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Bhaskar D. Rao ARO MURI Annual Review July. 28th, 2005

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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*

*MURI Supported

- 1. Overview of MIMO Systems With Finite Rate Feedback

- A. Importance of channel state information feedback.

- B. Nature of channel state information feedback.

- 2. Design of Finite Rate Feedback MIMO Systems

- A. Quantization of MIMO channels (codebook design criterion and algorithms).

- B. Multi-mode spatial multiplexing strategy and its performance.

- 3. Analysis of Finite Rate Feedback Systems (Source Coding Prespective)

- A. Difficulties associated with performance analysis of feedback systems.

- B. Connections between channel quantization and source coding.

- C. General vector quantization problem and its asymptotic analysis.

- D. Applications of the distortion analysis to finite rate feedback systems.

- Perfect CSIT

- NO CSIT

- Example: An MIMO system with M transmit and receive antennas,

- No CSIT, capacity can be achieved
- by some 2-D (space-time) code

- Pre-coder with perfect CSIT, system is
- equivalent to M parallel SISO channels

A. Improved system performance, in terms of capacity, SNR, BER, etc.

- Example: An MISO system with M transmit antennas and single receive antenna

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

interference.

Proper pre-coding is possible,

such as Zero-forcing, MMSE, etc

CSI Feedback

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.

Integer Index

Adaptive Transmitter

Channel Quantizer

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

Precoding Matrix

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.

Intuitive Explanation:

Inverse Water-Filling Power Allocation (Optimal)

water level

water level

power allocated

power allocated

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.

Similarity:

Quantization of the channel state information

Differences:

- 1> Not all channel information need to be quantized

- 2> The actual quantization objective and the channel instantiation may lie in different
- spaces and may have different dimensions

- 3> The additional information which is not to be quantized can be utilized as side
- information at the quantizer (or the receiver)

- 4> The distortion measure may be a general non-mean-square error function, and may be
- even parameterized by the side information

Final Motivation:

- Bridge the gap between these two fields, and introduce a new perspective to
- study finite rate feedback multiple antenna systems.

There should exist a strong connection between source coding and finite rate feedback systems

Additional

information

Quantization

Objective

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

Source distribution

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.

Point density:

describes the relative

density of the codepoints

Quantization levels

Story begins:

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:

Quantization resolution

Source dimension

Point density:

Describes the relative density

of the code points

Weighted moment of inertia:

Describes the relative distortion

of the quantizer w.r.t. the location

Source distribution

Problem Setup:

- with dimension

- Quantization objective:

- with dimension

- Side information:

Side information is: Available at the encoder (quantizer) but unavailable at the decoder (reproducer)

- Quantization scheme:

- General distortion:

Asymptotic Analysis:

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

- 1. Design of Finite Rate Feedback MIMO Systems

- A. Proposed efficient codebook design criterion and algorithms for MIMO systems.

- B. Proposed Multi-mode spatial multiplexing strategy.

- 2. Analysis of Finite Rate Feedback Systems

- A. It is very difficult to provide performance analysis by using conventional method.

- B. Found connections between channel quantization and source coding.

- C. Provided asymptotic analysis for a general vector quantization problem.

- D. Applied the distortion analysis to various finite rate feedback systems.

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