Multiple-Input-Multiple Output (MIMO) Systems

1. Multiple-Input-Multiple Output (MIMO) Systems

2. MIMO : Features A MIMO system has 3 basic features: Beam-forming (diversity gain), Diversity (diversity gain), Spatial multiplexing (increase data rates) A combination of the 3 features is also possible depending on the number of TX and Rx elements and number of independent data streams Research in MIMO Increasing the capacity of MIMO systems Developing systems that can operate close to the capacity Primary applications 4-th generation wireless and High capacity LANs ( 802.11…) 3GPP release 7 and 8 (LTE)

3. MIMO Channel Model Transmitter With Nt TX Antennas Total TX Power = P Receiver With Nr RX Antennas X • Channel can be deterministic, or slowly changing • Channel state information (CSI) may be • Unknown to both TX and RX • Known to RX only • Known to both RX and TX

4. Parallel Decomposition of MIMO Channels

5. Modeling of MIMO Fading Channel Channel Time-variance Time delay-spread • MIMO channel response on a slow fading model a and b are transmit and receive array factor vectors respectively. S is the complex gain that is dependant on direction and delay. g(t) is the transmit and receive pulse shaping impulse response • With suitable choices of array geometry and antenna element patterns, • H( ) = H which is an Mr x Mt matrix with complex Gaussian i. i. d random variables • Accurate for NLOS rich-scattering environments, with sufficient antenna spacing at transmitter and receiver with all elements identically polarized

6. MIMO Design Criterion Spatial Multiplexing Gain Diversity Gain • Maximize transmission rate (optimistic approach) • Use rich scattering/fading to advantage • Minimize Pe (conservative approach) • Go for Reliability / QoS etc • Counter fading • MIMO Systems can provide two types of gain • As expected, there is a tradeoff • System designs are carried out to achieve a little bit of both.

7. MIMO Diversity Gain : Beamforming • Beamforming takes advantage of interference to change the directionality of the array. • Beamformer controls the phase and relative amplitude of the signal at TX • At the Rx side, information from different sensors are combined to a preferentially observed radiation pattern • Beam formers are usually smart antennas: • Phased Array Systems (Switched Beamforming) with a finite number of fixed predefined patterns. • Adaptive Array Systems (AAS) (Adaptive Beamforming) with an infinite number of patterns adjusted to the scenario in real time.

8. MIMO Diversity Gain : Beamforming • Beamforming provides diversity gain by coherent combining of the multiple signal paths. y=u*Hvx + u*n • If H is known, the received SNR is optimized by choosing u and v as the principal left and right singular vectors of the channel matrix H. • Capacity for with beamforming is given as

9. Diversity in MIMO • Each pair of transmit-receive antennas provides a signal path from transmitter to receiver. By sending the SAME information through different paths, multiple independently-faded replicas of the data symbol can be obtained at the receiver end. • A diversity gain d implies that in the high SNR region, Pe decays at a rate of 1/SNRd as opposed to 1/SNR for a SISO system • The maximal diversity gain dmax is the total number of independent signal paths that exist between the transmitter and receiver • For an (MR,MT) system, the total number of signal paths is MRMT 1 ≤d≤ dmax= MRM The higher the diversity gain, the lower the Pe

10. Spatial multiplexing in MIMO: Equivalent Channel Model • We can convert a general MIMO model to an equivalent diagonal model of some rank r ≤ min( MT,MR) • For the sake of simplicity, let us assume that H is M by M and is of full rank ; Then • Now the channel can be modeled as • Equivalent form tells us that an (MT,MR) MIMO channel opens up m = min (MT,MR) independent SISO channels between the transmitter and the receiver The capacity equation becomes

