MIMO: What shall we do with all these degrees of freedom?

MIMO: What shall we do with all these degrees of freedom? Helmut Bölcskei Communication Technology Laboratory, ETH Zurich June 4, 2003 c H. Bölcskei, Communication Theory Group 1

Attributes of Future Broadband Wireless Networks Significantly higher data rates than UMTS Increase in spectral efficiency required High quality of service (QoS)/Availability Fixed broadband wireless access: QoS has to be DSL/cable like c H. Bölcskei, Communication Theory Group 2

The Wireless Channel 10 0-10 -20-30 -40 Multipath propagation causes severe fluctuations in signal level c H. Bölcskei, Communication Theory Group 3

Co-Channel Interference c H. Bölcskei, Communication Theory Group 4

Summary: Challenges in Wireless Communications The wireless propagation medium is very hostile Severe fluctuations in signal level (a.k.a. fading) Co-channel interference Signal dispersion in time and frequency Signal power falls off with distance (a.k.a. path loss) Bandwidth is a scarce and often very expensive resource Future wireless systems require Significantly higher spectral efficiency High quality of service/availability Low infrastructure cost c H. Bölcskei, Communication Theory Group 5

Multiple-Input Multiple-Output (MIMO) Systems TX H RX c H. Bölcskei, Communication Theory Group 6

Leverages from MIMO Wireless Systems Spatial Multiplexing (Paulraj & Kailath, 1994) a.k.a. BLAST (Foschini, 1996) yields substantial increase in data rate in wireless radio links. Receive diversity and transmit diversity (Alamouti, 1998, Tarokh et al., 1998) mitigate fading and significantly improve link quality. Array gain through coherent combining increases signal to noise ratio improved coverage. Reduction of co-channel interference increases cellular capacity. These goals are mutually conflicting. Clever balancing of competing goals required to maximize performance. c H. Bölcskei, Communication Theory Group 7

Spatial Multiplexing c H. Bölcskei, Communication Theory Group 8

Spatial Multiplexing Cont d Requires multiple antennas at both ends of radio link. Increase in data rate by transmitting independent information streams on different antennas. No channel knowledge at transmitter required. If scattering is rich enough (i.e. high rank channel H) several spatial data pipes are created within the same bandwidth. Multiplexing gain comes at no extra bandwidth or power. c H. Bölcskei, Communication Theory Group 9

Mitigation of Fading 0 0-5 Desired Signal -5 Desired Signal -10-10 db -15-20 db -15-20 Interferer -25-25 -30-35 Interferer -30-35 -40 0 10 20 30 40 50 60 70 80 90 Time (s) -40 0 10 20 30 40 50 60 70 80 90 Time (s) Antenna diversity stabilizes the link and reduces co-channel interference significantly. c H. Bölcskei, Communication Theory Group 10

Array Gain Tx Array Gain Rx Array Gain X X X X X X Array gain: log 10 N in receive or transmit for N 1 and 1 N arrays c H. Bölcskei, Communication Theory Group 11

Co-Channel Interference Reduction Tx CCI Avoidance Rx CCI Cancellation X X X X X X Can cancel N 1 interferers with N receive antennas. Can avoid N 1 interferees with N transmit antennas. c H. Bölcskei, Communication Theory Group 12

Summary: MIMO Gains MIMO wireless systems improve Spectral efficiency: Multiplexing gain Link reliability: Diversity gain Coverage: Diversity gain and array gain Cellular capacity: Co-channel interference reduction c H. Bölcskei, Communication Theory Group 13

Throughput in MIMO Cellular Systems 1 1 1 2 2 3 0.0Mbps 1.1Mbps 2.2Mbps 3.4Mbps 4.5Mbps 5.6Mbps 6.7Mbps 9.0Mbps 11.2Mbps13.5Mbps MIMO cellular systems offer increased coverage and capacity c H. Bölcskei, Communication Theory Group 14

Channel and Signal Models r = Hs + n Ergodic block-fading i.i.d. complex Gaussian H H is known at the receiver and unknown at the transmitter M T... number of transmit antennas M R... number of receive antennas Mutual information given by I = log 2 det ( I MR + ρ ) HH H M T bps/hz with ρ denoting the SNR per receive antenna. We restrict our attention to the high-snr case. c H. Bölcskei, Communication Theory Group 15

Ergodic Capacity For L = min(m T, M R ) and K = max(m T, M R ), the ergodic capacity is given by ( ) ρ C(ρ) = E{I} L log 2 M T + 1 ln 2 L j=1 K j p=1 1 p γl, where γ 0.577 (Euler s constant). The ergodic capacity grows linearly in the minimum of the number of transmit and receive antennas (Foschini, 1996, Telatar, 1995). c H. Bölcskei, Communication Theory Group 16

Definition: Multiplexing Gain Intuition: Multiplexing gain is the number of parallel spatial data pipes in the same frequency band between transmitter and receiver. We define the multiplexing gain as m = lim ρ C(ρ) log 2 ρ = L. c H. Bölcskei, Communication Theory Group 17

Definition: Diversity Order (SIMO Case) Intuition: Diversity order is the number of independently fading signal paths between transmitter and receiver. Fact: If the diversity order goes to infinity the fading channel approaches an AWGN channel (Jakes, 1974). For a SIMO system with M R receive antennas, we have σ 2 I (log 2 e) 2 M R. In the single-stream (m = 1) case, we define the effective diversity order as d(1) = (log 2 e) 2 σi 2 = M R. c H. Bölcskei, Communication Theory Group 18

