Joint Space-Division and Multiplexing: How to Achieve Massive MIMO - PowerPoint PPT Presentation

Communication Theory Workshop Joint Space-Division and Multiplexing: How to Achieve Massive MIMO Gains in FDD Systems Giuseppe Caire University of Southern California, Viterbi School of Engineering, Los Angeles, CA Phuket, Thailand, June 23-26, 2013

Channel estimation bottleneck on MU-MIMO • High-SNR capacity of N t × N r single-user MIMO with coherence block-length T [Zheng-Tse, 2003]: M ∗ = min { N t , N r , T/ 2 } C ( SNR ) = M ∗ (1 − M ∗ /T ) log SNR + O (1) , • Trivial cooperative bound: for large M = N t and N = KN r , the coherence block T is the limiting factor. • ⇒ Disappointing theoretical performance of “CoMP” (base station cooperation), in FDD. 18 γ =1, τ =1/32 16 γ =2, τ =1/32 γ =4, τ =1/32 14 γ =8, τ =1/32 Cell sum rate (bps/Hz) 12 10 8 cluster controller BS 3 6 4 BS 2 2 BS 1 0 0 5 10 15 20 25 B 1

Channel model with antenna correlation • In FDD, for large macro-cellular base stations, we have to exploit channel dimensionality reduction while still exploiting the large number of antennas at the BS. • Idea: exploit the asymmetric spatial channel correlation at the BS and at the UTs. • Isotropic scattering, | u − u ′ | = λD : � π E [ h ( u ) h ∗ ( u ′ )] = 1 e − j 2 πD cos( α ) dα = J 0 (2 πD ) 2 π − π • Two users separated by a few meters (say 10 λ ) are practically uncorrelated. 2

• In contrast, the base station sees user groups at different AoAs under narrow AS ∆ ≈ arctan( r / s ) . r s scattering ring ∆ ∆ θ region containing the BS antennas • This leads to the Tx antenna correlation model h = UΛ 1 / 2 w , R = UΛU H with � ∆ [ R ] m,p = 1 e j k T ( α + θ )( u m − u p ) dα. 2∆ − ∆ 3

Joint Space Division and Multiplexing (JSDM) • K users selected to form G groups, with ≈ same channel correlation. H = [ H 1 , . . . , H G ] , with H g = U g Λ 1 / 2 W g . g • Two-stage precoding: V = BP . • B ∈ C M × b g is a pre-beamforming matrix function of { U g , Λ g } only. • P ∈ C b g × S g is a precoding matrix that depends on the effective channel. • The effective channel matrix is given by   H H H H H H · · · 1 B 1 1 B 2 1 B G   H H H H H H · · · 2 B 1 2 B 2 2 B G H H =    . . . . ...  . . . . . . H H H H H H · · · G B 1 G B 2 G B G 4

• Per-Group Processing: If estimation and feedback of the whole H is still too costly, then each group estimates its own diagonal block H g = B H g H g , and P = diag( P 1 , · · · , P G ) . • This results in � y g = H H H H g B g P g d g + g B g ′ P g ′ d g ′ + z g g ′ � = g 5

Achieving capacity with reduced CSIT • Let r = � G g =1 r g and suppose that the channel covariances of the G groups are such that U = [ U 1 , · · · , U G ] is M × r tall unitary (i.e., r ≤ M and U H U = I r ). • Eigen-beamforming (let b g = r g and B g = U g ) achieves exact block diagonalization. • The decoupled MU-MIMO channel takes on the form H P g d g + z g = W H g Λ 1 / 2 y g = H g P g d g + z g , for g = 1 , . . . , G, g where W g is a r g × K g i.i.d. matrix with elements ∼ CN (0 , 1) . Theorem 1. For U tall unitary, JSDM with PGP achieves the same sum capacity of the corresponding MU-MIMO downlink channel with full CSIT. 6

Block Diagonalization • For given target numbers of streams per group { S g } and dimensions { b g } satisfying S g ≤ b g ≤ r g , we can find the pre-beamforming matrices B g such that: ∀ g ′ � = g, U H and rank( U H g B g ) ≥ S g g ′ B g = 0 • Necessary condition for exact BD Span( B g ) ⊆ Span ⊥ ( { U g ′ : g ′ � = g } ) . • When Span ⊥ ( { U g ′ : g ′ � = g } ) has dimension smaller than S g , the rank condition on the diagonal blocks cannot be satisfied. • In this case, S g should be reduced (reduce the number of served users per group) or, as an alternative, approximated BD based on selecting r ⋆ g < r g dominant eigenmodes for each group g can be implemented. 7

Performance analysis with regularized ZF • The transformed channel matrix H has dimension b × S , with blocks H g of dimension b g × S g . • For simplicity we allocate to all users the same fraction of the total transmit power, p g k = P S . • For PGP , the regularized zero forcing (RZF) precoding matrix for group g is given by P g, rzf = ¯ ζ g ¯ K g H g , where � � − 1 ¯ H g H H K g = g + b g α I b g and where S ′ ¯ ζ 2 g = . tr ( H H g K H g B H g B g K g H g ) 8

• The SINR of user g k given by S ¯ g k B g ¯ P ζ 2 g h g k | 2 g | h H K g B H γ g k , pgp = � � � j � = k ¯ g k B g ¯ g h g j | 2 + P j ¯ g k B g ′ ¯ j | 2 + 1 P ζ 2 K g ′ B H ζ 2 g | h H K g B H g ′ | h H g ′ h g ′ g ′ � = g S S • Using the “deterministic equivalent” method of [Wagner, Couillet, Debbah, Slock, 2011], we can calculate γ o g k , pgp such that M →∞ γ g k , pgp − γ o − → 0 g k , pgp 9

