Wireless Communication Multiple Input - Multiple Output Systems

1. Wireless CommunicationMultiple Input - Multiple Output Systems Sharif University of Technology Fall 1396 AfshinHemmatyar

2. Traditionally used in military systems for DOA (Direction Of Arrival) estimation. • Various DOA estimation techniques developed in 80’s (MUSIC, ESPRIT,..). • In early 90’s interest shifted toward commercial mobile technology. • Bell Labs (Winters) and Stanford (Paulraj) started this new wave, but little support for many years (little belief in practicality!). Introduction (1)

3. Entrance of Information theory community ignited new ideas and made the issue more interesting for both academia and industry (Foschini, Telatar, Tarokh, Alamouti, Jafarkhani, . . .). • As one of the first widespread applications, Space-Time codes included in 3G standard in 2003 (Alamouti code, 2 antennas at TX side). • Proposals in 2005 for 802.11n system focused on multiple antennas (at least two TX antenna, also MIMO cases considered). • Now quite active in WiMax systems as well. Introduction (2)

4. A multi-facet topic and hard to present a unified view. • Can be viewed from different aspects: • Diversity: Traditionally first use of multiple antennas, but were mainly used at RX side. • Beam-forming: Traditional military applications. • Coding: Mainly started by Foschini and Tarokh papers. • Multiple Antennas at • TX (MISO) • RX(SIMO) • Both TX and RX (MIMO) Multiple Antennas

5. MIMO results in four major performance improvements: • Array Gain • Diversity Gain • Spatial Multiplexing Gain • Interference Reduction Gain • In general it is not possible to take advantage of all above improvements due to conflicting demands. • One famous tradeoff is between Diversity and Multiplexing gain, known as DMT modulation (Discrete Multi-Tone). Performance Improvements (1)

6. Array Gain • Increase in average SNR due to coherent combining. • Requires channel knowledge of transmitter and receiver. • Depends on number of transmit and receive antennas. • Diversity Gain • Diversity mitigates fading in wireless links. • MIMO links composing independently fading MIMO channels can lead to higher order diversity as compared to SISO link (diversity order is slope of BER curve vs SNR in log-log plots). • Interestingly, can be achieved in the absence of channel knowledge at the transmitter by designing suitable transmit signals (Space-Time coding). Performance Improvements (2)

7. Spatial Multiplexing Gain • Transmit independent data signals from individual antennas. • Receiver can extract different streams under uncorrelated fading channel conditions (rich scattering). • A linear increase (in min(MT, MR)) in capacity for no additional power or bandwidth expenditure is obtained. • Interference Reduction • Differentiation between the spatial signatures of the desired channel and co-channel signals is exploited to reduce interference. • Requires knowledge of desired signal’s channel. • Smart antenna system: Beam-forming at transmitter through switched beam or adaptive array. • Aggressive frequency reuse and increase in multi-cell capacity. Performance Improvements (3)

8. More hardware and circuitry complexity. • More channel coefficients to be estimated (for coherent detection). • Requires the antennas to be sufficiently de-correlated. • Difficult to implement for mobile terminals. Disadvantages

9. Multiple RX antennas: L antennas sufficiently apart, get L independent Branches (one λ for mobile unit, 10s of λ for BTS). • Multiple antennas have two types of effects: • Power gain: with 2 antennas, we get 3db gain (even if signals are correlated over antennas). • Diversity gain: due to independent fading, probability of overall lower gain decreases. • A formal definition of diversity gain is: Receive Diversity

10. If we draw BER vs SNR on log-log plots, the slope at high SNRs is a measure of diversity. • The curve can be shifted by changing the power/coding gain. Diversity vs Power/Coding Gain (1)

11. Although for power gain by adding one antenna we only gain 3dB, when we move from L=1 to L=2 antennas, the performance improvement can be quite big, around 20dB at Pe=10-5. • But unlike power gain, this improvement reduces as we increase L (diminishing returns). Diversity vs Power/Coding Gain (2)

12. Now consider L transmit antennas and one receive antenna. • More popular in downlink of cellular systems. • If we know channel at TX side, we can do transmit beam-forming to get power gain. • If not, we can still get diversity gain by some simple options: • Send each symbol over a different antenna and turn off the others during that period. • Code signals over time with length L codes and then send them over antennas (one at a time). • Use specific transmit diversity codes, also known as Space-time codes. • Most widespread one: Alamouti code Transmit Diversity