11. MIMO System: A practical approach 1 2 R bits/symbol Space-Time Coding Channel coding Symbol mapping . . MT Non-redundant portion of symbols Spectral efficiency = (R*rc info bits/symbol)(rs)(Rs symbols/sec) w = Rrcrs bits/s/Hz assuming Rs = w rs is the parameter of concern : 0 ≤ rs≤ MT If rs = MT, - spatial multiplexing mode (max transmission rate) If rs ≤ 1, - diversity mode rs : number of different symbols N transmitted in T symbol periods rs = N/T • A MIMO system schematic is given below: Redundancy in time Coding rate = rc Space- time redundancy over T symbol periods Spatial multiplexing gain = rs

12. MIMO Channel Capacity Static Channels and Fading Channels Static Channels Capacity is given in terms of mutual information b/w channel input vector x and output vector y as C= max p(x) I(X;Y)=max p(x) [H(Y)-H(Y/X)] Mutual information of y depends on covariance matrix of y given as Ry=E [yyH]=HRxHh + IMr Entropy of y is maximized when Ry is maximized which in turn means that Rx has to be maximized [6] shows that Rx is maximized when x is a ZMCSCG random vector MIMO capacity obtained by maximizing Rx is given as C=max Rx:Tr(Rx)=p Blog2det[Imr + HRxHH]

13. MIMO Channel Capacity (Static Channels) Channel Known at Transmitter: Water filling H is known at the transmitter MIMO capacity with CSIT and CSIR is C=max pi:∑ipi≤pBlog2(1+σi2pi) p=P/σn2.Expressing capacity in terms of power Pi to i-th parallel channel C=max pi:∑ipi≤pBlog2(1+ σi2 P/σn2) =max pi:∑ipi≤pBlog2(1+ µi2 Pi/P) where µi= σi2P/ σn2 ( SNR associated with ith channel at full power) Solving above gives the water-filling power allocation for the MIMO channel Pi/P = 1/µ0-1/µi for µi > µ0and 0 otherwise The resulting capacity is then C =∑i: µi > µ0 Blog (µi/ µ0) Thus CSIT converts MIMO channel into non-interfering SISO channels

14. MIMO Channel Capacity (Static Channels) contd.. Channel Unknown at Tx: Uniform Power Allocation Tx does not know channel information which means Tx cannot optimize its power allocation or input covariance structure across antennas If the distribution of H follows ZMSW distribution (zero mean and identity covariance matrix) then allocate equal power to each transmit antenna Input covariance matrix will be scaled identity matrix Rx=p/Mt IMt [2] I= Blog2 det[IMr + p/Mt HHH] Using SVD of H, I= ∑i Blog2(1+µi/Mt) Average mutual information depends on the distribution of the singular values of H

15. MIMO Channel Capacity (Static Channels) contd.. In a static channel if the average mutual information is not known then the rate of transmission is unknown. In such case capacity is given by Outage capacity ( probability that transmitted data will not be received correctly) Pout=P(H: Blog2det[IMr + p/Mt HHH] < C) For ZMSW, according to law of large numbers Mt -> ∞ 1/Mt HHH =IMr., C= MrBlog2(1+p) or C= MBlog2(1+p) where M=min (Mr, Mt) Capacity grows linearly with M for large M [3] As SNR increases the capacity grows linearly with M [4] Thus even without CSIT there is a linear growth in capacity Capacity scales with number of Rx antennas and not Tx antennas [2] Cost of linear growth of capacity: Demodulation complexity If not ZMSW, then beamforming can achieve channel capacity

16. MIMO Channel Capacity (Fading Channels) Fading Channels Channel Gain matrix experiences flat fading (H varies with time) Channel known at Tx: Water-filling Capacity is given in terms of ergodic capacity Power allocation under ergodic capacity has two possibilities Short term power constraint (Power associated with each channel realization must equal average power constraint P ) C=EH [max Rx:Tr(Rx)=p Blog2det[Imr + HRxHH]] =EH [max pi:∑ipi≤pBlog2(1+ µi2 Pi/P) ] Long term power constraint (Different powers for different channel realizations) C=max pH:E(pH)=p EH [ max Rx: Tr(Rx)=pH Blog2 det[IMr + HRxHH] Short-term power constraint give rise to water-filling in space Long-term power constraint allows for two-dimensional water-filling across both space and time

