Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming

Transcription

1 Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming Master Universitario en Ingeniería de Telecomunicación I. Santamaría Universidad de Cantabria

2 Contents Introduction Multiplexing, Diversity, and Array Gains MIMO system model Capacity AWGN MIMO channel Fading MIMO channel MIMO beamforming Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 0/38

3 Introduction We consider point-to-point MIMO link (a.k.a. single-user MIMO or SU-MIMO) s [ n 1 ] s d [n] Tx x [ n 1 ] x M [n] M MIMO CHANNEL H n 1[ n ] y 1 [ n] n N [n] y N [n] Transmit antennas N Receive antennas Rx s ˆ1[ n ] ˆ [ n] The use of multiple antennas at the Tx and Rx sides opens up a new spatial dimension that can be exploited in several (sometimes conflicting) ways To increase the transmission rate Multiplexing gain To increase the robustness of the system Diversity gain To increase the SNR Array or coding gain Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 1/38 s d

4 Multiplexing gain The multiplexing gain is the slope of the capacity or rate with respect to the SNR at high SNR s r = lim SNR C(SNR) log(snr) Intuitively, it is the number of independent data stream that can be transmitted For a SISO system C(SNR) = log(1 + SNR), and then r = log(1 + SNR) log(snr) lim lim SNR log(snr) SNR log(snr) = 1 and thus the multiplexing gain of a SISO system is 1 The maximum multiplexing gain of a MIMO system is r = rank(h) = min(m, N) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming /38

5 C (bps/hz) Outage capacity curves for a fixed P e = 0.01 (outage probability) 0 15 MIMO (4# 4) MIMO (# ) slope =4 mult. gain 10 slope = mult. gain SNR (dbs) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 3/38

6 Diversity gain The diversity gain is the negative slope of the BER curve with respect to the SNR at high SNR s d = lim SNR log(p e (SNR)) log(snr) Intuitively, it is the number of independent channel realizations over which the signal has been trasnmitted For a SISO system and a Rayleigh fading channel P e (SNR) SNR 1, and then the diversity gain is d = 1 The maximum diversity gain of a MIMO system is d = MN Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 4/38

7 BER BER curves for a fixed spectral efficiency of bps/hz (fixed rate!) 10 0 # 1# slope =1 diversity gain 10-4 slope =4 diversity gain SNR (db) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 5/38

8 MIMO techniques There is a trade-off between the spatial and multiplexing diversities (which is in fact the usual trade-off between transmission speed and robustness). Consequently, there are two different families of MIMO techniques: Spatial Multiplexing techniques: We transmit simultaneously r = min(m, N) streams (layers) of data The goal is to increase the speed of transmission while keeping the robustness (diversity) constant Examples: Vertical-Bell Laboratories Layered Space-Time (V-BLAST), SVD-based precoding Spatial Diversity techniques: We transmit 1 stream of data The goal is to increase the the robustness (exploiting the spatial diversity), for a given transmission rate Examples: MIMO beamforming, Orthogonal Space-Time Block Coding (OSTBC) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 6/38

9 Array gain The array gain is the power gain obtained by using MIMO in comparison to a SISO system using the same transmitted power AG = E[SNR MIMO] E[SNR SISO ] As a reference, let us consider a SISO system transmitting a bandwidth W with power P and noise spectral density N 0 / over a Rayleigh channel with E[ h ] = 1 P h SNR SISO SNR SISO = PE[ h ] N 0 W = P N 0 W Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 7/38

10 Consider now a SIMO system using MRC (Maximum Ratio Combining): w = h s[n] P h h 1 h h N r [ n ] r 1 [ n] r N [n] w 1 w w N z[n] SNR SIMO MRC = PE[ wt h ] N 0 W = PE[ h ] N 0 W = PN N 0 W and the array gain is therefore AG = N (3 db when N = ) The array gain of a M N MIMO system can be bounded as follows 1 ( ) 1 AG N + M 1 J. Bach, Array gain and capacity for known random channels with multiple element arrays at both ends, IEEE Journal on Selected Areas on Comm., vol. 18, no. 11, pp , Nov Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 8/38

