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Channel Models, Favorable Propagation and Multi-Stage Linear Detection in Cell-Free Massive MIMO Laura Cottatellucci Dirk Slock Roya Gholami 2020 IEEE International Symposium on Information Theory (ISIT), 21-26 June 2020

Outline I. Objectives and Motivations II. System and Channel Model III. Analytical Conditions of Favorable Propagation IV. Mathematical Results for DAS and CF massive MIMO Analysis V. Favorable Propagation in Cell-Free Massive MIMO VI. Performance Analysis of Multistage Detectors VII. Simulation Results VIII. Summary and Conclusions R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO”

I. Objectives and Motivations Distributed Antenna Systems (DASs)  Access Points (APs) geographically distributed 𝑂 𝑆 antennas  APs connected to a CPU performing joint decoding Advantages:  Allowing accommodation of more users 𝑂 𝑈 users  Higher data rates  Reducing users ’ energy consumption Challenge:  Improving transmission quality • High receiver complexity R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 1

I. Objectives and Motivations In Massive MIMO Systems , as 𝑂 𝑆 → ∞ and 𝑂 𝑈 remains constant, the users’ channels become orthogonal Favorable Propagation Low complexity Matched Filters are optimum Cell-Free Massive MIMO Systems  DAS comprising a massive number of APs jointly serving a much smaller number of users Does Favorable Propagation hold in CF massive MIMO ? R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 2

II. System Model • Users and APs randomly and independently distributed 𝑂 𝑈 = 𝜍 T 𝑀 2 users • Users′ intensity 𝜍 T → 𝑀 → 𝑂 R = 𝜍 R 𝑀 2 APs • APs′ intensity 𝜍 R • All users transmit with same power 𝑄 Received signal at the processing unit Noise 𝑂 𝑺 × 1 𝒛 = 𝑄 𝑯 𝒚 + 𝒐 Channel Matrix Transmit Symbol Vector 𝑂 𝑆 × 𝑂 𝑼 𝑂 𝑼 × 1 R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 3

II. Channel Model  ( 𝑗, 𝑘 )-th element of the Path Loss Matrix ෠ 𝐇 𝛽 𝑒 0 if 𝑒 𝑗𝑘 > 𝑒 0 𝑕 ( 𝑒 𝑗𝑘 ) = ቐ 𝑒 𝑗𝑘𝛽 𝑕 𝑗,𝑘 = ො ො 1 otherwise 𝑕 ( 𝑒 ) • 2𝛽 : Path Loss exponent ො 𝑒 0 𝑒 𝑗𝑘 ) 𝛽 • ( 𝑒 0 : Reference distance • 𝑒 𝑗𝑘 = 𝒔 𝑗 − 𝒖 𝑘 2 : Euclidean distance between user 𝑘 and AP 𝑗 𝒔 𝑗 = ( 𝑠 𝑗,𝑦 , 𝑠 𝑗,𝑧 ) • • 𝒖 𝑘 = ( 𝑢 𝑘,𝑦 , 𝑢 𝑘,𝑧 ) 𝑒 0 𝑒 R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 4

II. Channel Model Channel Coefficients for Path Channel Coefficients for Loss and Rayleigh Fading Path Loss and LoS 𝑕 ( 𝑒 𝑗𝑘 ) 𝑓 −i 2𝜌 𝜇 −1 𝑒 𝑗𝑘 𝑕 𝑗,𝑘 = 𝑕 ( 𝑒 𝑗𝑘 ) = ො 𝑕 𝑗,𝑘 = 𝑕 ( 𝑒 𝑗𝑘 ) = ො 𝑕 ( 𝑒 𝑗𝑘 ) ℎ 𝑗,𝑘 Phase Rotation Path Loss Path Loss Rayleigh fading • • 𝜇 : Radio signal wavelength ℎ 𝑗,𝑘 : i.i.d complex Gaussian variables with zero mean and unit variance R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 5

III. Analytical Conditions of Favorable Propagation Eigenvalue moment of order 𝑚 of the channel covariance matrix 𝑫 = 𝐇 𝐼 𝐇 (𝑚) = ׬ 𝜈 𝑚 𝑒𝐺 𝑫 ( 𝜈 ) = 𝔽{ 1 𝑂 𝑈 trace (𝑫 𝑚 )} 𝑛 𝑫 • 𝜈 : eigenvalue, 𝐺 𝑫 ( 𝜈 ): empirical eigenvalue distribution of matrix 𝑫 Analytical conditions of Favorable propagation (𝑚) 𝑛 𝑫 ∀𝑚 ∈ 𝑂 + Moment Ratio : MR( 𝑚 ) = trace{ ( diag 𝑫 ) 𝑚 }/𝑂 𝑈 ≈ 1 Do Favorable Propagation Conditions Hold in CF Massive MIMO ? For LoS Channels ? And for Path Loss and Rayleigh fading ? R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 6