17. MIMO Channel Capacity (Fading Channels) Channel Unknown at Tx: Ergodic Capacity Ergodic Capacity: Optimization problem at Tx (Finding optimum input covariance matrix) For scalar channels, Rx=p/Mt IMt C=EH[Blog2 det[IMr + p/Mt HHH]] Like static channels, ergodic capacity also linearly scales with M for ZMSW channel

18. MIMO Channel Capacity (Fading Channels) Channel Unknown at Tx: Capacity with Outage Outage capacity can be improved by not allotting power to one or more Tx antennas, especially when outage probability is high

19. MIMO Channel Capacity (Fading Channels) • No CSI at the Tx or Rx • Linear growth in capacity as a function of number of antennas disappear • Capacity depends on the underlying channel model • Capacity depends on the structure of the fading process[5] • For general fading process, no multiplexing gain associated with multiple antennas when there is no Tx or Rx CSI

20. MIMO Channel Capacity (Fading Channels) • In the case of flat fading channel ergodic capacity or the outage capacity is used • Frequency selective fading capacity (more degrees of freedom) • Capacity Calculations are complex

21. MIMO APPLICATION: V-BLAST Vertical Bell Labs Layered Space-Time Architecture (BLAST) is a transmitter-receiver architecture used to implement multiplexing in MIMO. Transmitter: Split data into MT streams  maps to symbols  send. This process can be considered to be the encoding of the serial data into a vertical vector. Receiver: successive cancellation to recover signals. This architecture can achieve at most a diversity order of Mr, since each coded symbol is transmitted from one antenna and received by Mr antennas. Channel Data Source S/P CoderModulator CoderModulator CoderModulator Stream 2 Stream 1 Stream M

22. D-BLAST • In V-BLAST, there is no coding across these sub-channels: outage therefore occurs whenever one of these sub-channels is in a deep fade and cannot support the rate of the stream using that sub-channel. • receiver first estimates x11 obtained without interference  MMSE estimate of x12 is obtained by suppressing the interference from x21 combine x11 and x12 to decode the x1  x1 is cancelled and the process restarts with x2 • By coding across the sub-channels, D-BLAST can average over the randomness of the individual sub-channels and get better outage performance. The D-BLAST scheme suffers from a rate loss because in the initialization phase some of the antennas have to be kept silent.

23. Space-Time Coding • Receiver-diversity: maximum ratio combining & selection combining Transmitter- diversity: space-time coding • Space-Time Codes can be designed in two different ways: (1) Space-Time Block Code or STBC; (2) Space-Time Trellis Code or STTC. The first code is the easiest way to achieve Spatial Diversity and is widely used. The second code is more complex and expensive nowadays. • STBC is designed such that the vectors transmitted in any pair of transmitters are orthogonal. The result of this is simple, linear, optimal decoding at the receiver.

24. Space-Time Coding contd.. • The Alamouti Space-Time Block Code for 2 Tx antennas • It is the only orthogonal STBC that achieves rate 1. That is to say that it is the only STBC that can achieve its full diversity gain without needing to sacrifice its data rate. • For more than two antennas there are several Pseudo-Alamouti Codes. They can not achieves rate 1. • Some Quasi-Orthogonal STBC improves the data rate but permits some Inter-Symbol-Interferences (ISI). Despite this, the bit error rate (BER) is still within the tolerance range. None of these codes are able to achieve full code rate like Alamouti and the receiver become much more complex.

25. MIMO Standards • Overview of all current MIMO standards and their technologies. • Spatial multiplexing techniques makes the receivers very complex, and therefore it is typically combined with OFDM, where the problems created by multi-path channel are handled efficiently.

