Compressed-Sensing Based Multi-User Millimeter Wave Systems: How Many Measurements Are Needed? Ahmed Alkhateeb*, Geert Leus #, and Robert W. Heath Jr.* * Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin # Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology (TU Delft) See more on mmwave @ School of ICASSP talk on Wednesday www.profheath.org
Why MIMO at mmwave? millimeter wave band possible bands used for cellular 1.3 GHz 2.1 GHz 7 GHz (unlic) 10 GHz several GHz of spectrum is promising but found in many separate bands 28 GHz 37 / 42 GHz spatial multiplexing & beamforming isotropic radiator mmwave aperture 60GHz E-Band just beamforming to 300 GHz multiple data streams TX RX sub-6ghz aperture Beamforming for antenna gain Spatial multiplexing for spectral efficiency Shu Sun, T. Rappapport, R. W. Heath, Jr., A. Nix, and S. Rangan, `` MIMO for Millimeter Wave Wireless Communications: Beamforming, Spatial Multiplexing, or Both?,'' IEEE Communications Magazine, December 2014. 2
Analog beamforning Baseband DAC Chain Chain ADC Baseband Phase shifters u Low power consumption (1 chain) u Beamforming gain achieved using narrow beams u MmWave specific constraints ª Constant gains: Only phases are typically adjusted ª Quantized phases: Fixed set of steering directions is allowed De-facto approach in IEEE 802.11ad / WiGig and Wireless HD Only provides single stream MIMO beamforming 3
Hybrid analog/digital architecture Baseband Baseband Precoding Precoding 1-bit DAC ADC Chain Beamforming + + Combining Chain 1-bit ADC ADC Baseband Baseband Combining Precoding 1-bit DAC ADC Chain Beamforming + Combining Chain 1-bit ADC ADC u Compromise on power consumption & complexity (# ADCs << # Antennas) u Enables spatial multiplexing and multi-user MIMO u Digital can correct for analog limitations [Aya 14] Hybrid architecture is one viable approach for multi-stream MIMO o [Aya 14] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. Heath, Spatially sparse precoding in millimeter wave MIMO systems, IEEE Transactions on Wireless Communications, vol. 13, no. 3, pp. 1499 1513, March 2014 4
MmWave channel estimation u MmWave channel estimation is challenging ª Large transmit and receive arrays ª Low signal-to-noise ratio before beam forming ª Limited number of chains imposes constraints on training signal design ª Channel is seen through the lens u Prior work (e.g., [Wan 09], [Hur 13]) ª Avoids explicit channel estimation ª Relies on beam training with no prior channel knowledge ª Limitations: Supports only single-stream transmission o o Need to design new mmwave channel estimation algorithms [Wan 09] J. Wang, Z. Lan, C. Pyo, T. Baykas, C. Sum, M. Rahman, J. Gao, R. Funada, F. Kojima, H. Harada et al., Beam codebook based beamforming protocol for multi-gbps millimeter-wave WPAN systems, IEEE Journal on Selected Areas in Communications, vol. 27, no. 8, pp. 1390 1399, 2009. [Hur 13] S. Hur, T. Kim, D. Love, J. Krogmeier, T. Thomas, and A. Ghosh, Millimeter wave beamforming for wireless backhaul and access in small cell networks, IEEE Transactions on Communications, vol. 61, no. 10, pp. 4391 4403, 2013. 5
MmWave channels are sparse 1 1 2 2 3 3 u MmWave channel estimation can be formulated as a sparse problem [Alk 14] ª Channels are sparse in the angular domain, a few paths exist [Akd 13] u Sparse formulation motivates using (adaptive) compressed sensing tools ª Limitation: Training overhead scales with the # of users (repeated per user) [Alk 14] o o [Akd 13] M. Riza Akdeniz, Y. Liu, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkip, Millimeter Wave Channel Modeling and Cellular Capacity Evaluation, IEEE Journal on Selected Areas on Communications, vol. 32, no. 6, pp. 1164-1179, June 2014 [Alk 14] A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath Jr, Channel estimation and hybrid precoding for millimeter wave cellular systems. IEEE J. Selected Topics in Signal Processing (JSTSP), vol. 8, no. 5, May 2014, pp. 831-846 6
Contribution u Propose and evaluate a new mmwave system operation based on CS ª CS-based downlink channel training ª Data transmission with beamforming based on estimated channel u Evaluate system achievable rate as a function of the # of CS measurements ª Provides insights into the performance of CS-based mmwave systems ª Compares the performance with the exhaustive search solution 7
System model F precoder H u w U w 1 w u combiner N MS combiner w u Chain Limited Feedback uth mobile station N BS N MS u U users, antennas at BS and antennas at each MS ª Analog precoding at the base station and analog combining at each user u BS has # of chains = # of users ª Phase shifters are assumed to have a constant modulus & quantized angles 8
