Feedback Benefits in MIMO Communication Systems

1. Feedback Benefits in MIMO Communication Systems David J. Love Center for Wireless Systems and Applications School of Electrical and Computer Engineering Purdue University djlove@ecn.purdue.edu

2. Transmitter Receiver • • • • • • Multiple Antenna Wireless Systems • Multiple-input multiple-output (MIMO) using multiple antennas at transmitter and receiver • Antennas spaced independent fading • Offer improvements in capacity and reliability

3. time space Space-Time Signaling • Design in space and time • Transmit matrices – transmit one column each transmission • Sent over a linear channel Assumption: is an i.i.d. complex Gaussian matrix

4. Role of Channel Knowledge • Open-loop MIMO [Tarokh et al] • Signal matrix designed independently of channel • Most popular MIMO architecture • Closed-loop MIMO [Sollenberger],[Telatar],[Raleigh et al] • Signal matrix designed as a function of channel • Dramatic performance benefits

5. Transmitter Channel Knowledge • Fundamental problem: How does the transmitter find out the current channel conditions? • Observation: Receiver knows the channel • Solution: Use feedback

6. Limited Feedback Problem • Solution: Send back feedback [Narula et al],[Heath et al] • Feedback channel rate very limited • Rate  1.5 kb/s (commonly found in standards, 3GPP, etc) • Update  3 to 7 ms (from indoor coherence times) Feedback amount around 5 to 10 bits

7. FX X H Open-Loop Space-Time Encoder Receiver F … … … … H Choose F from codebook Low-rate feedback path Update precoder Solution: Limited Feedback Precoding • Use open-loop algorithm with linear transformation (precoder) • Restrict to • Codebook known at transmitter/receiver and fixed • Convey codebook index when channel changes bits

8. Example 1: Limited Feedback Beamforming unit vector • Convert MIMO to SISO • Beamforming advantages: • Error probability improvement • Resilience to fading Complex number r

9. Challenge #1: Beamformer Selection • Nearest neighbor union bound [Cioffi] • Instantaneous channel capacity [Cover & Thomas] [Love et al]

10. channel term codebook term Challenge #2: Beamformer Codebook • Want to maximize on average • Average distortion • Using sing value decomp & Gaussian random matrix results [James 1964] ( ) where is a uniformly distributed unit vector

11. Grassmannian Beamforming Criterion [Love et al]: Design by maximizing Bounding of Criterion Grassmann manifold radius2 metric ball volume [Love et al]

12. Simulation 3 by 3 QPSK Error Rate (log scale) 0.6 dB SNR (dB)

13. Example 2: Limited Feedback Precoded OSTBC • Require • Use codebook:

14. Challenge #1: Codeword Selection Channel Realization H Codebook matrix • Can bound error rate [Tarokh et al] • Choose matrix from from as [Love et al]

15. Challenge #2: Codebook Design • Minimize loss in channel power Grassmannian Precoding Criterion [Love & Heath]: Maximize minimum chordal distance • Think of codebook as a set (or packing) of subspaces • Grassmannian subspace packing

16. Simulation 8 by 1 Alamouti 16-QAM Open-Loop 16bit channel Error Rate (log scale) 9.5dB 8bit lfb precoder SNR (dB)

17. Example 3:Limited Feedback Precoded Spatial Multiplexing • Assume • Again adopt codebook approach

18. Channel Realization H Codebook matrix Challenge #1: Codeword Selection • Selection functions proposed when known • Use unquantized selection functions over • MMSE (linear receiver) [Sampath et al], [Scaglione et al] • Minimum singular value (linear receiver) [Heath et al] • Minimum distance (ML receiver) [Berder et al] • Instantaneous capacity [Gore et al]

19. Challenge #2: Distortion Function • Min distance, min singular value, MMSE (with trace) [Love et al] • MMSE (with det) and capacity [Love et al]

20. Codebook Criterion Grassmannian Precoding Criterion [Love & Heath]: Maximize Min distance, min singular value, MMSE (with trace) – Projection two-norm distance MMSE (with det) and capacity – Fubini-Study distance

