Title: PRECODING OF ORTHOGONAL STBC WITH CHANNEL COVARIANCE FEEDBACK
1PRECODING OF ORTHOGONAL STBC WITH CHANNEL
COVARIANCE FEEDBACK
- Yi Zhao, Raviraj Adve and Teng Joon Lim
- University of Toronto
-
- September 6th, 2004
2Outline
- Background
- Optimal Linear Transformation
- Simplified Schemes
- Summary
3 I. Background
4Space-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.
5Correlated fading
- Performance of STBC over correlated fading
channels. - Correlation matrices are based on the one-ring
model.
6Correlated 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?
7II. Optimal Linear Transformation
8Linear Transformation of STBC
- Transmitter and receiver of the proposed system
9Linear 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.
10MISO 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.
11Discussions
- 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. -
12Discussions
- 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.
13Performance
Performance of the optimal transformation scheme,
M2 Performance is not sensitive to the power
allocation
14Performance
Performance of the optimal transformation scheme,
M4 Performance is sensitive to the diversity
order
15MIMO 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.
16Performance
Performance of the second-order waterfilling
scheme, the numerical problem is solved by
fmincon() function in Matlab.
17III. Simplified Schemes
18Transmit 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.
19Ignoring Receive Correlation
20Switching 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. -
21Switching Scheme
- Another correlation model assumes same
correlation between any antenna pair - This correlation matrix has only two eigenvalues,
thus there is NO transition region.
22Switching Scheme
- Performance of the switching scheme with the
all equal correlation model,
23EPA 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.
24EPA Scheme
Performance of the EPA scheme with one ring model
much better than switching
25IV. Summary
Performance Criterion
waterfilling
2nd order waterfilling
switching
EPA
26Thank You!
27Illustration 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 .
28Diversity 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
29Performance
- Performance of the Alamoutis scheme
- Same diversity gain as the MRC scheme
- 3dB loss
30Encoding 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
31Capacity-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,
32Relation 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.
33Summary
- 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.