11 Example Consider a N N MIMO system that applies beamforming at the Tx and Rx sides (MIMO beamforming) s[n] w Tx,1 w Tx, w Tx, N H w Rx, 1 w Rx, z[n] w Rx, N The N N MIMO channel has i.i.d. Rayleigh entries, with E[ h ij ] = 1 Find the array gain when the beamformers are given by w Tx = w Rx = 1/ N. 1/ N Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 9/38

12 MIMO system model Consider a MIMO system with M transmit and N receive antennas At time instant n, we transmit d min(m, N) information symbols s[n] = (s 1 [n],..., s d [n]) T The transmit symbols are usually normalized so that [ E s[n]s[n] H] = I The Tx converts the d information symbols into M signals to be applied (after upconversion) at the M transmit antennas. In general, the transformation is linear x[n] = Ts[n] where T is a M d precoding matrix Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 10/38

13 For instance, if we apply beamforming at the Tx side: d = 1 and the precoding matrix will become a vector x 1 [n] w 1. =. s 1 [n], x M [n] w M but this model also encompasses more general systems for which mixtures of the symbols are transmitted through all antennas The transmitted signal has covariance matrix Q = E[x[n]x[n] H ] The total power transmitted by the MIMO system is M E[ x i [n] ] = tr(q) i=1 where tr(q) denotes the trace of Q Typically, we have an average power constraint tr(q) P Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 11/38

14 H C N M is the MIMO channel (assumed here as flat fading) h 11 h 1M H =..... h N1 h NM where h ij is the flat fading SISO channel from the j-th transmit antenna to the i-th receive antenna (a complex scalar) n[n] C N 1 is the Gaussian noise with covariance matrix R n = σ I (recall that σ = N 0 W ) Finally, skipping the time index for notational simplicity, the received signal can be modeled as y = Hx + n Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 1/38

15 Outline We study the capacity of SU-MIMO systems This will serve as an example of a spatial multiplexing MIMO technique We study MIMO beamforming: single stream (d = 1) transmissions by means of optimal Tx-Rx beamforming This will serve as an example of a spatial diversity MIMO technique Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 13/38

16 Capacity SU-MIMO with CSIT We first consider an AWGN channel (constant) which is known at the transmitter side (CSIT) We also assume that M = N The capacity is C = ( max log det R n + HQH H) log det (R n ), Q : tr(q) P and it is achieved by Gaussian signaling We assume the noise is white so R n = σ I The problem of getting the optimal transmit covariance matrix Q that maximizes capacity subject to a power constraint is convex, and admits a closed-form solution Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 14/38

17 Solution First, the optimization problem can be written as max log det (I + 1σ ) Q : tr(q) P HQHH, (1) Second, let s introduce the singular value decomposition (SVD) of the full-rank N N MIMO channel N H = UΛV H = λ i u i vi H. where U = (u1,..., u N ) is a unitary matrix (U H U = UU H = I) containing the left singular vectors, V = (v 1,..., v N ) is a unitary matrix (V H V = VV H = I) with the right singular vectors, and Λ = diag(λ 1,..., λ N ) is a diagonal matrix with the singular values (which are real and positive λ i 0) along the diagonal Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 15/38 i=1

18 Substituting the SVD in (1) we have max log det (I + 1σ ) Q : tr(q) P UΛVH QVΛU H, The solution of the problem can be written as Q = VPV H, where P is a diagonal matrix containing the powers allocated to each transmitted mode P = diag(p 1,, P N ) Substituting Q = VPV H in the logdet expression we obtain max P : log det (I + 1σ ) N i=1 P i P UΛ PU H, Writing I = UU H and defining the diagonal matrix Σ = I + 1 σ Λ P the optimization problem becomes Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 16/38

19 ( max P : log det UΣU H) = max N i=1 P i P P : N i=1 P i P N i=1 ( log 1 + P iλ ) i σ which can be viewed as the capacity of N parallel SISO channels, where the power channel gain of each SISO channel is λ i The optimal powers Pi therefore are obtained via waterfilling ) + Pi = (µ σ where x + = max(x, 0) and µ is the water level chosen to satisfy the power constraint N i=1 P i = P Optimal solution The optimal transmit directions are matched to the channel matrix (right singular vectors) The optimal power allocation is waterfilling Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 17/38 λ i

