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– There is spatial information in the channel norm, especially when it is strong. – This motivates beamforming communication with channel norm feedback. BEAMFORMING UTILIZING CHANNEL NORM FEEDBACK IN MULTIUSER MIMO SYSTEMS Emil Björnson, David Hammarwall and Björn Ottersten School of Electrical Engineering Royal Institute of Technology (KTH) Stockholm, Sweden – The channel norm provides substantial spatial information. – Two simple channel norm feedback strategies are proposed that outperform opportunistic beamforming with the same amount of feedback. Spatial Correlation, example What information is gained of v i if the norm (v 1 ,v 2 ) 2 = v 2 1 + v 2 2 is observed? Figure 1: The shaded area shows how the probability mass is distributed over the circle, representing a given norm. The inner arrows indicate the standard deviation. Observe that the probability mass is more focused for larger norms. Consider two real-valued Gaussian distributed variables v i ∈N (0,σ 2 i ), i =1,2. System Model Limited Feedback Strategy Opportunistic beamforming: • The beamformer is chosen at random. • Each user feeds back its SNR. • Advantage: Knowledge of the SNR, Disadvantage: Neglects spatial information. Channel norm supported eigenbeamforming: • Each user feeds back its channel norm ρ k . • The beamformer is determined using conditional channel statistics. • Advantage: Exploits spatial information, Disadvantage: Requires SNR estimation. • This strategy is analyzed in the paper. ρ k Pilot/Feedback New channel realization Scheduling Data k ) /BF SN R k (ρ Antenna Selection Simulation Results Figure 2: The CDF (over scenarios) of the cell throughput in a system with a transmitting eight-antenna UCA and four receive antennas per user. Opportunistic beamforming is compared to the proposed Feed- back supported eigenbeamforming for different receive strategies: antenna selection and receive beamforming (increasing perfor- mance). The angular spread is 15 degrees. Figure 3: The mean cell throughput for different angular spreads. The proposed channel norm supported eigenbeamforming with antenna selection (circles) and receive beamforming (diamonds) is compared to the corresponding opportunistic beamforming. 0 10 20 30 5 6 7 8 9 10 11 Angular spread [degrees] Mean cell throughput [bits/symbol] Eigen−BF Opp. BF 6 7 8 9 10 11 0 0.2 0.4 0.6 0.8 1 Cell throughput [bits/symbol] 6 8 10 12 0 0.2 0.4 0.6 0.8 1 Cell throughput [bits/symbol] Eigen−BF Opp. BF • Elevated base station, n T antennas. • Multiple users, n R antennas. • Partial channel information at base station. Some directions are more favorable to a user: System Operation Channel information: Only R k and ρ k = max i h k ,i 2 , for each user k , is known to the base station. • Only the strongest receive antenna is used. • The channel norm is ρ k . MMSE estimation of SNR: E SNR k ρ k = E h H k ,strongest w T k 2 σ 2 k ρ k = w H T k R(ρ k )w T k σ 2 k V SNR k ρ k = E h H w T 4 ρ k − (w H T R(ρ k )w T ) 2 σ 4 k Closed-form expressions for R(ρ k ) and E h H w T 4 ρ k are previously known. • The antennas are combined to maximize the SNR. • The norm of the strongest receive channel is ρ k . • The other channel norms are known to be smaller. MMSE estimation of SNR: Receive Beamforming A strategy is needed to determine: • The beamforming vector for user k . • The corresponding SNR (and supported rate). • Proportional fair scheduling. • Rich scattering around the users. Capacity estimation based on: with α chosen to achieve outage probability 5%. Channel model: Rayleigh fading MIMO channel to user k : H k =[h k ,1 ,...,h k ,n R ] H with independent h k ,i ∈ CN (0,R k ). The received vector (user k ): y k (t)= H k w T k s k (t)+ n k (t), with beamforming vector w T k , symbol s k (t) and AWGN n k (t) ∈ CN (0,σ 2 k I). E SNR k ρ k = w H T k R(ρ k )w T k σ 2 k +( n R − 1) w H T k R(0 ≤ ρ ≤ ρ k )w T k σ 2 k V SNR k ρ k = V SNR k ρ k +( n R − 1)V SNR k 0 ≤ ρ ≤ ρ k Closed-form expressions for R(0 ≤ ρ ≤ ρ k ) and V SNR k 0 ≤ ρ ≤ ρ k are derived. E SNR k ρ k − α V SNR k ρ k ,
