E7220: Radio Resource and Spectrum Management. Lecture 4: MIMO

Transcription

1 E7220: Radio Resource and Spectrum Management Lecture 4: MIMO 1

2 Timeline: Radio Resource and Spectrum Management (5cr) L1: Random Access L2: Scheduling and Fairness L3: Energy Efficiency L4: MIMO L5: UDN Exe 1 Exe 2 Exe 3 Exe 4 Exe 5 2

3 Warm up discussion (5min) How does this session about MIMO communication relate to other topics of this course? 3

4 Schedule of Today s Lecture 12:15-12:45 Introduction 12:45-13:00 Group Discussion (6+6 minutes) Break 13:10-13:25 Preliminaries of matrix, and a short discussion 13:25-14:00 MIMO channel and Massive MIMO 4

5 Massive MIMO a new technology for 5G Linear user 1 Distributed user 2 Cylindrical Rectangular Base station user K 5

6 MIMO Antennas: 3 approaches MIMO beamforming: Tx or/and Rx utilize beamforming. Achieved by adaptive weights. Maximize SINR. Effectiveness: LoS. MIMO diversity: multiple antennas are used at TX and RX, where the same signal is sent over all antennas and coherently combined at the receiver. MIMO channel, where Tx and Rx use multiple antennas to exploit the low correlation between paths in multipath environment. Achieved by multiplexing and filtering. Higher data rate through parallel transmission. Effectiveness: NLoS (e.g. Rayleigh fading channel). 6

7 An Example of Receive Beamforming Resolvability of the signals. 1/bandwidth measures the signal resolvability in the time domain. 1/L r 1/L r measures the resolvability in the angular domain. The signals arriving with angles much less than can not be resolved. TX antenna 1 TX antenna 2... Receive antenna array L r = n r r normalized length of the receive antenna array. n r the number of antenna elements. r the normalized (to the unit of the wavelength) receive antenna separation. further reading Tse s book Chapter 3 7

8 Resolvability in the angular domain f(ω) receive antennas f(ω) receive antennas Ω is the directional cosine, i.e., f(ω) = cos(ω) Ω 16 receive antennas Ω sinc function f(ω) f(ω) Ω Ω L r = n r r =8 8

9 Receive beamforming patterns aimed at receive antennas receive antennas receive antennas receive antennas receive antennas receive antennas receive antennas receive antennas L r =2 L r =4 9

10 Antenna Diversity Receive diversity: using multiple receive multiple antennas (single input multiple output (SIMO) channels). Transmit diversity: using multiple transmit multiple antennas (multiple input single output (MISO) channels). In addition, basics on space-time coding are introduced. Degrees of freedom*: MIMO channels provide additional degrees of freedom in addition to diversity for communication. (the rank of the random channel matrix.) * DoF of a channel: the dimension of the received signal space. 10

11 Receive Diversity Assume a flat fading channel with 1 transmit antenna and L receive antennas. The channel model reads y l [m] =h l [m]x[m]+w l [m], l =1,,L where antennas, and w l [m] CN(0,N 0 ) x[m] is independent across the is the transmitted signal. 11

12 Receive Diversity We aim to detect x[m] basedony 1 [m],y 2 [m],,y L [m] The channels h l [m], l are considered to be independent, if the receive antennas are spaced sufficiently far away. So that there is a diversity gain of L. Assume BPSK modulation, the error probability conditional on the channel gains is given by Q( 2 h 2 SNR) The total received SNR reads LSNR 1 L h 2 LSNR corresponds to a power gain (also or array gain): by having multiple receive antennas and coherent combing at the receiver, the effective total received signal power increases linearly with L. A detailed explanation could be found at page 72 of the book Tse, D. & Viswanath, P. Fundamentals of Wireless Communication Cambridge University Press, 2005 SNR (per complex symbol time) : average received signal energy/noise energy 12

13 Receive Diversity 1 reflects the diversity gain: by averaging over multiple independent L h 2 signal paths, the probability that the overall gain is small is decreased. h l [m], l If the channel gains are fully correlated across all antennas, only the power gain is obtained but no diversity gain as L increases. Even when all the are independent, there is a diminishing marginal return as L increases: due to the Law of Large Numbers, h l 1 L h 2 = 1 L L l=1 The power gain does not suffer this limitation. h l [m] 2 1 * Assuming each of the channel gains is normalized with unit variance. 13

14 Transmit Diversity L transmit antennas and 1 receive antenna. To achieve the diversity gain: transmit the same symbol over the L different antennas during L symbol times. At any one time, only one antenna transmits and the others are silent. 14

15 Transmit Diversity Coding gain over repetition code: using one antenna at a time and transmitting the coded symbols of the time diversity code successively over the different antennas. (quite wasteful of degrees of freedom) Coding gain using space-time coding: e.g. Alamouti scheme proposed in several third-generation cellular standards. Alamouti scheme: is designed for two transmit antennas (it is possible, to some extent, to be generalized to more than two antennas.) 15

