The Role of SNR in Achieving MIMO Rates in Cooperative Systems

1. The Role of SNR in Achieving MIMO Rates in Cooperative Systems Chris T. K. Ng, Stanford University J. Nicholas Laneman, U. of Notre Dame Andrea J. Goldsmith, Stanford University

2. 2 Cooperation in Wireless Networks MIMO systems achieve full multiplexing gain. But a mobile device may not be able to accommodate multiple antennas. Cooperation has been proposed to improve wireless network reliability and capacity. Each node has a single antenna. Nodes that are close together can cooperate by exchanging messages to form a virtual MIMO. Is cooperation effective in improving capacity?

3. 3 Background:Cooperative Multiplexing Gain Multiplexing gain: Defined at asymptotically high SNR. The pre-log factor: MIMO multiplexing gain: Cooperative systems N transmitter nodes, N receiver nodes [Host-Madsen& Nosratinia�05]. Cooperative multiplexing gain was conjectured to be 1. Cooperative systems lack full multiplexing gain.

4. 4 Introduction: Non-asymptotic Cooperative Capacity Gain We consider cooperative capacity gain in the non-asymptotic regime. Moderate SNR. Fixed number of cooperating antennas. Cooperative system performs at least as well as a MIMO system with isotropic inputs. Up to an SNR threshold. The SNR threshold depends on network geometry, and the number of antennas.

5. 5 System Model: Motivation A general M-transmitter cluster and M-receiver cluster Gaussian network. Multi-dimensional capacity region intractable. Modeled as a multiple-antenna Gaussian relay channel. Optimistic model: performance upper bound as some of the nodes can cooperate at no cost.

6. 6 System Model: Multiple-antenna Relay Channel Discrete-time frequency-flat block-fading AWGN. Phase fading: All nodes have perfect CSI; Tx can adapt to the channel. Normalize transmit power per antenna: Power at the transmit cluster: P (SNR of the system).

7. 7 Capacity of the Cooperative System Gaussian multiple-antenna relay channel. Capacity is an open problem; bounds in [Wang et al.�05]. Cut-set bound and DF rate: Optimal input covariance matrix depends on the channel realization, and is hard to compute. We derive channel-independent upper and lower relay channel capacity bounds. Compare to the MIMO channel capacity. Characterize cooperative capacity gain in low-SNR, and high-SNR regions.

8. 8 Optimal input covariance matrix hard to compute. Non-convex, depends on channel realization. Cut-set Bound

9. 9 Capacity Upper Bound Details Channel-independent capacity upper bound: Upper bound is loose.

10. 10 Decode-and-forward Achievable Rate Decode-and-forward rate Relay is close to the transmitter. Relay fully decodes the message. Optimal input covariance matrix: Depends on channel realization. Hard to compute.

11. 11 DF Rate Lower Bound Lower bound by choosing a particular input covariance matrix: Isotropic inputs (equal power, uncorrelated). Input covariance matrix = identity matrix IM . Numerical results show that the lower bound is tight.

12. 12 Low-SNR and High-SNR Regions MIMO-gain Region SNR threshold lower bound: Cooperative capacity is at least as high as isotropic-input MIMO. Coordination-limited Region SNR threshold upper bound: Cooperative capacity is strictly less than orthogonal MIMO.

13. 13 Numerical Results: SNR Thresholds Numerically solve M-th degree polynomial. The SNR threshold bounds are almost equal as g is large. Large g : extends MIMO-gain region. Large M : coordination-limited region sets in at a lower SNR.

14. 14 Numerical Example: Capacity of a 2x2 Cooperative System Tx, relay: single-antenna. Rx: 2 antennas. Relay close to Tx (g =100). Numerically optimize input covariance. Relay cut-set bound and DF rate are nearly equal. SNR lower and upper thresholds are nearly equal.

15. 15 Low SNR Region Cooperative capacity is higher than isotropic MIMO. Cooperative capacity scales more favorably with SNR than non-cooperative capacity.

16. 16 High SNR Region Cooperative capacity is strictly less than orthogonal MIMO capacity. Limited by communication between the cooperating nodes. Multiplexing gain = 1.

17. 17 Conclusion Cooperation is efficient. Up to an SNR threshold. Beyond threshold capacity is limited by the cooperation channel. SNR threshold Depends on network geometry and the number of cooperating antennas. MIMO-gain region. Large g : extends MIMO-gain region. Large M : coordination-limited region sets in at a lower SNR.

18. 18 Future Work Consider fading channel magnitude. E.g., Rayleigh fading. Cooperative systems with dominant coordination cost at the receiver cluster: Multiple-antenna relay near the receiver, the source is a multiple-antenna transmitter. Clusters with transmitter and receiver cooperation costs: One relay cluster is near the source, and a second relay cluster is near the destination.