13. The basic Alamouti code: Two TX and one RX antenna • Channel: flat fading: [h1, h2] • Take two complex symbols: u1, u2 • At time 1:Send u1 over antenna 1 and u2 over antenna 2 • At time 2:Send –u2* over antenna 1 and u1* over antenna 2 • Equivalently, we can rewrite received signal at RX side, at times 1 and 2 as: • Columns are orthogonal, so we are sending the two symbols in two orthogonal directions, [h1h2*] and [h2 -h1*], thus we have two parallel channels. Alamouti Space-Time Code (1)

14. Consequently, we have two independent scalar detection problems at RX side and a linear receiver, with the knowledge of channel, can detect u1 and u2: • u1= [h1h2] [y1y2*]T and u2= [h2* -h1*] [y1y2*]T • We also have a diversity of 2 with Alamouti code: Alamouti Space-Time Code (2)

15. Alamouti Space-Time Code (3) Alamouti Code vs Repetition Code • If we send one bit per symbol, during two symbols, we send 2 bits. • If we want to send two bits with repetition code, we need to use two-bit symbols (for example 4-PAM) and repeat them over the two antennas (one antenna off at a time). • With same minimum distance, we need 5 times more energy per symbol (by going from BPSK to 4-PAM), but use half energy at a time, thus 2.5 times more energy for same rate repetition code. • Thus we have 4 dB improvement with Alamouti code.

16. Alamouti Code vs Repetition Code • by Repetition code we will have: • Significant loss at high SNR • Negligible loss at low SNR Alamouti Space-Time Code (4)

17. Simple extension to two RX antenna. • If we assume enough antenna spacing at TX and RX sides, we will have 4 independent channels hij(i,j =1, 2). • Therefore, a maximum diversity gain of 4 can be expected. • If we use repetition code as before, at time 1, we get [h11 h12] and at time 2, we get [h21 h22] at RX side. • Therefore, MRC receiver will get Σ|hij|2 and so a diversity gain of 4. • However, we are only sending one symbol during 2 symbols period. • By using Alamouti code, we send two symbols during same period and also 2 independent channels with gain in each channel equal to Σ|hij|2 , thus we have diversity gain of 4. Extension to 2x2 MIMO Case

18. Although Alamouti achieves very nice features in terms of diversity and decoding, we are not efficiently using the two antennas to increase the rate. • One option to improve the rate is to multiplex the input stream into two independent paths and send them over two antennas: • At time 1: send u1 and u2 over two antennas • At time 2: send u3 and u4 over two antennas Spatial Multiplexing (V-BLAST) (1)

19. Since on TX side, we are sending one symbol only through one antenna, we only get at most a diversity of 2 from RX side (for 2x2 MIMO case). • However, we have two times packing of bits and so a better coding gain. • Error bounds (effect of Coding Gain and Diversity): • Alamouti: 1600/SNR4 • V-BLAST: 4/SNR2 Spatial Multiplexing (V-BLAST) (2)

20. Here, since detection of two symbols does not separate, we have to use joint detection of two symbols. • ML solution: keeps diversity of 2 and also no performance degradation ; but has exponential complexity with number of antennas. • One suboptimal solution is to use channel inversion:y = Hx+w  r = H-1y (also called decorrelator or ZF receiver) • This will remove interference from second antenna by projecting h1 over the direction orthogonal to h2. • This will result in a scalar number and so diversity reduces to 1. • This will also make noise correlated and so performance is not as good as ML detector. Spatial Multiplexing (V-BLAST) (3)

21. V-BLAST can also be seen as a multiuser scheme where each transmit antenna sends stream of different users. • For example in uplink, where multiple antennas are only used at RX side, which also called interference nuller. • As mentioned before, this will reduce diversity to 1 but will null interference. • However, by using ML we can support two users and also a diversity of two. • There are methods known as successive interference cancellation (SIC) that are used for multiuser receivers and can be re-used for V-BLAST as well. Spatial Multiplexing (V-BLAST) (4)

