Quantization Algorithms and their Analysis in Finite Rate Feedback MIMO Systems. Bhaskar D. Rao ARO MURI Annual Review July. 28th, 2005. GSR ’ s: J. C. Roh, J. Zheng, C. R. Murthy and Y. Isukapalli*. * MURI Supported. 1. Overview of MIMO Systems With Finite Rate Feedback.
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Quantization Algorithms and their Analysis in Finite Rate Feedback MIMO Systems
Bhaskar D. Rao
ARO MURI Annual Review
July. 28th, 2005
GSR’s: J. C. Roh, J. Zheng, C. R. Murthy and Y. Isukapalli*
A. Improved system performance, in terms of capacity, SNR, BER, etc.
B. Reduced implementation complexity
C. Enables exploitation of multi-user diversity
With CSIT, effective selection of active users and route selection can be made.
Example: A multi-user MISO broadcasting channel with M transmit and single receive antenna
users are not allowed
to cooperate, and hence
cause serious multi-user
Proper pre-coding is possible,
such as Zero-forcing, MMSE, etc
E. Improve the robustness of the communication link (QoS requirements)
Power and rate control is possible when CSIT is available and the network throughput is increased.
D. Greatly increase the system capacity region as well as the sum capacity
For example: An MIMO system with M transmit antennas and N receive antennas, .
It is not reasonable to feedback total 2MN real numbers of continuous values.
Each index represents a particular mode of the channel, which corresponds to a particular transmission strategy
Channel state information CSI is a complex vector or matrix of continuous values
Practical Feedback Schemes:
A. Design of Optimal Quantizers (at the receiver) & Optimization of the Codebook?
1) The quantizer (or the encoder) should be simple as well as effective.
2) The quantizer and the codebook should be designed to match both the channel
distribution and the system performance metrics, such as capacity, SNR, BER, etc.
B. Performance Analysis of Finite Rate Feedback Multiple Antenna Systems
1) To understand the effects of the finite rate feedback on the system performance,
to be specific, performance metricvsfeedback rate.
2) Shed insights on the choice of the feedback schemes as well as the quantizer design.
MIMO Channel System Model:
Equal Power Allocation
Channel Model With Quantized Feedback:
With Quantized CSI Feedback
With ideal CSI Feedback
The first n eigen-values
Generalized Weighted Matrix Inner Product
between and .
The codebook is designed to minimize the system mutual information rate loss
Under the high resolution assumptions, it can be approximated as
For given code matrices ,
the optimum partitions are given by:
For given partitions ,
the optimal code matrices are given by:
Shifting new centers
Nearest Neighborhood Condition (NNC):
partitioning the regions
Centroid Condition (CC):
Multi-mode SP transmission strategy:
1) The number of data streams n is determined by the system SNR:
2) In each mode, the simple equal power allocation over n spatial channels is employed.
Inverse Water-Filling Power Allocation (Optimal)
Case II: High SNR
Case I: Low SNR
Ideal CSI Feedback
Quantized CSI Feedback
Some Interesting Questions:
1. Finite Rate Effects: What is the performance (capacity, SNR, BER) versus the feedback rate ?
2. Mismatched Analysis: What happens if a codebook designed for one system is used in another system?
3. Transferred Code: The codebook for a particular system is transferred from another system through
a linear or non-linear operation. What is the performance? & How to design?
4. Feedback With Error: What happens if the feedback information also suffers from error (delay)?
5. Quantization of Imperfect CSI: What happens if CSI to be quantized suffers from estimation error?
Generally speaking, analysis of finite rate feedback MIMO system is very difficult.
For example, an MISO i.i.d. fading channel with transmit antennas and single receive antenna
with ideal CSI
with quantized CSI
It is very hard to characterize the distribution of the inner product .
By approximating the inner product to a truncated beta distribution, we can obtain a system
performance analysis, [Roh and Rao, 2004].
