PRECODING OF ORTHOGONAL STBC WITH CHANNEL COVARIANCE FEEDBACK

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PRECODING OF ORTHOGONAL STBC WITH CHANNEL
COVARIANCE FEEDBACK

• Yi Zhao, Raviraj Adve and Teng Joon Lim
• University of Toronto
• September 6th, 2004

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Outline

• Background
• Optimal Linear Transformation
• Simplified Schemes
• Summary

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I. Background
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Space-Time Block Coding

• Advantages
• Simple Encoding
• Full transmit diversity
• Linear-complexity ML decoding
• Space-Time Coding prefers independent fading.
• Fading correlation results in a performance loss.

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Correlated fading

• Performance of STBC over correlated fading
  channels.
• Correlation matrices are based on the one-ring
  model.

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Correlated Fading

• With or without correlation, Space-Time Codes
  still guarantees the maximum achievable diversity
  gain.
• However, the advantage of diversity is weakened
  by fading correlation.
• Without channel information at the transmitter,
  full diversity is still the best choice.
• If channel information is available at the
  transmitter, it can be used to improve
  performance. HOW?

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II. Optimal Linear Transformation
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Linear Transformation of STBC

• Transmitter and receiver of the proposed system

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Linear Transformation of STBC

• The goal is to minimize the decoding error
  probability.
• A performance criterion for transformation matrix
  W is given, without solution for the optimal
  matrix.
• In general, it is extremely difficult to find the
  optimal solution for arbitrary MIMO systems.

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MISO Systems

• The performance criterion is simplified since
  only one antenna is used at the receiver.
• Define a new matrix , the optimization
  problem can be translated into the same form as
  the waterfilling problem in information theory.
  
• Theorem For MISO channel with correlation matrix
  , the optimal linear transformation is
  , where is a scalar related
  to SNR, is the eigenvector of , and
  is a diagonal matrix obtained by using
  waterfilling algorithm.

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Discussions

• Optimal transformation modulates channel symbols
  with the eigenvectors of the transmit correlation
  matrix.
• The waterfilling process determines power
  distribution among the eigen-channels. It puts
  more power into stronger channel.

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Discussions

• The diversity order of the transmission is
  determined by the number of active
  eigen-channels.
• STBC and eigen beamforming are two extreme cases
  of the transformation.

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Performance
Performance of the optimal transformation scheme,
M2 Performance is not sensitive to the power
allocation
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Performance
Performance of the optimal transformation scheme,
M4 Performance is sensitive to the diversity
order
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MIMO systems

• A widely used assumption about channel
  correlation is that it equals the product of the
  transmit and receive correlations.
• In matrix form, we have
• Similar to the MISO case, the optimal
  transformation matrix is still
• while is chosen to maximize
• under trace constraint.
• This is a second-order waterfilling problem. It
  can be
• solved by numerical schemes, such as SQP
  algorithm.

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Performance
Performance of the second-order waterfilling
scheme, the numerical problem is solved by
fmincon() function in Matlab.
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III. Simplified Schemes
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Transmit and Receive Correlation

• The transmitter is elevated and unobstructed. The
  receiver is surrounded by a scattering ring.
• The downlink of a wireless communication system
• Receive correlation is much smaller than the
  transmit ones.
• By ignoring the receive correlation, waterfilling
  solution for MISO system is also optimal for MIMO
  systems.

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Ignoring Receive Correlation
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Switching Scheme

• This simplified scheme switches between
  beamforming and STBC according to the SNR level
  and channel correlation.
• The complexity is dramatically reduced.
• No performance loss at low SNR. Beamforming is
  optimal.
• Slight loss at high SNR. Full diversity is
  optimal.
• Large loss in the transition region. The
  diversity order is wrong.

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Switching Scheme

• Another correlation model assumes same
  correlation between any antenna pair
• This correlation matrix has only two eigenvalues,
  thus there is NO transition region.

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Switching Scheme

• Performance of the switching scheme with the
  all equal correlation model,

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EPA Scheme

• Switching scheme can only provide diversity order
  of 1 or M.
• A better scheme is introduced as Equal Power
  Allocation (EPA) scheme.
• Use the same diversity order as the optimal one.
• Use equal power for each active eigen-channel.

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EPA Scheme
Performance of the EPA scheme with one ring model
much better than switching
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IV. Summary
Performance Criterion
waterfilling
2nd order waterfilling
switching
EPA
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Thank You!
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Illustration of Waterfilling Process

• The process resembles pouring water into a
  vessle.
• The unshaded bar represents the inverse of the
  eigenvalues
• The shadowed bar represents water.
• is the water level, set to satisfy the trace
  constraint.
• The total amount of water is proportional to SNR .

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Diversity Schemes

• Diversity is essential in wireless communication
  systems to compensate multipath fading.
• Space diversity (antenna diversity) improves the
  performance dramatically without extra bandwidth.
• Receive diversity can be achieved by employing
  MRC receiver.
• Transmit diversity schemes

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Performance

• Performance of the Alamoutis scheme
• Same diversity gain as the MRC scheme
• 3dB loss

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Encoding Algorithm

• The encoding process can be illustrated using a
  encoding Matrix
• Each element is a combination of the channel
  signals and conjugates.
• Each row represents a time slot
• Each column represents a transmitting antenna
• Rate loss

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Capacity-Achieving Scheme

• With channel covariance matrix available at the
  transmitter, the capacity-achieving transmission
  vector for a correlated MIMO system is a
  correlated zero-mean Gaussian vector. Its
  covariance matrix is determined by the following
  theorem
• Theorem The capacity achieving transmission
  covariance matrix for a correlated MIMO channel
  has eigenvalue decompostion
• where,

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Relation to Capacity Analysis

• Transmission over the eigenvectors of the
  transmit correlation matrix is optimal
• Beamforming is optimal at high correlation/ low
  SNR
• Diversity order increases with SNR
• Full diversity is optimal in uncorrelated
  channels/ high SNR.

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Summary

• Contribution of the thesis
• Provide a waterfilling solution for the optimal
  linear transformation of STBC for MISO systems.
• Present a numerical second-order waterfilling
  solution for MIMO systems.
• Introduce three simplified scheme (Ignoring,
  Switching, EPA) to reduce the complexity.