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

1. 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

2. 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. Presentation Outline

3. 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 Importance of CSI Feedback 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

4. 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. Importance of CSI Feedback D. Greatly increase the system capacity region as well as the sum capacity

5. 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 CSI Feedback Channel state information CSI is a complex vector or matrix of continuous values Practical Feedback Schemes:

6. Considerations in Feedback Systems 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.

7. MIMO Channel System Model: Equal Power Allocation Precoding Matrix Channel Model With Quantized Feedback: MIMO Channel Quantizer

8. With Quantized CSI Feedback With ideal CSI Feedback The first n eigen-values Generalized Weighted Matrix Inner Product between and . Codebook Design Algorithm (Criterion) The codebook is designed to minimize the system mutual information rate loss Under the high resolution assumptions, it can be approximated as

9. For given code matrices , the optimum partitions are given by: For given partitions , the optimal code matrices are given by: Shifting new centers Codebook design using the Lloyd Algorithm Nearest Neighborhood Condition (NNC): partitioning the regions Centroid Condition (CC):

10. 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 Multi-mode Spatial Multiplexing

11. Ideal CSI Feedback Quantized CSI Feedback Performance of Multi-mode S-M

12. Performance Analysis 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?

13. 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]. Direct Approach (Case Specific) Note, it is not straightforward to extend the results to more complicated systems, such as MIMO.

14. 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. Related Publications

15. 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. Source Coding Perspective There should exist a strong connection between source coding and finite rate feedback systems

16. 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 A Simple Example (Motivation) MISO system channel model: with finite rate feedback transmit beamforming:

17. 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 Important Source Coding Results 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.

18. 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 Extension to Feedback MIMO Systems 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:

19. The codebook is designed optimally such that the Voronoi regions of each quantization cell can lead to a minimum moment of inertia profile Intuitive Explanation of The Theory 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

20. t = 3 transmit antennas r = 1 receive antenna Application to MISO Systems MISO beamformingsystem with finite rate feedback over i.i.d. Fading channels: Exactly the same as the “Direct Approach” [Roh and Rao, 2004]

21. 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: Application to MIMO Case

22. Numerical Examples (MIMO)

23. 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 Additional Results 2) Mismatched analysis of finite rate feedback MIMO systems

24. 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 ! Some More Interesting Results 3) Design and Analysis of systems Transformed Codebook

25. 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. Summary

26. 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 Ongoing and Future Work Goal: How to best use MIMO (Feedback) in Ad-Hoc Networks