Low-Overhead Receiver-side Channel Tracking for mmWave MIMO Karthik Upadhya † , Sergiy A. Vorobyov † , and Robert W. Heath, Jr. ∗ Department of Signal Processing and Acoustics, Aalto University † Wireless Networking and Commmunications Group, University of Texas at Austin ∗ Introduction ◮ mmWave transceivers are expected to employ large antenna arrays. ◮ mmWave channels are sparse in the angular domain. ◮ The communication link is susceptible to changes in the AoA or AoD. ◮ In hybrid beamforming architectures, the transceiver can look only in a few directions. ◮ In use cases such as hand-held transceivers, drones etc., the AoAs change, but the AoDs remain constant. Main Contribution ◮ Algorithm for blind subspace estimation at the receiver. ◮ AoAs are obtained from the estimated subspace. ◮ Useful for low-latency communication since a very low overhead is required. System Model and Initial Channel Estimation ◮ Received observations in symbol k of downlink y [k ]= W H HFs [k ]+ W H q [k ] W ∈ C N UE ×N s : Receive combiner, F ∈ C N AP ×N s : Transmit precoder H ∈ C N UE ×N AP : Channel matrix. ◮ Channel model (assuming ULA) H = P −1  p =0 α p a UE (φ p ) a H AP (ψ p )= A UE DA H AP ≈ ¯ A UE ¯ D ¯ A H AP a UE (·) and a AP (·) : steering vector at the UE and AP. α p , φ p , and ψ p : Path gain, AoA, and AoD of path p . ¯ A AP , ¯ A UE : Matrix of steering vectors containing quantized angles. ¯ D ∈ C G UE ×G AP : sparse matrix with non-zero locations corresponding to the AoA and AoD pairs. ◮ M AP training symbols transmitted by AP. UE makes M UE measurements for each training symbol. ◮ J  M AP M UE received observations for training : Y = W H HF + Q ≈ W H ¯ A UE ¯ D ¯ A H AP F + Q ◮ Sparse recovery of channel :  d = min d ‖d ‖ 0 subject to ‖y − Ψd ‖ 2 ≤ ǫ y  vec (Y ), d  vec ( ¯ D ) , Ψ  F T ¯ A ∗ AP ⊗ W H ¯ A UE Proposed Method ◮ Coherence block assumed to be divided into M AP × M UE sub blocks. ◮ During sub-block (m , n ), AP uses precoder F m and UE uses combiner W n . ◮ Hybrid architecture at the AP and UE ◮ N RF−T UE (N RF−T AP ) out of N RF UE (N RF AP ) reserved for channel estimation at the UE (AP). = ⇒ W n   W d , W t n  and F m   F d , F t m  ◮ The covariance matrix of the observations within sub-block (m , n ): R m,n  E  y m,n [k ] y H m,n [k ]  = W H n HF m F H m H H W n + σ 2 W H n W n ◮ Summed over all m : R n  M AP  m=1 R m,n = W H n XW n + σ 2 M AP W H n W n X  HFF H H H F  [F 1 ,..., F M AP ] Proposition Let H = U s Σ s V s . Then, span {X } = span {U s } if and only if F is chosen such that V H s F has full row-rank. = ⇒ Basis vectors of span {H } can be obtained without knowing F as long as F is such that AP transmits in the directions of all the AoDs of the channel. Blind Subspace Estimation ◮ X is low-rank, therefore can be estimated using matrix completion methods. ◮ Alternatively, X can be sparsified using a dictionary and recovered using sparse reconstruction. ◮ Vectorizing {R n } M UE n =1 and stacking r      r 1 . . . r M UE     =     Ψ 1 . . . Ψ M UE     vec  ¯ DG ¯ D H  +     q 1 . . . q M UE     G  ¯ A H AP FF H ¯ A AP Ψ n    ¯ A H UE W n  T ⊗ W H n ¯ A UE  ◮ vec  ¯ DG ¯ D H  is sparse. ◮ The columns of ¯ A UE corresponding to non-zero rows of ¯ D span the column space of H . Choice of F t m and W t n ◮ F t m has to be chosen such that V H s F is full rank. ◮ To avoid interference to transmitted data, F t m = Π ⊥ F d ¯ F t m =  I − F d  F d H F d  −1 F d H  ¯ F t m ◮ We have chosen W t n to have random values = ⇒ diffused beams in random directions. Updating W d ◮ Given a basis B for span {U s }, W d can be chosen to satisfy ZF condition, i.e., (W d ) H HF d = I . ◮ Resulting W d = BP † where P  B H HF d . ◮ P can be estimated using N RF−T UE pilot symbols. Simulation Results ◮ N AP = 64 antennas, N UE = 32 antennas, N RF AP = N RF UE = 4, N RF−T UE = 1. ◮ M AP = 12 symbols, M UE = 3 symbols for initial channel estimation. ◮ For subspace estimation M AP = 1 and M UE = 20. ◮ Each block has 256 symbols. So, channel is constant for 5120 symbols. ◮ Channel has an LOS path with φ = 90 ◦ and NLOS cluster with 100 paths with angular spread 10 ◦ and mean angle φ = 45 ◦ . ◮ NLOS is at 10dB lower power than LOS path. ◮ Angular difference between each block of 5120 symbols is distributed as CN (0,σ 2 φ ). ◮ Path amplitudes varies across coherence blocks according to Gauss Markov model with factor 0.8. 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 Block index ℓ Average achievable rate (bps/Hz) Proposed method No tracking Method in [2] Figure: Plot of the average achievable rate vs block index at SNR = 0 dB, σ φ =2 ◦ and σ ψ =0 0.10.20.30.40.50.60.70.80.91 4 5 6 7 8 9 σ φ in degrees Average Achievable Rate (bps/Hz) Proposed method No Tracking Method in [2] Figure: Plot of the average achievable rate vs σ φ at SNR = 0 dB and σ ψ = 0 at the ℓ = 50th block. Conclusion ◮ Proposed a blind channel tracking algorithm for mmWave MIMO. ◮ Possible research directions : Design F t and W t adaptively, extend to the multi-user case, and remove the requirement of dedicated RF chain for training are possible research directions. References [1] R. W. Heath, N. Gonzlez-Prelcic, S. Rangan, W. Roh and A. M. Sayeed, An Overview of Signal Processing Techniques for Millimeter Wave MIMO Systems IEEE J. Sel Topics Sig. Process., vol. 10, no. 3, pp. 436-453, April 2016. [2] N. Garcia, H. Wymeersch, and D. T. M. Slock, Optimal robust precoders for tracking the AoD and AoA of a mm-Wave path. ArXiv, 2017, http://arxiv.org/abs/1703.10978.