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
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
BackgroundCooperative 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 Nosratinia05.
• Cooperative multiplexing gain was conjectured to
  be 1.
• Cooperative systems lack full multiplexing gain.

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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
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.

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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).

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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.

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Cut-set Bound

• Optimal input covariance matrix hard to compute.
• Non-convex, depends on channel realization.

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Capacity Upper Bound Details
Chiurtu et al.00, Shen et al.05

• Channel-independent capacity upper bound
• Upper bound is loose.

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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.

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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.

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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.

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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.

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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.

SNR thresholds
Relay capacity
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Low SNR Region

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

MIMO-gain Region
Relay capacity
Isotropic MIMO
No cooperation
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High SNR Region

• Cooperative capacity is strictly less than
  orthogonal MIMO capacity.
• Limited by communication between the cooperating
  nodes.
• Multiplexing gain 1.

Coordination-limited Region
Orthogonal MIMO
Tx-Relay capacity
Relay capacity
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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
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.