Channel model 1 1 2 2 3 3 Robert W. Heath Jr. (2015) H u = N BS N MS α u a MS (θ u ) a BS (φ u ) u MmWave channel assumptions ª Single-path channels (for simplicity, it can be extended to multi-path channels) ª Array response vector is known, generally non-uniform e.g. UPA ª Paths may be LOS/NLOS path gain array response (includes path-loss) vectors angles of arrival/ departure (AoA/AoD) o [Akd 13] M. Riza Akdeniz, Y. Liu, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkip, Millimeter Wave Channel Modeling and Cellular Capacity Evaluation, IEEE Journal on Selected Areas on Communications, vol. 32, no. 6, pp. 1164-1179, June 2014 9
Downlink CS-based channel training u BS use M 1 training beamforming vectors P u Each MS uses M 2 combining vectors Q for each basestation training vector u Received signal at user u can be written as Y MS = P Q H H u P + N u After vectorization & neglecting angle grid errors y MS = P ( P T Q H) ( A BS A MS ) z u + v, Vectorized received signal Training beamforming/combining matrices (realized using analog-only or hybrid in general) Sparse vector with a single nonzero at the AoA/AoD location Dictionary whose columns are BS/MS array response vectors with quantized grid angles 10
Downlink CS-based channel training u Vectorized signal can be written as y MS? P Measurement matrix Φ = P T Q H z u ` v Dictionary matrix Ψ = A BS A MS u Random compressed sensing measurements (similar for P & Q) [P] m,n = e jφ 2π m,n {0,,..., ( N BS Q 1 )2π m,n 2 } NQ BS NQ BS Different designs can be investigated # quantized angles zu u AoA/AoD estimation (equivalent to detecting the non-zero location of ) supp(z u ) = arg max Ψ H Φ H y MS Different support recovery algorithms can be investigated 11
Downlink data transmission F precoder H u Limited Feedback w U w 1 w u combiner u BS uses conjugate beamforming based on the estimated AoDs at the MS s ( ) ( ) ( )] F = [a BS ˆφ1, a BS ˆφ2,...,a BS ˆφU u MS s use maximum ratio combining (matched to the estimated AoA s) ) w u = a MS (ˆθu 12
Calculating the achievable rate u Key assumptions ª All channels are line-of-sight (generalization to NLOS is possible) ª Arrays are uniform (ULAs/UPAs) ª AoAs/AoDs are taken from a grid with critical quantization u The achievable rate of user u can be written as R u = log 2 1+ {ˆθ u =θ u } {ˆθ u =θ u,φ u = ˆφ u } U r=1 {φ u = ˆφ r } + 1 SNR U N BSN MS α u 2 Estimated AoAs/AoDs using the recovery algorithm 13
Achievable rate u The rate can be bounded as where { } [ R u E [ Ru { ( R u 1 U u Accounting for the channel coherence time, the rate is bounded by ( R u,eff R u 1 U )( 1 M ) ϵ (1 ϵ) N BS L C ( r u(φ u ˆφ r)} {ˆθ u =θ u,φ u = ˆφ u } { } { ) P { } N {ˆθu =θ u,φ u = ˆφ u } BS R u = log 2 (1+ SNR U N BSN MS α u 2) Single-user rate ] Robert W. Heath Jr. (2015) Probability of support recovery Single-user rate with no interference and perfect channel knowledge Channel coherence time (# symbols) # of CS measurements with support recovery error 14
Simulation results Setup: BS has ULA with 64 antennas MS has ULA s with 32 antennas Operating at 28 GHz Bandwidth 50 MHz TX-RX separation is 500 m Average transmit power of 37 dbm 2 bits for phase shifter angle quantization Random phase shifted P, Q OMP recovery Achievable Rate (bps/ Hz) 9 8 7 6 5 4 3 2 1 for a given, best M will be less than this Robert W. Heath Jr. (2015) Analog Beamforming Perfect Channel Knowledge Analog Beamforming CS Based Channel Estimation Lower Bound in (12) with 0.95 recovery success probability 0 0 50 100 150 200 250 300 350 400 450 500 Number of Measurements (M BS x M MS ) u Need for 200-300 random measurements to approach optimal rate u Order of magnitude less than exhaustive search ~ 64x32 u Optimizing the measurement matrices design should improve the performance u With N chains at the receiver, overhead will be (200-300)/N 15
Simulation results Training that maximizes the achievable rate 5 Setup: BS has ULA with 64 antennas 4 MS s have ULA s with 32 antennas Operating at 28 GHz 3 Bandwidth 50 MHz TX-RX separation is 500 m 2 Average transmit power of 37 dbm 2 bits for phase shifter angle quantization Effective Achievable Rate (bps/ Hz) 6 1 L C = 600 symbols L C = 400 symbols L C = 200 symbols 0 0 50 100 150 200 250 300 350 400 Number of Measurements (M BS x M MS ) u More CS measurements does not necessarily means better performance u CS measurements need to be adapted with the channel coherence time u Having more chains at the receivers will increases the efficiency 16
Conclusion u Compressed sensing based mmwave systems are promising ª An order of magnitude less training overhead vs exhaustive search solutions ª All users can simultaneously estimate their channels (similar codebooks) ª With hybrid receivers, chains can be used to reduce training overhead u Future work ª Investigate different CS beamforming/measurement matrices design ª Evaluating the performance for different sparse recovery algorithms (OMP, ) ª Extend to multi-path & wideband channels ª Include the impact of AoAs/AoDs quantization errors 17
Questions? Professor Robert W. Heath Jr. Wireless Networking and Communications Group Department of Electrical and Computer Engineering The University of Texas at Austin www.profheath.org 18