Ergodic Capacity of MIMO

Relay Channel

Bo Wang and Junshan Zhang

Dept. of Electrical Engineering

Arizona State University

Anders Host-Madsen

Dept. of Electrical Engineering

University of Hawaii

CISS’ 2004

Outline

Introduction Review of capacity bounds for fixed channel case Bounds on ergodic capacity over Rayleigh fading Discussions on achievability on ergodic capacity:

High SNR case Scalar channel case

Numerical results Conclusion

MIMO Relay Channel Model

Vector relay channel -- source, destination and/or relay are equipped with multiple antennas

Cont’d

Signal model

, and : , and independent matrices SNR parameters:

Power constraints: Noise vectors:

2H 3H

Capacity Bounds of Relay Channel Upper bound (max-flow min-cut)

General channel [Cover & El Gamal 79]

Degraded channel: achieve upper bound Lower bound – achievable rates

MIMO Relay Channel Capacity Challenges

Non-degraded Vector channel: maximization over matrices

We study capacity bounds [Wang-Zhang03] Fixed Channel case Rayleigh fading channel case

Upper Bound: Fixed Channel Case Theorem 1: An upper bound on capacity of MIMO

relay channel is given by

where and

Capacity Bounds: Rayleigh Fading Case Upper bound on ergodic capacity over Rayleigh

fading (receiver CSI only)

Theorem 2:

a). An upper bound on ergodic capacity is given by

Cont’d

b). A lower bound on ergodic capacity is given by

Some Intuition

Upper bound and lower bound can “meet” under certain conditions

Ergodic capacity can be characterized exactly; previously, this was shown only for degraded relay channel (fixed channel case)

Independent codebooks at source and relay Channel uncertainty (randomness) at transmitters

make and independent Relaying improves capacity by achieving MAC gain

and BC gain

1X 2X

Some Intuition ( cont’d)

Question: sufficient conditions for achieving ergodic capacity? Recall upper bound and lower bound: common term Ergodic Capacity can be Observation: If and , upper bound meets

lower bound at

RR CC 12 RR CC 32

RC2RC2

RC2

Outline of Proof: Upper Bound on Ergodic Capacity Apply Gaussian codebooks

,

Cont’d Choosing

maximizes :

The same distributions maximize

Thanks Dr. Kramer for his comments on this proof.

Outline of Proof: Lower Bound on Ergodic Capacity Without relay: single-user MIMO channel With relay, following rate can be achieved

Consider fading:

Independent input signals maximize above rates

Conditions on Capacity Achievability Numbers of antennas = 2 in all cases Case I:

Case II:

Case III:

Case I: 321

Case II: and 21 23 10

Case III : and , upper bound meets lower bound

21

32 21 10

Discussions on Capacity Achievability Assume numbers of antennas Upper bound is given by , iff Remains to find conditions for An upper bound on

Study two cases: High SNR Regime Scalar Channel case

N

RR CC 32

RC2

RC2

High SNR Regime

Approximate by

Approximate upper bound on by

High SNR Regime (cont’d)

Example:

Sufficient conditions for achieving capacity can be viewed as a generalization of “degradedness” to fading channels

Scalar Channel

Compute and

Compare them to find sufficient conditions

Conclusion and Future Work Study upper bounds and lower bounds on capacity of

MIMO relay channel over Rayleigh fading (full version at www.eas.asu.edu/~junshan/)

For equal numbers of antenna cases: Find sufficient conditions for achieving ergodic capacity Sufficient conditions can be viewed as a generalization of

“degradedness” to fading channels Future work on correlated fading channel and partial CSI cases, and

study sufficient conditions for achieving ergodic capacity

Other results: ergodic capacity and power allocation for relay channel over Rayleigh fading, for both full-duplex and time-division cases. (Host-Madsen & Zhang03)

Questions?

& Thank You!

Single-user MIMO Channel Capacity Channel Model: Capacity of fixed channel:

Ergodic capacity: time varying, receiver CSI only [Telatar 99, Foschini & Gans 98]