Example • M = 100 , G = 6 user groups, Rank ( R g ) = 21 , effective rank r ∗ g = 11 . • We serve S ′ = 5 users per group with b ′ = 10 , r ⋆ = 6 and r ⋆ = 12 . • For r ∗ g = 12 : 150 bit/s/Hz at SNR = 18 dB: 5 bit/s/Hz per user, for 30 users served simultaneously on the same time-frequency slot. 350 350 Capacity Capacity ZFBF, JGP ZFBF, JGP RZFBF, JGP RZFBF, JGP ZFBF, PGP ZFBF, PGP 300 300 RZFBF, PGP RZFBF, PGP 250 250 Sum Rate 200 Sum Rate 200 150 150 100 100 50 50 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 SNR (in dBs) SNR (in dBs) 10

Training, Feedback and Computations Requirements • Full CSI: 100 × 30 channel matrix ⇒ 3000 complex channel coefficients per coherence block (CSI feedback), with 100 × 100 unitary “common” pilot matrix for downlink channel estimation. • JSDM with PGP: 6 × 10 × 5 diagonal blocks ⇒ 300 complex channel coefficients per coherence block (CSI feedback), with 10 × 10 unitary “dedicated” pilot matrices for downlink channel estimation, sent in parallel to each group through the pre-beamforming matrix. • One order of magnitude saving in both downlink training and CSI feedback. • Computation: 6 matrix inversions of dimension 5 × 5 , with respect to one matrix inversion of dimension 30 × 30 . 11

Non-ideal CSIT • Parallel downlink training in all groups: a scaled unitary training matrix X tr of dimension b ′ × b ′ is sent, simultaneously, to all groups in the common downlink training phase. • Received signal at group g receivers is given by � Y g = H H H B g ′ X tr + Z g . g X tr + H g g ′ � = g • Multiplying from the right by X H tr and letting ρ tr denote the power allocated to training, we obtain � tr = ρ tr H H Y g X H H B g ′ + Z g X H g + ρ tr tr . H g g ′ � = g 12

• The relevant observation for the g k -th user effective channel is:   � h g k = √ ρ tr h g k + √ ρ tr � B H  h g k + � z g k . g ′ g ′ � = g • The corresponding MMSE estimator is given by � � � � − 1 � H H � h g k � h g k � � h g k = E h E h h g k g k g k     − 1 G G � � = √ ρ tr �  B H   ρ tr B H  g ′ R g B g ′′ + I b ′ h g k g R g B g ′ g ′ =1 g ′ ,g ′′ =1 R g O T � � � − 1 � 1 R g O T + 1 M g ˜ O ˜ � = h g k √ ρ tr I b ′ ρ tr 13

g h g k , and we introduced the b ′ × b block where we used the fact that h g k = B H matrices = [ 0 , . . . , 0 , , 0 , . . . , 0 ] M g I b ′ ���� block g = [ I b ′ , I b ′ , . . . , I b ′ ] . O • Notice that in the case of perfect BD we have that R g B g ′ = 0 for g ′ � = g . Therefore, the MMSE estimator reduces to � � − 1 1 R g + 1 � ¯ ¯ � h g k = h g k √ ρ tr R g I b ′ ρ tr where ¯ R g = B H g R g B g . 14

• Also in this case, the deterministic equivalent approximations of the SINR terms for RZFBF and ZFBF precoding can be be computed. • Eventually, the achievable rate of user g k is given by � � � � 1 − b ′ γ o × log R g k , pgp , csit = max T , 0 1 + � . g k , pgp , csit 15

Tradeoff parameter b ′ • b ′ large yields better conditioned matrices, but it “costs” more in terms of training phase dimension. SNR = 30 dB SNR = 10 dB 230 220 80 210 RZFBF, PGP, ICSI RZFBF, PGP, ICSI ZFBF, PGP, ICSI 70 ZFBF, PGP, ICSI 200 RZFBF, PGP RZFBF, PGP ZFBF, PGP ZFBF, PGP 190 Sum Rates 60 Sum Rates 180 170 50 160 40 150 140 30 130 4 6 8 10 12 14 16 4 6 8 10 12 14 16 b‘ b‘ (a) S’ = 4, SNR = 10dB (b) S’ = 8, SNR = 30dB 16

Impact of non-ideal CSIT 300 400 Full CSI, RZFBF Full CSI, RZFBF Full CSI, ZFBF Full CSI, ZFBF JGP, RZFBF JGP, RZFBF JGP, ZFBF JGP, ZFBF PGP, RZFBF PGP, RZFBF 350 PGP, ZFBF PGP, ZFBF 250 PGP ICSI, RZFBF PGP ICSI, RZFBF PGP ICSI, ZFBF PGP ICSI, ZFBF 300 200 250 Sum Rate Sum Rate 150 200 150 100 100 50 50 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 SNR (in dBs) SNR (in dBs) (c) S’ = 4 (d) S’ = 8 17

Discussion: is the tall unitary realistic? • For a Uniform Linear Array (ULA), R is Toeplitz, with elements � ∆ [ R ] m,p = 1 e − j 2 πD ( m − p ) sin( α + θ ) dα, m, p ∈ { 0 , 1 , . . . , M − 1 } 2∆ − ∆ • We are interested in calculating the asymptotic rank, eigenvalue CDF and structure of the eigenvectors, for M large, for given geometry parameters D, θ, ∆ . • Correlation function � ∆ r m = 1 e − j 2 πDm sin( α + θ ) dα. 2∆ − ∆ 18