22. So, just in comparing between Alamouti and V-BLAST, we see different metrics for comparison: Diversity Gain and Coding Gain. • Obviously, for two codes with same diversity, coding gain is a good comparison measure, but if diversity is different, how we can compare them? Diversity vs Multiplexing Gain (1)

23. Full channel decomposition maximizes spatial multiplexing at the expense of no diversity/array gains. • Full diversity/array gains with all antennas used for diversity, no spatial multiplexing gain. • Compromise possible by using some antennas for diversity and some for spatial multiplexing. • Adapting diversity and multiplexing gains to channel conditions is an interesting possibility. • Good channels: more antennas for spatial multiplexing • Poor channels: more antennas for diversity Diversity vs Multiplexing Gain (2)

24. For block fading channels in the limit of asymptotically high SNR transmission scheme achieves multiplexing gain “r“ and diversity gain “d”, if BER“Pe” and data rate “R” (bps) satisfy: • For each r, optimal diversity gain, achievable by any scheme, is denoted by dopt (r). Diversity vs Multiplexing Gain (3)

25. Diversity vs Multiplexing Gain (4)

26. MIMO: t transmit antennas connected to r receive antennas via a wireless fading channel, with the following options: • Space-time block coding: no CSI at TX, diversity gain, robust to interference. • Space-time trellis coding: no CSI at TX, diversity & coding gains, robust to interference. • Spatial multiplexing: reliable CSI at TX, multiplexing gains, susceptible to interference. • Beam-forming: CSI at TX, power gains, minimizing interference. Multiple Input- Multiple Output (1)

27. So far, the V-BLAST and Alamouti codes did not use CSI information at TX side. • To use channel information at TX side, we can go back to the original input-output equation and use knowledge of channel at transmitter side. • In OFDM case, we used SVD (Singular Value Decomposition) of channel matrix to achieve a set of parallel channels. • We saw that by using Cyclic Prefix, the SVD of the resulting circulant matrix contained DFT matrices. • In MIMO case, if we assume full CSIT, we can translate the problem into a general framework by using the SVD of Hnrxnt. Performance Improvement with CSIT (1)

28. Parallel Channels in OFDM • In fact, in OFDM, by using CP, we obtained cyclic convolution relationship between input-output of channel: y=h⊗x • In matrix form, we would then have: Y=HX where, H was a circulant matrix with SVD of the form: H=U-1ΛU where U is a unitary DFT matrix with elements: and diagonal elements of Λ(λi’s) are DFTs of channel vector (or eigen-values of the channel) • In this way, we obtained a set of parallel channels in terms of DFT coefficients as Yi = Hi Xi Performance Improvement with CSIT (2)

29. Y = HX + W H is nr by nt, fixed channel matrix. • H = UΛV* U and V are unitary matrices (UU*=1 , VV*=1) and Λ is a diagonal matrix with entries λi (singular values of H), i=1, . . ., nmin = min(nr, nt). • By defining X=VX’, X’=U*X ,y’=U*Y, we will get: Y’= U*Y=U*(UΛV*)VX’+ U*W = ΛX’+ W’ • In this way, our channel will reduce to a set of parallel independent channels again! • Note: In OFDM, U was the DFT matrix and so independent of channel, but in MIMO, U and V depend on channel matrix H. Parallel Channels in MIMO (1)

30. The number of parallel channels is at most nmin=rank(H) • We have seen the capacity of such parallel channels before for fixed frequency selective and also fading channels. • Now we can use water-filling in spatial domain to find optimum distribution of power among input streams to maximize the rate. • Can decompose the MIMO channel into a bounch of orthogonal sub-channels. • Can allocate power and rate to each sub-channel according to water-filling. Parallel Channels in MIMO (2)

31. Parallel Channels in MIMO (3)

32. In the case of parallel channels, the capacity for MIMO is given by: where Pi’s are equal to: • In this way, each eigen-mode of channel can support an independent data stream, so MIMO can support spatial multiplexing of multiple streams in an optimum way. Parallel Channels in MIMO (4)

33. For high SNR, equal power level is close to optimal: • So capacity is proportional to k=rank(H) (in a “rich” channel, k = nmin) • For low SNR, optimal policy is to assign power to strongest channels, which in this case is the strongest singular value (ln(1+x) ≅x): • In this case, rank is not important and we are actually using antennas with rank one: also known as beam-forming which gives us optimum power gain. Parallel Channels in MIMO (5)