16 Alamouti Scheme Assume a flat fading channel with two transmit antennas and one receive antenna, y[m] =h 1 [m]x 1 [m]+h 2 [m]x 2 [m]+w[m] Alamouti scheme: transmit 2 complex symbols and over 2 symbol time: u 1 u 2 At time 1, x 1 [1] = u 1, x 2 [1] = u 2. At time 2, x 1 [2] = u 2,x 2[2] = u 1. denotes the complex conjugate. 16

17 Alamouti Scheme Assume that the channel remains constant over 2 symbol h 1 = h 1 [1] = h 1 [2] and h 2 = h 2 [1] = h 2 [2]. Then [ ] [ ] [ u y[1] y[2] = h1 h 1 u ] 2 2 u 2 u + [ w 1 [1] w 2 [2] ] 1 times, i.e. Rewrite the previous equation [ ] y[1] y[2] = [ h1 h 2 h 2 h 1 ][ u1 u 2 ] Here the diversity gain is 2 for the detection of each symbol. 2 symbols are transmitted over 2 symbol times with half the power in each symbol (assuming that the total transmit power as using repetition code.) + Orthogonal matrix [ ] w[1] w[2] 17

18 A 2x2 MIMO Example 2 transmit antennas and 2 receive antennas. signature 1 signature 2 Four paths: Maximum diversity gain 4. (h 11,h 21 ) and (h 12,h 22 ) If are linearly independent, the signal space dimension is 2, i.e., DoF per symbol time is 2. DoF of a channel: the dimension of the received signal space. 18

19 A 2 2 MIMO Example source: Tse, D. & Viswanath, P., Fundamentals of Wireless Communication, Cambridge University Press,

20 Group Discussion (6+6 minutes) First discuss in each group and present the results on board. During walk-gallery, presenting your results to a new group everyone in the group will present the results How do multi-antennas provide diversity gains? How do receive antennas provide power gains? How can transmit antennas provide power gains? 20

21 Preliminaries Complex Gaussian (CG) vector : entries are CG random variables (RV); real and imaginary parts of each entry are real Gaussian RVs. mean E[x] and covariance matrix E x C n [(x E[x]) (x E[x]) ] Circularly Symmetric (CS) property: is CS if has the same distribution of x for any. e jθ x x, θ x C n θ e jθ x Denote a CSGR vector with covariance matrix x C n V = E[xx ] as CN(0, V) Unitary matrix: UU = I denotes complex conjugate transpose. 21

22 Preliminaries Singular value decomposition (SVD): any matrix H C m n can be written as H = UDV Isotropic: if the random vector, then its real w CN(0, I) w and imaginary components are i.i.d., and i.e., for any unitary matrix. U, Uw w is isotropic, projections of w onto orthogonal directions are i.i.d. CN(0, 1). For a nonnegative definite matrix A, (with equally when it is diagonal), det(a) k A ii 22

23 Discussion (5 minutes) You could use all resources (you are responsible for the correctness), e.g., google, wikipedia, books. Eigenvalues vs Singular values. Matrix rank. Positive-definite matrix. 23

24 MIMO Channel Model Time-varying composite MIMO channel response: h 1,1 (τ,t) h 1,2 (τ,t) h 1,NT (τ,t) H N R N T h 2,1 (τ,t) h 2,2 (τ,t) h 2,NT (τ,t) (τ,t)= h NR,1 (τ,t) h NR,2 (τ,t) h NR,N T (τ,t) h ij (τ,t) denotes the channel, between the i-th receive antenna and the j-th transmit antenna, response at time t to a impulse at time t τ a vector of H(τ,t) : patio-temporal signature. 24

25 MIMO Channel Model Received signal at i-th receive antenna (time-varying) reads y i (t) = N T j=1 h ij (τ,t) s j (t)+n i (t), i =1,,N R Focusing on the Time-Invariant & Flat Gaussian Channels Matrix form: y i = N T j=1 h ij s j + n i, i =1,,N R y = Hs + n, n CN(0,σ 2 I) 25

26 Eigen Channel Model H = UDV Recall SVD:. Then ỹ = D s + ñ y = Hs + n = UDV s + n = where ỹ = U y, s = V s, and ñ = U n CN(0,σ 2 I) Assume the channel state information (CSI) is available only at the receiver, the Gaussian codebook is used at the transmitter. We have the achievable rate (capacity) C =logdet ( I + HQH /σ 2) = min(n T,N R ) k=1 * Deterministic channels. How about the case of fading channels? log(1 + snrλ k ), where Q = E [ ss ] and λ k is the kth eigenvalue of HH * details see, e.g., Telatar, I. E. Capacity of multi-antenna Gaussian channels Eur. Trans. Telecomm., 1999, 10, * SVD: singular value decomposition 26