Note, it is not straightforward to extend the results to more complicated systems, such as MIMO.
J. C. Roh and B. D. Rao, “Performance Analysis of Multiple Antenna Systems with VQ Based Feedback,” Thirty Eighth Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, Nov. 2004
J. C. Roh and B. D. Rao, “Vector Quantization Techniques for Multiple Antenna Channel Feedback,” International Conference on Signal Processing and Communications (SPCOM), Bangalore, India, Dec. 2004
J. C. Roh and B. D. Rao, “MIMO Spatial Multiplexing Systems and Limited Feedback,” IEEE International Conference on Communications, Seoul, Korea, May 2005.
Quantization of the channel state information
There should exist a strong connection between source coding and finite rate feedback systems
The quantization objective has constraints:
If we take the capacity loss as the distortion measure, then it is a general function
MISO system channel model:
with finite rate feedback transmit beamforming:
Extension to vector quantization:
A. Gersho, “Asymptotically optimal block quantization”, IEEE Trans. Info. Theory, 1979.
Moment of inertia:
describes the relative
distortion w.r.t. its location
Further extension to general distortion functions:
W.R. Gardner and B.D. Rao, “Theoretical analysis of the high-rate vector quantization of LPC parameters”,
IEEE Trans. Speech Audio Processing, 1995.
For the order Euclidean quantization errors :
S. Na and D.L. Neuhoff, “Bennett’s Integral for vector quantizers”, IEEE Trans. Info. Theory, 1995.
describes the relative
density of the codepoints
W. R. Bennett, “Spectra of quantized signals”, Bell System Tech. Journal, 1948.
Provide asymptotic analysis of the system mean square (quantization) error in terms of the source
distribution, quantization rate, and compander slope.
Tightasymptotic distortion lower bound:
Describes the relative density
of the code points
Weighted moment of inertia:
Describes the relative distortion
of the quantizer w.r.t. the location
Side information is: Available at the encoder (quantizer) but unavailable at the decoder (reproducer)
The codebook is designed optimally
such that the Voronoi regions of each
quantization cell can lead to a minimum
moment of inertia profile
The code points are distributed (or located)
“smartly” such that the relative code point
density matches the source distribution as
well as the distortion function.
Dense in important areas
Sparse in unimportant areas
t = 3 transmit antennas
r = 1 receive antenna
MISO beamformingsystem with finite rate feedback over i.i.d. Fading channels:
Exactly the same as the “Direct Approach” [Roh and Rao, 2004]
Application of the general asymptotic distortion analysis:
An i.i.d. Rayleigh fading MIMO channels of size , using beamforming vectors
as pre-coding matrix with equal power allocation, the channel quantization rate is Bits.
The system asymptotic capacity loss in High SNR regimes is given by,
A very interesting observation:
1) Asymptotic analysis of correlated MIMO channels
Special case:Correlated MISO channel with channel covariance matrix , the capacity loss satisfies
white channel is the worst
channel to quantize
Special case I:MISO channel, sub-optimal MMSE criterion is used for codebook generation
Special case II:MISO channel, codebook for i.i.d. channels is used in systems with channel correlation
can not be worse than i.i.d channels
2) Mismatched analysis of finite rate feedback MIMO systems
Special case:Correlated MISO channel with channel correlation matrix , suppose its codebook
is obtained from the following transformation from codebook of i.i.d channels,
Obtained results:If is chosen as , then with a large and in high-SNR and low-SNR regimes,
Transformed codebook can be
as good as the optimal one !
3) Design and Analysis of systems Transformed Codebook
Complete the Analytical Work on Quantized Feedback
Robust Feedback Strategies in the presence of Mobility
CSI feedback and transmission schemes in multi-link systems
Influence on routing and MAC
CSI estimation overhead (Blind and Semi-blind schemes)
Low complexity variants and tradeoffs involved
Goal: How to best use MIMO (Feedback) in Ad-Hoc Networks