20 Waterfilling µ PP PP PP 3 1 σ λ 1 σ λ σ σ λ 4 λ3 Water level The capacity with CSIT is C = λ i /σ >1/µ log ( µλ i σ ) = N i=1 ( ( µλ )) + log i σ Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 18/38

21 Maximum capacity scheme with CSIT The Tx knows the channel and computes the SVD of the channel: H = UΛV H The optimal Tx directions are the column vectors of V The optimal powers are obtained via waterfilling P = diag(p 1,..., P N ) The vector of N symbols to be transmitted, s = (s 1,..., s N ) T, is precoded at the Tx side with VP 1/ x = VP 1/ s Notice that, since V is unitary, the power constraint is satisfied E[xx H ] = E[VP 1/ ss H P 1/ V H ] = VP 1/ E[ss H ]P 1/ V H = VPV H, the total transmitted power is tr(q) = tr(p) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 19/38

22 The Rx also computes the SVD of the channel and uses U H as decoding matrix z = U H (Hx + n) = U H UΛV H VP 1/ s + U H n = ΛP 1/ s + n }{{} y After decoding the noise is also Gaussian and white, n CN(0, σ I), because U H is unitary and therefore does not change the noise distribution x = VP 1/ s y = H x + n s Precoder 1/ VP Channel H = UΛV H Decoder H U z n Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 0/38

23 By means of the SVD precoding-decoding we have diagonalized the MIMO channel N parallel SISO channels, which are called sometimes channel eigenmodes z 1 λ P s 1 z. = 0 λ P... 0 s z N λ N P N s N s 1 s sn * P 1 λ1 λ λ N σ * P * P N σ σ Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 1/38 z 1 z z N n 1 n. n N

24 M N If the number of transmit antennas, M, is different than the number of receive antennas, N, everything remains pretty much the same, but The maximum number of transmitted streams (multiplexing gain) is r = min(m, N) We use the compact SVD For instante, if M = 4 and N = then r = and the compact SVD is [ ] h11 h 1 h 13 h 14 = UΛV H = [ ] [ ] [ ] λ u h 1 h h 3 h 1 u 1 0 v H λ v H }{{} H where U is and V is 4 Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming /38

25 Capacity SU-MIMO with CSIR only Now, we consider the case when only the Rx has channel state information (CSIR) In this case, the capacity is achieved by transmitting isotropically Gaussian codewords and it is given by Q = P M I ( C = log det I + P ) Mσ HHH = N i=1 ( log 1 + P ) Mσ λ i where we have assumed that the the channel matrix is square and full-rank N = rank(h) E. Telatar, Capacity of multi-antenna Gaussian channels, European Trans. on Telecom., vol. 10, no. 6, pp , Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 3/38

26 SU-MIMO: Multiplexing gain Comparing the capacity when the Tx knows the channel C = N i=1 ( ( µλ )) + log i σ and when it does not C = N i=1 ( log 1 + P ) Mσ λ i we observe that in both cases the multiplexing gain is the same, the only difference comes from the optimal power allocation The multiplexing gain of a SU-MIMO system with or without CSIT is r = min(m, N) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 4/38

27 The value of CSIT Rayleigh MIMO channel with i.i.d. entries CSIT+CSIR CSIR Capacity (bps/hz) x4 x SNR (dbs) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 5/38

28 Capacity (bps/hz) Impact of correlation on capacity 4 4 MIMO channel CSIT+CSIR CSIR i.i.d. MIMO channel Tx & Rx correlation d= SNR (dbs) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 6/38

29 Example: Power allocation in MIMO channels Consider the following 3 3 MIMO channel with SVD i H = i i i i }{{}}{{} 6 U Λ 1 i i 3 i i 6 } {{ } V H The noise variance is σ = 1 and the total available power at the Tx is P = 1. Obtain the optimal transmit covariance matrix Q with CSIT. Obtain the capacity (in bps/hz) with CSIT and compare it with the capacity without CSIT. Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 7/38

30 Fading channels (Recap) Remember that in fading channels we have two different situations that give rise to different capacity concepts 1. Fast fading channels Ergodic capacity. Block fading channels Outage capacity L H 1 AWGN L H H1 1 L Instantaneous capacity H1 H L Ergodic (fast fading) H L H 1 H L H L Ergodic capacity Block fading L H 1 L H L H n Outage capacity Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 8/38