27 MIMO Receivers Maximum Likelihood (ML) Receiver with diversity order (d.o.) N T N R : minimising the error probability Linear Receivers: ŝ = arg min y Hs 2 s N R N T +1 Zero-forcing (ZF) with d.o. (each stream) : Front-end of the ZF receiver: G ZF = ( H H H ) 1 H H Output of the ZF receiver: y ZF = s + ( H H H ) 1 H H n 27

28 MIMO Receivers Linear Receivers (Cont.) Minimum-mean Square Error (MMSE) Front-end of the MMSE receiver: G MMSE = low snr G MMSE =snrh H ( H H H + 1 ) 1 snr I H H high snr G MMSE = G ZF Successive cancellation + ZF or MMSE 28

29 Multi-User (MU) MIMO user 1 Key advantages over single-user (SU) MIMO: user 2 Gain in multiple access capacity due to MU multiplexing. Base station More immune to propagation limitations (Why?): channel rank loss or antenna correlation. line of sight propagation is no longer a problem. Spatial multiplexing gain at the BS side with single-antenna devices. user K Gesbert, D.; Kountouris, M.; Heath, R.; Chae, C.-B. & Salzer, T. Shifting the MIMO Paradigm IEEE Signal Process. Mag., 2007, 24,

30 Massive MIMO Hundreds of antennas simultaneously serve tens of users. Energy-efficient, secure, robust, and spectrum-efficient. Linear Massive MIMO realise on phasecoherent Distributed Cylindrical Rectangular Examples of Massive MIMO configurations 30

31 Massive MIMO: advantages Utilising spatialdivision multiplexing: different data streams share the same frequencies and times. stream 1 stream 2 stream K Channel state information Precoding 1 M Downlink M-antenna base station stream 1 user 1 stream 2 user 2 stream K user K 10s times capacity increment Channel state information Uplink stream 1 user times improvement on radiated energy efficiency stream 1 stream 2 stream K Decoding 1 M stream 2 user 2 stream K user K M-antenna base station Marzetta, T. Massive MIMO: An Introduction Bell Labs Technical Journal, 2015, 20,

32 Theoretical background: An example Example: 2 single-antenna users, and a N-antenna base station. The received signal model of user 1 (user 2 is similar) Interference y 1 = h 1 x 1 + h 2 x 2 + n where h i CN(0, I), and n CN(0, I) user 1 user 2 Base station 1 N [h 1 h 2] 1 N [h 1 almost surely N n] almost surely N E[h 1 h 2]=0 E[h 1 n]=0 Interference is vanishing Noise is vanishing? Law of Large Number 1 N [h 1 y] almost surely N E[h 1 h 1]x 1 = x 1 (Matched filter output) We use Matlab to show this. 32

33 Theoretical background: RMT Example: K single-antenna users, and N-antenna base station. Assume H N K CN(0, I) K,N, and N/K = c>0 1 K HH a.s. I N However, the eigenvalues do not converge to 1 instead they are spreading around 1. Due to Let us have a look on the eigenvalues of for N=500, K = HH RMT: random matrix theory a.s.: almost surely, i.e., with probability 1 33

34 Theoretical background: RMT N = 500, K = 4000 N = 100, K = 1000, 5000, N/K =0.125, N = 500, 1000, 5000 histogram(x, Normalization, pdf ) to plot an estimate of the probability density function of x. 34

35 Research problems Pilot contamination and channel raging. Cost of channel reciprocity calibration. Low-cost hardware challenges: hundreds of RF chains, frequency up/down converters, A/D and D/A converters, etc. Total power consumption including baseband signal processing. Higher layers challenges, for instance, new user arrivals. Marzetta, T. Massive MIMO: An Introduction Bell Labs Technical Journal, 2015, 20, Larsson, E.; Edfors, O.; Tufvesson, F. & Marzetta, T. Massive MIMO for next generation wireless systems IEEE Commun. Mag., 2014, 52,

36 Some references Seminal paper on MIMO capacity Telatar, I. E., Capacity of multi-antenna Gaussian channels, Eur. Trans. Telecomm., 1999, 10, (cited by since 1999) Seminal paper on Massive MIMO Marzetta, T., Noncooperative Cellular Wireless with Unlimited Numbers of Base Station Antennas, IEEE Trans. Wireless Commun., 2010, 9, (cited by 2143 since 2010) Classical textbook Tse, D. & Viswanath, P. Fundamentals of Wireless Communication, Cambridge University Press, 2005 (available online) Research website on Massive MIMO Short video lectures about Massive MIMO:

37 Reading assignment for Exercise Session Marzetta, T. Massive MIMO: An Introduction Bell Labs Technical Journal, 2015, 20, Exercise session: Gallery Walk (details will be provided later). Students will work in groups. If you are not able to participate in the exercise session, a minimum 1.5- page (A4, font size 12, line spacing 1.5) is needed. Please be aware that this essay should be about your understanding instead of a summary. 37