Definition: Diversity Order (MIMO Case) Given a multiplexing gain of m, what is the effective diversity order experienced by the individual streams? We define the per-stream diversity order as m j=1 d(m) = (log 2 e) 2 σ 2 I /m m 1 1 K j+1 where we used a result by N. R. Goodman, Ann. Math. Stat., 1963., c H. Bölcskei, Communication Theory Group 19

Operational Meaning Mutual information obtained by coding over N independently fading blocks I (N) = 1 N I k N where the I k are i.i.d. with I k log 2 det k=1 ( I MR + ρ ) HH H. M T Ergodic capacity achieved by coding over infinitely many independently fading blocks. C = lim N I(N) c H. Bölcskei, Communication Theory Group 20

Operational Meaning Cont d Variance of I (N) given by σ 2 I (N) = 1 N σ2 I determines level to which mutual information fluctuations are stabilized to ergodic capacity (level of channel hardening). Higher per-stream diversity order requires coding over fewer independently fading blocks to achieve a certain level of channel hardening. c H. Bölcskei, Communication Theory Group 21

Stream Separation Penalty Simple example: M T = 2 and M R 2 so that m = 2. Question: What is the per-stream diversity order? Wrong answer: Total number of degrees of freedom is 2M R. Number of independent streams is 2 Per-stream diversity order is M R. Correct answer: The per-stream diversity order is given by d(2) = M R }{{} orthogonal muxing 2M R 2 2M R 1 < M R. Reduction in per-stream diversity order can be attributed to stream separation penalty. c H. Bölcskei, Communication Theory Group 22

Stream Separation Penalty Cont d 0.7 MT=2 MT=3 MT=4 0.6 0.5 Separation penalty s 0.4 0.3 0.2 0.1 0 4 6 8 10 12 14 16 18 20 Number of receive antennas MR Stream separation penalty as a function of M R for M T = 2, 3, 4. c H. Bölcskei, Communication Theory Group 23

The Multiplexing-Diversity Tradeoff Curve Multiplexing-diversity tradeoff curve tells us how much diversity each stream can get if a multitude of independent streams is spatially multiplexed. The multiplexing-diversity tradeoff curve is given by where K = max(m T, M R ). d(m) = m m j=1 1 1 K j+1 L. Zheng and D. Tse, 2001, describe a multiplexing-diversity tradeoff using different definitions for multiplexing and diversity gains (based on outage capacity and error probability)., c H. Bölcskei, Communication Theory Group 24

Multiplexing-Diversity Tradeoff Curve Cont d 1 K=20 K=10 0.9 Normalized per stream diversity order d(m)/k 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 2 4 6 8 10 12 14 16 18 20 Multiplexing gain m Multiplexing-diversity tradeoff curve for K = 10, 20. c H. Bölcskei, Communication Theory Group 25

Low Loading Diversity Spatial multiplexing Adding extra receive antennas offers additional diversity gain beyond self-diversity. c H. Bölcskei, Communication Theory Group 26

Full Loading Diversity Spatial multiplexing Only self-diversity is available. c H. Bölcskei, Communication Theory Group 27

Overloading Diversity Spatial multiplexing Beyond full loading extra antennas at the transmitter can only increase diversity gain. c H. Bölcskei, Communication Theory Group 28

Multiplexing-Diversity Tradeoff for Fixed M R 16 14 Per stream diversity order d(m) 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 Number of transmit antennas MT Multiplexing-diversity tradeoff curve for M R = 10. c H. Bölcskei, Communication Theory Group 29

Multiplexing-Diversity Tradeoff for ZF Receiver 10 9 Optimum ZF 8 Normalized per stream diversity order d(m) 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Multiplexing gain m Multiplexing-diversity tradeoff curve for M R = 10. c H. Bölcskei, Communication Theory Group 30

Impact of Co-Channel Interference Assume co-channel interference such that r = Hs + i + n The interfering signal is assumed to be I-dimensional with large interference-to-noise ratio in each dimension. Interference is Gaussian and unknown at the receiver. c H. Bölcskei, Communication Theory Group 31

Multiplexing-Diversity-Interference Canceling Tradeoffs In the presence of an I-dimensional interferer, the multiplexing gain is given by m = min(m T, M R I) The multiplexing-diversity tradeoff curve is obtained as d(m) = m where K = max(m T, M R I). m j=1 1 1 K j+1, The I-dimensional interferer essentially knocks off I receive antennas. c H. Bölcskei, Communication Theory Group 32

Multiplexing-Interference Canceling Tradeoff 12 10 8 Multiplexing gain m 6 4 2 0 1 2 3 4 5 6 7 8 9 10 Dimensionality of interferer I Multiplexing gain as a function of I for M R = 15 and M T = 10. c H. Bölcskei, Communication Theory Group 33

Diversity-Interference Canceling Tradeoff 9 8 7 Per stream diversity order d(m) 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Dimensionality of interferer I Per-stream diversity order as a function of I for M R = 15 and M T = 10. c H. Bölcskei, Communication Theory Group 34

Conclusion MIMO channels offer multiplexing gain, diversity gain, interference canceling gain, and array gain. MIMO system design requires careful balancing between these gains. We introduced a simple information-theoretic framework for quantifying the fundamental tradeoffs between MIMO gains. Our approach can easily be generalized to encompass the frequency-selective and/or spatially correlated cases. c H. Bölcskei, Communication Theory Group 35