31 Ergodic channels If the transmitted signal undergoes a fast fading channel so that a codeword experiences multiple fading states, the capacity is the mathematical expectation of the AWGN capacity For instance, when only CSIR is available the ergodic capacity is [ ( C = E[C(H)] = E log det I + P )] Nσ HHH To compute the ergodic capacity we have to integrate over the joint distribution of HH H, which in general is a complicated task Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 9/38

32 Block fading channels Under a block fading channel model each packet experiences only one channel state, therefore we cannot code over multiple fading states and averaging the capacity makes no sense Strictly, the Shannon capacity is zero We use in this case the outage capacity C out,p = r Pr(r > C(H)) = p Again, computing the outage capacity for MIMO channels can be a tough problem Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 30/38

33 MISO and SIMO beamforming MISO SIMO w Tx,1 w Tx, h h 1 h 1 h w Rx, 1 w Rx, w Tx, N h N h N w Rx, N We have seen that for MISO and SIMO channels we can apply beamforming to get: 1. Full spatial diversity: N. Maximum array gain: N The optimal beamformer in both cases is w = h / h, which yields an equivalente SISO channel y[n] = h s[n] + r[n] Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 31/38

34 MIMO beamforming The problem now is how to choose the optimal Tx and Rx beamformers for a MIMO channel s[n] w Tx,1 w Tx, w Tx, N MIMO H r [ n] r [ n 1 ] r N [n] w Rx, 1 w Rx, w Rx, N We assume that both beamformers have unit norm w Tx = w Rx = 1 The equivalent SISO channel is z[n] = wrx T Hw Tx s[n] + r[n] }{{} h Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 3/38 z[n]

35 Problem The problem is to find Tx & Rx beamformers maximizing the array gain or SNR and achieving full spatial diversity N MIMO beamforming problem maximize w Rx,w Tx w T Rx Hw Tx subject to w Tx = w Rx = 1 The solution is obtained through the SVD of the MIMO channel matrix H = UΛV H = N λ i u i vi H i=1 Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 33/38

36 Remember that Λ = diag (λ 1, λ,..., λ N ) with λ 1 λ... λ N contains the singular values, and U, and V are unitary matrices with the right and left singular vectors Channel eigenmodes The solution therefore is given by w Tx = v 1 and w Rx = u 1 The equivalent SISO channel after MIMO beamforming is z[n] = λ 1 s[n] + r[n] The SNR is SNR = Pλ 1 WN 0 The array gain is: AG = E[λ 1 ] Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 34/38

37 The case s[n] w Tx,1 w Tx, h h MIMO 11 1 [ ] h11 h H = 1 h 1 h h1 h r [ n] r [ n 1 ] w Rx, 1 w Rx, = [ ] [ ] [ ] λ u 1 u 1 0 v H 1 0 λ v H z[n] Assume h ij CN(0, 1), with E[ h ij ] = 1, then the average SNR for a SISO channel would be SNR SISO = PE[ h ij ] WN 0 = P WN 0 Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 35/38

38 Using w Tx = v 1 and w Rx = u 1, the equivalent SISO channel after MIMO beamforming is z[n] = λ 1 s[n] + r[n] Denoting µ 1 = λ 1, the SNR after MIMO beamforming is SNR MIMO beam = PE[λ 1 ] WN 0 = PE[µ 1] WN 0 and the array gain is E[µ 1 ] What is the distribution of µ 1? f (µ 1 ) = e µ 1 ( µ 1 µ 1 + ) e µ 1, µ 1 0 with E[µ 1 ] = 3.5, so the array gain is AG = 3.5 (5,4 db) Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 36/38

39 p(snr) Simulated MIMO-beamforming (x) SISO SNR Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 37/38

40 Example: MIMO beamforming Consider the following 3 3 MIMO channel with SVD i H = i i i i }{{}}{{} 6 U Λ 1 i i 3 i i 6 } {{ } V H The noise variance is σ = 1 and the total available power at the Tx is P = 1. Obtain the optimal Tx-Rx beamformers and the capacity of the resulting MIMO beamforming scheme. Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming 38/38