Evaluating Finite-Rate Feedback in Multi-Antenna Systems

1. Evaluating Finite-Rate Feedback in Multi-Antenna Systems Feeding Back the `Input Covariance Matrix’ Rajesh T Krishnamachari July 14, 2010

2. Overview of Talk

3. Multiple-Input Multiple-Output (MIMO) Link Channel 1 • Higher data rates than single-antenna systems. • UMTS LTE mandates Nt ≥ 2, Nr ≥ 2. 1 NOISE 2 2 + RECEIVER TRANSMITTER Nr Nt

4. MIMO Link Channel Equations Nt Single User Nr X Nt N1 N1 N1 Nt Nr Nr Block 1 Block 2 Block L H=H2 H=H1 H=HL i.i.d. N(0,I) Real Complex CN(0,I) Block Fading Model independent independent …

5. MIMO Multiple-Access Channel • … • … User 30 k30 1 • Cell has U users. • User i has kiantennas. k20 1 User 20 User 40 k40 1 • … • … Base Station k50 1 k10 User10 User 50 1 • ... • … k1 1 User 1

6. MIMO Basics • Assumption of Rayleigh fading : Hij ~ CN(0,1). • Input Covariance Matrix : Q = E (xxH). Computed by Telatar in 1995 for Rayleigh H.

7. Feedback Basics • CSIR assumption is practically sound. • Why → Tx can send pilot signals. • CSIR Capacity is • Recall → From CSITR, EH and max were exchanged. • Use feedback to convey CSI. • Why → Channel remains constant over block.

8. Grassmannian Quantization Single Rank Comprehensive Analysis Higher Rank Quantize Subspace on GNt, s Nt i.i.d. messages Transmit Signal

9. Main Ideas of Grassmann Quantization

10. Overview of Talk

11. Idea of Input Covariance Feedback Feedback Rates Circuit Complexity Why not feed back the Input Covariance Matrix itself ?

12. Water-filling Algorithm P units water Power Allocated ---> 11

13. Water-filling to find Qopt P units water Power Allocated ---> • Distribution of Q not known even for Rayleigh-faded H.

14. Rank Convergence theorem Theorem 1 For the optimal input covariance matrix, determined by water-filling, the ratio of its rank to its size converges almost surely to a deterministic function of the signal-to-noise ratio as the number of transmit and receive antennas grow to infinity with their ratio approaching a finite constant.

15. Rank(Q) Converges to Function of SNR

16. Block Fading Model N1 N1 N1 … Block 1 Block 2 Block L H=H1 H=H2 H=HL … N2 N2 N2 Rank changes along with H. Case One: γ=γ2 γ=γL γ=γ1 OR Rank is relatively constant. N2 Case Two: γ=γ1 γ=γ1 γ=γ1

17. Definition of Unconstrained-Rank Manifolds

18. Dual Loop Feedback Outer– Loop Control Obtain System Mode Feedback System Mode Inner – Loop Control Reconstruct Input Covariance Matrix Quantize Input Covariance Matrix CHANNEL Data Rank Matrix Index

19. Definition of Rank-Constrained manifolds

20. MIMO Multiple Access Channel (MAC) • Channel Equation . • Use Iterative Water-filling to compute • Want to feedback Q using Nfbits.

21. Manifolds • Sum power constraint • Individual power constraint Shortened as Shortened as

22. Lack of Study of P.S.D. matrix Feedback

23. Overview of Talk

24. Manifold Dimension Theorem 2A

25. Manifold Co-ordinates • View ‘svec’ as an operation extracting coordinates. Theorem 2B

26. Distance Metric • Use in quantization through codebooks. • Log-det metric is not suitable. • For Int and Int , • . • For ,

27. Manifold Volume Theorems Theorem 3

28. Ball Volume Theorems • Geodesic Ball • Normalized Ball Volume Theorem 4

29. Idea Of A Codebook • Code • Quantization Rule Q2 Q1 Q3 Q7 Q0 Q6 Q4 Q5 VORONOI CELL

30. Codebook Analysis Parameters

31. The Sphere Packing Code • Constructing optimal codebook is difficult. • Capacity difference is bounded by a factor ~ Δmax . Sphere Packing

32. The Random Code • Generalize idea of Random Vector Quantization. • If , generate i.i.d. Qi ~ Unif (M). • Numerical construction technique found.

33. Random Code Distortion Theorems For sufficiently large code size K, Theorem 5A • Compare with Dai-Liu-Rider, 2008 for Grassmann manifold, Uniform distribution and Chordal distance.

34. Random Codes [Continued] • Fast Convergence Nt =4, Ratio ≤1.01 • Asymptotically Tight Theorem 5B • Flat Manifold case improvement • Asymptotic optimality for quantizing uniform sources: Theorem 5C Theorem 5D

35. Overview of Talk

36. Results on Coding Theoretic Bounds • When δ is sufficiently small, then there exists a code in the following manifolds of size K and minimum distance δ, such that Theorem 6A Gilbert-Varshamov Lower Bound Hamming Upper Bound • The density of this codebook is given by Theorem 6B

37. Theorem For P1 Q P2 Related to Hamming Bound dmin P3 Theorem 7

38. Intuition Behind Capacity Difference Results • Assume γ=1. • Define • If Q and Q0 are close by, Depends on system strategy chosen. Evaluate this term’s behavior w.r.t. Nf Quantization Error Theorem 8

39. Capacity Difference Theorem Theorem 9 When the covariance matrix to be fed back is quantized using , the expected difference in the achievable information rate between the infinite and finite rate feedback case varies with the number of feedback bits used to quantize the covariance feedback as • Beats the previous Dabbagh-Love result with a 2-line computation. • → Compute dimension of their quantization manifold.

40. Immediate Application of result Theorem 10 To limit the capacity drop w.r.t. the maximal CSITR value to some X bps/Hz while using the code , we would require

41. Capacity Difference Theorem Theorem 11 • Capacity Difference is bounded as • To limit this difference to X bps/Hz, Theorem 12

42. Overview of Talk

43. New Paradigm in Finite-Rate Feedback Analysis

44. Extensions to covariance feedback analysis Trace = ρ2 Trace ≤ ρ2 Rank = s Rank ≤ s • Many other questions in report

45. Other forms of feedback Stiefel Quantization

46. Stiefel Quantization Based on a map of Seven Falls, CO Springs.

47. Interference Alignment Scheme Tx1 Tx2 Txk Rx1 Rx2 Rxk K-user SISO IC: CSITR dsum = K/2 [Cadambe-Jafar, 2008]

48. Our work on interference alignment • CSI lies on Composite Grassmann manifold. • K-user R X 1 L-taps channel, CSI lies in . • scaling is sufficient to simulate CSITR dsum performance using IA. Conjecture : If CSI lies on manifold M, then is Nf= dim (M)/2 log P scaling sufficient to attain the same dsum as the CSITR case ?

49. summary • Studied finite-rate feedback in MIMO link and MAC. • Feedback spaces include , , and manifolds. • Rank of input covariance matrix converged fast to a function of SNR. • Used ball volume results to bound distortion of two different codebooks. • The difference in capacity between CSITR and finite rate feedback was shown to be