21. Simulation 4 by 2 2 substream 16-QAM 16bit channel Perfect Channel Error Rate (log scale) 6bit lfb precoder 4.5dB SNR per bit (dB)

22. Conclusions • Limited feedback allows closed-loop MIMO • Beamforming • Precoded OSTBC • Precoded spatial multiplexing • Large performance gains available with limited feedback • Limited feedback application • IEEE 802.16e • IEEE 802.11n

23. Codebook as Subspace Code • is a subspace distance – only depends on subspace not vector • Codebook is a subspace code • Minimum distance [Sloane et al] set of lines

24. Beamforming Summary • Contribution #1: Framework for beamforming when channel not known a priori at transmitter • Codebook of beamforming vectors • Relates to codes of Grassmannian lines • Contribution #2: New distance bounds on Grassmannian line codes • Contribution #3: Characterization of feedback-diversity relationship More info: D. J. Love, R. W. Heath Jr., and T. Strohmer, “Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems,” IEEE Trans. Inf. Th., vol. 49, Oct. 2003. D. J. Love and R. W. Heath Jr., “Necessary and Sufficient Conditions for Full Diversity Order in Correlated Rayleigh Fading Beamforming and Combining Systems,” accepted to IEEE Trans. Wireless Comm., Dec. 2003.

25. Outline • Introduction • MIMO Background • MIMO Signaling • Channel Adaptive (Closed-Loop) MIMO • Limited Feedback Framework • Limited Feedback Applications • Beamforming • Precoded Orthogonal Space-Time Block Codes • Precoded Spatial Multiplexing • Other Areas of Research

26. Orthogonal Space-Time Block Codes (OSTBC) • Constructed using orthogonal designs [Alamouti, Tarokh et al] • Advantages • Simple linear receiver • Resilience to fading • Do not exist for most antenna combs (complex signals) • Performance loss compared to beamforming

27. Precoded OSTBC save at least bits compared to beamforming! Feedback vs Diversity Advantage • Question: How does feedback amount affect diversity advantage? Theorem [Love & Heath]: Full diversity advantage if and only if bits of feedback Proof similar to beamforming proof.

28. Precoded OSTBC Summary • Contribution #1: Method for precoded orthogonal space-time block coding when channel not known a priori at transmitter • Codebook of precoding matrices • Relates to Grassmannian subspace codes with chordal distance • Contribution #2: Characterization of feedback-diversity relationship More info: D. J. Love and R. W. Heath Jr., “Limited feedback unitary precoding for orthogonal space time block codes,” accepted to IEEE Trans. Sig. Proc., Dec. 2003. D. J. Love and R. W. Heath Jr., “Diversity performance of precoded orthogonal space-time block codes using limited feedback,” accepted to IEEE Commun. Letters, Dec. 2003.

29. Outline • Introduction • MIMO Background • MIMO Signaling • Channel Adaptive (Closed-Loop) MIMO • Limited Feedback Framework • Limited Feedback Applications • Beamforming • Precoded Orthogonal Space-Time Block Codes • Precoded Spatial Multiplexing • Other Areas of Research

30. Spatial Multiplexing [Foschini] { • True “multiple-input” algorithm • Advantage: High-rate signaling technique • Decode Invert (directly/approx) • Disadvantage: Performance very sensitive to channel singular values Multiple independent streams

31. Limited Feedback Precoded SM[Love et al] • Assume • Again adopt codebook approach

32. Channel Realization H Codebook matrix Challenge #1: Codeword Selection • Selection functions proposed when known • Use unquantized selection functions over • MMSE (linear receiver) [Sampath et al], [Scaglione et al] • Minimum singular value (linear receiver) [Heath et al] • Minimum distance (ML receiver) [Berder et al] • Instantaneous capacity [Gore et al]

33. Challenge #2: Distortion Function • Min distance, min singular value, MMSE (with trace) [Love et al] • MMSE (with det) and capacity [Love et al]