26. MIMO Standards contd.. • The IEEE 802.16e standard incorporates MIMO-OFDMA. The IEEE 802.11n standard, which is expected to be finalized soon, recommends MIMO-OFDM. MIMO is also planned to be used in Mobile radio telephone standards such as recent 3GPP and 3GPP2 standards. In 3GPP, High-Speed Packet Access plus (HSPA+) and Long Term Evolution (LTE) standards take MIMO into account. Moreover, to fully support cellular environments MIMO research consortia including IST-MASCOT propose to develop advanced MIMO techniques, i.e., multi-user MIMO (MU-MIMO).

27. MIMO and OFDM • Both of MIMO & OFDM need precoding (a matrix channel into a set of parallel independent sub-channels). In the OFDM setting, the matrix channel is given by the circular matrix C, defined by the ISI channel together with the cyclic prefix added onto the input symbols. In fact, the decomposition in is the SVD decomposition of a circular matrix C, with and V = Q. • The important difference between the ISI channel and the MIMO channel is that, for the former, the U and V matrices (DFTs) do not depend on the specific realization of the ISI channel while for the latter, they do depend on the specific realization of the MIMO channel.

28. MIMO and OFDM contd.. • OFDM is adapted for multi-path propagation in wireless systems. The length of the OFDM-frames is determined by the Guard Interval (GI). This Gurad Interval restricts the maximum path delay and therefore the expansion of the network area. MIMO also uses the multi-path propagation. • OFDM is a wideband system with many narrowband sub-carriers. The mathematical MIMO channel model (see chapter 2) is based on a narrow band non-frequency selective channel. The latter is supported by OFDM as well. Fading effects in wideband systems normally occur only at particular frequencies and interfere with few sub-carriers. The data is spread over all carriers, so that only a small amount of bits get lost, and these can be repaired by a forward error correction (FEC). OFDM provides a robust multi-path system suitable for MIMO. At the same time OFDM provides high spectral efficiency and a degree of freedom in spreading the time dimension of Space-Time Block Codes over several sub-carriers. This results in a stronger system based on the principle described previously.

29. MIMO-OFDM From: Tetsushi Abe, Takahiro Asai and Hirohito Suda, “A Practical Throughput Comparison of MIMO-CDMA and MIMO-OFDM”, Vehicular Technology Conference, 2004.

30. MIMO Capacity: References • [1] Andrea Goldsmith “Wireless Communication” text book • [2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T-Bell Labs Internal Memo., pp. 585–595,June 1995. • [3] A.L. Moustakas, S.H.Simon, A.M. Sengupta, “MIMO capacity through correlated channels in the presence of correlated interferers and noise: a (not so) large N analysis,” IEEE Trans. Inform. Theory, vol. 48, pp. 2545 - 2561, Oct. 2003. • [4] G. J. Foschini, “Layered space-time architecture for wireless communication in fading environments when using multi-element antennas,” Bell Labs Techn. J., pp. 41–59, Autumn 1996. • [5] A. Lapidoth and S. Moser, “On the fading number of multi-antenna systems over flat fading channels with memory and incomplete side information,” Proc. Intl. Symp. Inform. Theory, p. 478, July 2002. • [6] E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. on Telecomm. ETT, vol. 10,pp. 585–596, Nov. 1999. • [7] Goldsmith. A, Jafar S.A, Jindal N, Vishwanathan.S “Capacity limits of MIMO channels", Selected Areas in Communication IEEE Journal, vol. 21, pp 684-702, June 2003 • [8] Prof.Sam Shanmugan’s charts on MIMO and MUD • [9] R. U. Nabar A. J. Paulraj, D. A. Gore and H. B¨olcskei, “An overview of MIMO communications—a key to gigabit wireless,” Proceedings of the IEEE, vol. 92, no. 2, pp. 198–218, Feb. 2004.