34. With no info at TX side, for a “rich” channel H, optimal transmit is by putting equal power in all eigen-mode directions: • At high SNR, capacity is proportional to nminlog(SNR) bit/s/Hz: • So the capacity of a 1xnr system is not much different from a 1x1 SISO system and we only have a power gain. Capacity without CSIT and Fast Fading (1)

35. At low SNR, capacity is proportional to nr*SNR*log2e, also known as receive beam-forming power gain. • Even with nt TX antennas, we do not get much benefit from them, since we do not know channel at TX side. • Effectively, the system is performing similar to a single transmit antenna case. • To summarize, for n=nt=nr case, at all SNRs, capacity will be linearly proportional to n. Capacity without CSIT and Fast Fading (2)

36. In slow fading, performance characterized by outage probability and outage capacity. • At low SNRs, transmitter effectively uses only one antenna and performance similar to multiple antennas only at RX side. • At high SNRs, MIMO channel provides high diversity gain of the order of nrnt (number of independent random variables in channel H). • In this case V-BLAST is only sub-optimal, and can not achieve full diversity. Capacity without CSIT and Slow Fading (1)

37. In V-BLAST, for each TX antenna stream, diversity is at most nr, but due to interference between streams, diversity is lower (nr-(nt-k)) for kth antenna stream decoded. • In V-BLAST, in slow fading, if random SINR of any stream can not support rate allocated to it , then V-BLAST will be in outage: log2(1+SINRk) < R • Need to code across transmit antennas to improve performance: (D-BLAST). Capacity without CSIT and Slow Fading (2)

38. Earlier we have discussed multiuser diversity for SISO systems. • The question is how to intelligently exploit multiuser diversity for MIMO scenarios? • We have seen before that scheduling algorithms exploit the nature-given channel fluctuations by hitting the peaks for the best user. • In MIMO case, we have one extra degree of freedom to form beams and intentionally change such beams over time. • So, if there are not enough fluctuations, we can purposely induce them! Multiuser MIMO (1)

39. Slow Fading Channel Multiuser MIMO (2)

40. Inducing Change in the Channel • The information bearing signal at each of the transmit antenna is multiplied by a random time-varying phase. Multiuser MIMO (3)

41. Channel after Induced Change Multiuser MIMO (4)

42. Beam-forming Omni-directional Antenna Antenna Array (Beam-forming) • Beam-forming direction is controlled by the relative phase θ(t). • Multiple antennas sweep a beam over all directions. Multiuser MIMO (5) 42

43. Random Beam-forming ( Single User) • For the single user scenario, if we just randomly form a beam, most of the time, the beam is not near the user. Multiuser MIMO (6)

44. Random Beam-forming (Many Users) • However, in a large system with many users, it is likely that there is a user near the beam at any time. • By transmitting to that user, close to true beam-forming performance is achieved without knowing the locations of the users. Multiuser MIMO (7)

45. Opportunistic vs. True Beam-forming • If the gain h1k and h2k are known at the transmitter, then True beam-forming can be performed: • Dumb antennas randomly sweep out a beam opportunistically sends data to the user closest to the beam. • Opportunistic beam-forming can approach the performance of true beam-forming when there are many users in the system, but with much less feedback and channel measurements. Multiuser MIMO (8)

46. Opportunistic Transmission vs. Space-Time Coding • Space-time codes improve reliability of point-to-point links but reduce multiuser diversity gain. • Random BF adds fluctuations to point-to-point links but increases multiuser diversity gains. Multiuser MIMO (9)

47. Opportunistic Transmission vs. Conventional M.A. Multiuser MIMO (10)

48. MIMO systems have multiple transmit and receiver antennas. • With perfect channel estimates at TX and RX, decomposes into r independent channels. • r-fold capacity increase over SISO system • Just TX or RX antennas will not lead to additional capacity (only power gain which is helpful at low SNRs). • But, with MIMO, we can improve capacity as well (especially at high SNRs). • With n TX and n RX antennas, capacity can increase by a factor of n as well. Summary