34. Codebook Criterion Grassmannian Precoding Criterion [Love & Heath]: Maximize Min distance, min singular value, MMSE (with trace) – Projection two-norm distance MMSE (with det) and capacity – Fubini-Study distance

35. Simulation 4 by 2 2 substream 16-QAM 16bit channel Perfect Channel Error Rate (log scale) 6bit lfb precoder 4.5dB SNR per bit (dB)

36. Precoded Spatial Multiplexing Summary • Contribution #1: Method for precoding spatial multiplexing when channel not known a priori at transmitter • Codebook of precoding matrices • Relates to Grassmannian subspace codes with projection two-norm/Fubini-Study distance • Contribution #2: New bounds on subspace code density More info: D. J. Love and R. W. Heath Jr., “Limited feedback unitary precoding for spatial multiplexing systems,” submitted to IEEE Trans. Inf. Th., July 2003.

37. Outline • Introduction • MIMO Background • MIMO Signaling • Channel Adaptive (Closed-Loop) MIMO • Limited Feedback Framework • Limited Feedback Applications • Beamforming • Precoded Orthogonal Space-Time Block Codes • Precoded Spatial Multiplexing • Other Areas of Research

38. Multi-Mode Precoding • Fixed rate • Adaptively vary number of substreams • Yields • Full diversity order • Rate growth of spatial multiplexing >98% Capacity Ratio >85% SNR (dB) D. J. Love and R. W. Heath Jr., “Multi-Mode Precoding for MIMO Wireless Systems Using Linear Receivers,” submitted to IEEE Transactions on Signal Processing, Jan. 2004.

39. Space-Time Chase Decoding • Decode high rate MIMO signals “costly” • Existing decoders difficult to implement • Solution([Love et al] with Texas Instruments): Space-time version of classic Chase decoder [Chase] • Use linear or successive decoder as “initial bit estimate” • Perform ML decoding over set of perturbed bit estimates D. J. Love, S. Hosur, A. Batra, and R. W. Heath Jr., “Space-Time Chase Decoding,” submitted to IEEE Transactions on Wireless Communications, Nov. 2003.

40. Visually important Diversity 4 … Diversity 2 Visually unimportant Diversity 1 Assorted Areas • MIMO channel modeling • IEEE 802.11N covariance generation • Joint source-channel space-time coding

41. Future Research Areas • Coding theory • Subspace codes • Binary transcoding • Reduced complexity Reed-Solomon • UWB & cognitive (or self-aware) wireless • Capacity • MIMO (???) • Multi-user UWB • Cross layer optimization (collaborative) • Sensor networks • Broadcast channel capacity schemes

42. Conclusions • Limited feedback allows closed-loop MIMO • Beamforming • Precoded OSTBC • Precoded spatial multiplexing • Diversity order a function of feedback amount • Large performance gains available with limited feedback • Multi-mode precoding & Efficient decoding for MIMO signals

43. Beamforming Criterion • [Love et al] • Differentiation maximize

44. Precode OSTBC Criterion • Let

45. Precode OSTBC – Cont. • [Barg et al] • Differentiation maximize

46. Precode Spat Mult Criterion – Min SV • Let • Differentiation maximize

47. Precode Spat Mult Criterion – Capacity • Let • Differentiation maximize

48. SM Susceptible to Channel • Fix • Condition number Decreasing

49. Vector Quantization Relationship • Observation: Problem appears similar to vector quantization (VQ) • In VQ, • 1. Choose distortion function • 2. Minimize distortion function on average • VQ distortion chosen to improve fidelity of quantized signal Can we define a distortion function that ties to communication system performance?

50. Grassmannian Subspace Packing • Complex Grassmann manifold • set of M-dimensional subspaces in • Packing Problem • Construct set with maximum minimum distance • Distance between subspaces • Chordal • Projection Two-Norm • Fubini-Study Column spaces of codebook matrices represent a set of subspaces in 1 2