October 2012

1. Codebook Design for Limited Feedback Multiple Input Multiple Output Systems • AshvinSrinivasan • Supervisor: Prof. Olav Tirkkonen • Instructor: R-A Pitaval • October 2012

2. Outline • Research Background • Research Objectives • Conceptual Framework • Literature Review • Results and Discussion • Conclusions and Future Work • References

3. Research Background • Traditionally SISO systems were used, but need for higher system throughputs led to MIMO systems. • MIMO systems use spatial multiplexing to achieve higher data rates without wasting frequency/time resources. • Several technologies like LTE advanced, CDMA 2000, WIMAX, UMTS use FDD systems in which perfect CSIT is not feasible due to absence of channel reciprocity. • Partial CSIT made available via precoding by using pre-determined codebooks at Tx and Rx.

4. Research Objectives Research Problem • Constructing codebooks for limited feedback MIMO systems. Codebook consists of codewords in the form of matrices. Codebook construction reduces to a quantization problem on certain manifolds. Grassmann and permutation invariant flag manifolds are considered. Research Scope • Codebook impact on the system capacity.

5. Conceptual Framework Precoding Codebook Design for LF-MIMO Systems Review Review Review MIMO wireless systems Codebook design Manifolds Quantization Outcome Outcome Increase in system capacity with Tx precoding.

6. Literature Review MIMO systems • MIMO system model. for the antenna. For the entire model ; ; ; ; • Assumptions. 1) Channel coefficients are modeled as complex Gaussian random variables, i.e. (0,I). 2) w is Gaussian white noise vector (I)

7. Differentiable Manifolds. • A manifold is a curved space(for e.g. sphere), which locally resembles Euclidean space. A manifold endowed with globally defined differential structure is a differential manifold. This allows us to perfom calculus on these manifolds. For e.g. Riemannian manifold. • Riemannian manifold is a differentiable manifold equipped with a metric, which is the collection of all the inner products defined in the tangent space for every point in the manifold. • Grassmann and Permutation invariant flag manifolds are special cases of Riemannian manifolds. Grassmann manifold is formally defined as the quotient space: Where is a unitary group of dimension n. Grassmann manifold is the space containing those points which are p-dimensional sub-spaces in a n-dimensional space. Points belonging to Grassmann span different sub-spaces, are basically non-equivalent. Chordal distance is used to obtain this non-equaivalent points given by[1] Where is an orthonormal matrix.

8. P.I. flag manifold is formally defined as the quotient space represented by: where contains sets of matrices invariant under permutation and phase rotation. The corresponding distance function is given by[2]: Where belongs to the unitary group of dimension n.

9. Precoding Quantizer Structure 1) Encoder: at Rx side 2) Decoder: at Tx side , where has left singular vectors, has right singluar vectors with as a diagonal matrix containing ordered singular values.

10. Lloyd Algortihm • Initially, generate many random points on an n- dimensional space, and set MQE= . 2) Associate the random points to a particular cluster based on NN condition. Where distance functions for Grassmann and flag manifold are given on the previous slides. 3) Compute the centroid for each cluster on the non Euclidean space. For Grassmann manifold[1],

11. For flag manifold, each column vector of the unitary matrix lies in , as per the formula, For , the unitary centroid point is obtained via polar decomposition of 4) Continue the iterations until decrease in overall distortion at current iteration relative to previous is less than a threshold value. iteration iteration iteration iteration

12. Results and Discussion Distortion/Performance Measure Semi analytical bounds • Critical radius estimation: • The critical radius, is numerically found to be 1,1.2,1.4 for following A.P. • Bounds for flag manifold[3]:

13. Monte Carlo Simulations Cluster size • Increase in cluster size , i.e., Increase in the feedback bits, • With increase in cluster size leads to a lesser overall distortion. This physically translates to more feedback bits, or a wider range of codewords to choose from. Spatial Sub-Streams • For unitary precoding, and Grassmann precoding, • As increases, the overall distortion worsens quite drastically.

14. Brute Force Method • Proposed codebook vs codebook obtainedvia brute force. • For unitary points distributed uniformly over the space. From a small sample size of these points, select that point as a centroid candidate, which minimizes the overall distortion. • Increase the sample size until all unitary points are considered. • We observe for all cases, the proposed centroid fairs better than the brute force centroid candidate.

15. Capacity Analysis of LF-MIMO systems ZF receiver General Capacity Equation with ZF receiver[4] Where ; is the equivalent correlation matrix. is the effective channel as seen by the receiver, with as its precoder. MIMO open loop capacity with perfect CSIR

16. Capacity Analysis with Unitary Precoding Feedback bits • The increase in feedback bits tends to improve the system’s spectral efficiency since Tx. gets a better channel estimate. • Intuitively, we see that system capacity is a function of overall distortion. Spatial Substreams • For a 2 by 2 system, we approach perfect precoding with 4-5 feedback bits. However, for higher order systems, it tends to saturate, requiring infinite feedback to approach perfect precoding. • Increase in results in increase of the manifold’s volume making quantization a non-trivial task.

17. Capacity Analysis with Grassmann Precoding • The effect of feedback and spatial sub streams are similar to Unitary precoding. • , outperforms with 3 dB gain. This is because of increase in average received power commonly referred to as beamforming/array/power gain.

18. Precoder Partitioning • Codebook partitioned between orthogonal codebook() and Grassmann codebook (). • It is shown that with 4 feedback bits outperforms the partitioned space between and of 2 bits eachwhich outperforms with 4 feedback bits.

19. Conclusions and Future Work • Brief overview on MIMO wireless systems, Grassmann and flag manifolds, and precoding were presented. • Codebooks were constructed by quantizing flag and Grassmann manifolds and their impact were analyzed on system capacity. • Future work could include capacity bounds as a function of distortion. A good suggestion would be to analyze the capacity by optimally distributing power at Tx. based on water filling principle.

20. References [1] B. Mondal, S. Dutta, and R. Heath, “Quantization on the Grassmann Manifold,” IEEE Transactions on Signal Processing, vol. 55, pp. 4208-4216, Aug. 2007. [2] I. Kim, S. Park, D. Love, and S. Kim, ”Improved multiuser MIMO unitary precoding using partial channel state information and insights from the Riemannian manifold,” IEEE Transactions on Wireless Communications, vol. 8, pp. 4014-4023, Aug. 2009. [3] R. T. Krishnamachari,, “Analyzing Finite-Rate Feedback of the General Covariance Matrix: Rate-Distortion and Capacity Difference Aspects.” PhD thesis, University of Colorado - Boulder, 2010. [4] H.-L. Maattanen, K. Schober, O. Tirkkonen, and R. Wichman, ”Precoder partitioning in closed-loop MIMO systems,” IEEE Transactions on Wireless Communications, vol. 8, pp. 3910-3914, Aug. 2009.

21. Thank You