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

MIMO: What shall we do with all these

degrees of freedom?

Helmut B¨olcskei

Communication Technology Laboratory, ETH Zurich

Attributes of Future Broadband Wireless Networks

• Significantly higher data rates than UMTS

– Increase in spectral efficiency required

• High quality of service (QoS)/Availability

The Wireless Channel

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

Multiple-Input Multiple-Output (MIMO) Systems

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.

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.

Mitigation of Fading

Desired Signal Desired Signal

“Antenna diversity stabilizes the link and reduces co-channel interference significantly.”

Array Gain

Tx Array Gain Rx Array Gain

Array gain:

Co-Channel Interference Reduction

Tx CCI Avoidance Rx CCI Cancellation

• Can cancel N−1 interferers with N receive antennas.

Summary: MIMO Gains

MIMO wireless systems improve

• Spectral efficiency: Multiplexing gain

• Link reliability: Diversity gain

• Coverage: Diversity gain and array gain

Throughput in MIMO Cellular Systems

1 × 1 1 × 2 2 × 3

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}

– MT ... number of transmit antennas

MR ... number of receive antennas

• Mutual information given by

I = log₂ det I_M R + ρ MT HHH bps/Hz

with ρ denoting the SNR per receive antenna.

Ergodic Capacity

• For L = min(MT, MR) and K = max(MT, MR), the ergodic capacity is given by C(ρ) = E{I} ≈ L log₂ ρ MT + 1 ln 2   L X j=1 K−j X p=1 1 p − γL  ,

where γ ≈ 0.577 (Euler’s constant).

• The ergodic capacity grows linearly in the minimum of the number of

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)

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 MR receive antennas, we have

σ_I2 ≈ (log2 e)

2

MR

.

• In the single-stream (m = 1) case, we define the effective diversity

order as

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

d(m) =∆ (log2 e) 2 σ_I2/m ≈ m 1 Pm j=1 K−1j+1 ,

Operational Meaning

• Mutual information obtained by coding over N independently fading

blocks I(N) = 1 N N X k=1 Ik where the Ik are i.i.d. with

Ik ∼ log₂ det I_M R + ρ MT HHH .

• Ergodic capacity achieved by coding over infinitely many

independently fading blocks.

Operational Meaning Cont’d

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

Stream Separation Penalty

• Simple example: MT = 2 and MR ≥ 2 so that m = 2.

Question: What is the per-stream diversity order?

• Wrong answer: Total number of degrees of freedom is 2MR. Number

of independent streams is 2 ⇒ Per-stream diversity order is MR.

• Correct answer: The per-stream diversity order is given by

d(2) = MR |{z} orthogonal muxing 2MR − 2 2MR − 1 < MR.

Stream Separation Penalty Cont’d

Number of receive antennas MR

Separation penalty s

MT=2 MT=3 MT=4

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

d(m) = m_Pm 1

j=1 K−1j+1

,

where K = max(MT, MR).

• L. Zheng and D. Tse, 2001, describe a multiplexing-diversity tradeoff

Multiplexing-Diversity Tradeoff Curve Cont’d

Normalized per−stream diversity order d(m)/K

Multiplexing gain m

K=20 K=10

Low Loading

Spatial multiplexing Diversity

Full Loading

Spatial multiplexing Diversity

Overloading

Spatial multiplexing Diversity

Multiplexing-Diversity Tradeoff for Fixed

_M

Number of transmit antennas MT

Per−stream diversity order d(m)

Multiplexing-Diversity Tradeoff for ZF Receiver

Normalized per−stream diversity order d(m)

Optimum ZF

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.

Multiplexing-Diversity-Interference Canceling Tradeoffs

• In the presence of an I-dimensional interferer, the multiplexing gain

is given by

m = min(MT, MR − I)

• The multiplexing-diversity tradeoff curve is obtained as

d(m) = m_Pm 1

j=1 K−1j+1

,

where K = max(MT, MR − I).

Multiplexing-Interference Canceling Tradeoff

Diversity-Interference Canceling Tradeoff

Per−stream diversity order d(m)

Dimensionality of interferer I

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