decoding using QPSK for M = 4 four times more complex Not good goodes! Luca Sanguinetti (IET) MIMO April, 2009 79 / 93

Space-time block codes RQO-STBCs How about achieving full rate and full diversity? Possible solutions in [Yuen, 2003], [Sharma, 2003] and [Su, 2004] Main idea different constellations for different time slot. Luca Sanguinetti (IET) MIMO April, 2009 80 / 93

Space-time block codes Probability of error Encoding rule X J = x 1 x 2 x 3 x 4 x 2 x 1 x 4 x 3 x 3 x 4 x 1 x 2 x 4 x 3 x 2 x 1 with x k = e jθ x k Same complexity of QO-STBCs complex symbol pair-wise decoding Known as Rotated QO-STBCs (RQO-STBCs) Luca Sanguinetti (IET) MIMO April, 2009 81 / 93

Space-time block codes RQO-STBCs 10 0 Rayleigh Fading Channel 2 bit/s/hz, N = 1 10-1 10-2 BER 10-3 10-4 Alamouti Tarokh QO-STBC RQO-STBC 10-5 -3 0 3 6 9 12 15 18 21 24 db Luca Sanguinetti (IET) MIMO April, 2009 82 / 93

Space-time block codes The main plot Luca Sanguinetti (IET) MIMO April, 2009 83 / 93

Space-time coding Space-time trellis code Conventional coding techniques Convolutional coding (CC) Trellis-coded modulation (TCM) Luca Sanguinetti (IET) MIMO April, 2009 84 / 93

TCM Set partitioning Luca Sanguinetti (IET) MIMO April, 2009 85 / 93

Space-time coding Space-time trellis code [Tarokh, 1998] Extension of TCM to multiple antennas Designed according to rank and determinant criteria optimal rate, diversity, trellis complexity not known how to provide optimal coding gain for a given rate, diversity, trellis complexity Advantages Better performance Higher coding gain Disadvantages Code design: very difficult! Optimum decoding: Viterbi algorithm! Luca Sanguinetti (IET) MIMO April, 2009 86 / 93

Space-time coding Space-time trellis code Each STTC is characterized by a trellis Designed by hand The number of nodes corresponds to the number of states Each nodes has Q groups of symbols to the lefts Each group consists of M entries The M entries correspond to the symbols to be transmitted from the M antennas Increasing the number of states higher coding gain much complexity as the decoder! Luca Sanguinetti (IET) MIMO April, 2009 87 / 93

Space-time trellis code 4-PSK 4-State with M =2 Luca Sanguinetti (IET) MIMO April, 2009 88 / 93

Space-time coding Space-time trellis code Design rules for two transmit antennas Transitions from the same state should differ in the second symbol Transitions merging to same state should differ in the first symbol Luca Sanguinetti (IET) MIMO April, 2009 89 / 93

Space-time trellis code 8-PSK 8-State with M =2 Luca Sanguinetti (IET) MIMO April, 2009 90 / 93

Space-time coding Ongoing research activity Space-time coding for frequency selective channels OFDM, MC-CDM,... Space-time coding for informed transmitters linear or non-linear pre-coding,... Space-time coding in a multiuser environment Luca Sanguinetti (IET) MIMO April, 2009 91 / 93

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