IV. Mathematical Results for DAS and CF massive MIMO Analysis 𝐇 decomposable as ෡ 𝐇 = 𝛚 𝑆 ෡ Euclidean matrix ෡ 𝐼 𝑼 𝛚 𝑈 𝜐 ( 𝜄 2 × 𝜄 2 ) : deterministic depending on ො • ෡ 𝑼 𝑕 ( 𝑒 ) • 𝛚 𝑆 (𝑂 𝑆 × 𝜄 2 ) : depending on Rx locations • 𝛚 𝑈 ( 𝑂 𝑼 × 𝜄 2 ) : depending on Tx locations 𝑀 = 𝜐 𝜄 How matrices 𝛚 𝑆 , 𝛚 𝑈 , and ෡ 𝑼 are obtained? Consider a 𝜄 2 × 𝜄 2 channel matrix ෡ 𝓗 of a system with 𝜄 2 transmit and receive nodes regularly spaced on a grid.  ෡ 𝓗 is a symmetric block Toeplitz matrix of 𝜄 × 𝜄 blocks 𝓗 admits an eigenvalue decomposition based on the 𝜄 2 × 𝜄 2 Fourier matrix F as 𝜄 2 → ∞ : ෡ 𝓗 = F ෡ 𝑼 F 𝐼  ෡ 𝛚 R and 𝛚 T obtained by extracting uniformly at random 𝑂 R and 𝑂 T rows from matrix F . R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 7

IV. Mathematical Results: Channel Matrix with Path Loss and LoS 𝐼  Decomposition of channel matrix for path loss and LoS channel: 𝐇 = 𝛚 𝑆 𝑼 𝛚 𝑈  Approximation by ෩ 𝐼 𝐇 = 𝛠 𝑆 𝑼 𝛠 𝑈 where 𝛠 𝑆 and 𝛠 𝑈 consist of i.i.d. Gaussian elements of zero mean and variance 𝜄 −2 . Fundamental Results 𝐇 𝐼 ෩  Recursive algorithm to compute the eigenvalue moments of ෩ 𝑫 = ෩ 𝐇 , as 𝑂 𝑆 𝑂 𝑈 𝜄 2 , 𝑂 𝑆 , 𝑂 𝑈 → ∞ with 𝜄 2 → 𝛾 𝑆 and 𝜄 2 → 𝛾 𝑈 (𝑚) only on 𝛾 𝑆 , 𝛾 𝑈 , and 𝑛 𝑼  Dependence of eigenvalue moments 𝑛 ෩ 𝑫 (𝑚) , ∀𝑙 , 𝑚 , ሚ (𝑚) = 𝑛 ෩ (𝑚) being the diagonal elements of matrix ෩ ሚ 𝑫 𝑚  𝐷 𝑙𝑙 𝐷 𝑙𝑙 𝑫 R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 8

IV. Mathematical Results: Channel with Path Loss and Rayleigh Fading Eigenvalue Moments 𝐇 𝐼 ෩ As L → ∞ , the eigenvalue moments of matrix ෩ 𝑫 = ෩ 𝐇 converge to a deterministic value 𝑚 𝑚 − 1 𝑚 (𝑚) = 1 𝑙 𝛾 𝑆 2 𝑚−1 𝑚−𝑙 σ 𝑙=0 𝑛 ෩ 𝑛 ෡ 𝛾 𝑈 ; 𝑚 ≥ 1 𝑫 𝑼 𝑙 𝑙 𝑙+1 (𝑚) depend only on 𝛾 𝑈 , 𝛾 𝑆 , and 𝑛 ෡ 2 The eigenvalue moments 𝑛 ෩ 𝑼 𝑫 R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 9

V. Favorable Propagation Conditions in CF massive MIMO with Rayleigh fadin g Τ For any 𝑚 , as 𝛾 𝑆 → ∞ and 𝛾 𝑈 is kept constant, i.e., for 𝛾 𝑈 𝛾 𝑆 → 0 , and 𝛾 𝑈 > 0 (𝑚) 𝑀→∞ 𝑛 ෩ 𝑫 MR(𝑚) = 1 trace{ ( 𝑒𝑗𝑏𝑕 ෩ 𝑫 ) 𝑚 }/𝑂 𝑈 Favorable Propagation Conditions are satisfied in CF Massive MIMO with Path Loss and Rayleigh fading Low Complexity Matched Filters are almost optimum R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 10

V. Favorable Propagation Conditions in CF massive MIMO with Path Loss and LoS Τ For 𝑚 = 2 , 3 : as 𝛾 𝑆 → ∞ and 𝛾 𝑈 > 0 , i.e., for 𝛾 𝑈 𝛾 𝑆 → 0 (4) 𝑀→∞ 1 + 𝛾 𝑈 𝑛 𝑈 MR(2) (2) ) 2 ( 𝑛 𝑈 (4) (6) 𝑀→∞ 1 + 3𝛾 𝑈 𝑛 𝑈 𝑛 𝑈 2 MR(3) (2) ) 2 + 𝛾 𝑈 (2) ) 3 ( 𝑛 𝑈 ( 𝑛 𝑈 Favorable propagation does not hold in CF massive MIMO with LoS Linear multistage detectors are expected to provide substantial gains compared to Matched Filter R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 11

VI. Multistage Detectors (𝑚) and 𝑛 ෩ (𝑚) , Cottatellucci et al. ‘05  Asymptotic analysis and design based on ሚ 𝐷 𝑙𝑙 𝑫 • For an 𝑁 -stage detector define matrix 𝑌 (2) + 𝜏 2 𝑌 (1) 𝑌 (𝑁+1) + 𝜏 2 𝑌 (𝑁) … 𝑌 (3) + 𝜏 2 𝑌 (2) 𝑌 (𝑁+2) + 𝜏 2 𝑌 (𝑁+1) … S M ( 𝑌 ) = ⋮ ⋱ ⋮ 𝑌 (𝑁+1) + 𝜏 2 𝑌 (𝑁) 𝑌 (2𝑁) + 𝜏 2 𝑌 (2𝑁−1) … and a vector s M ( 𝑌 ) = [𝑌 1 , 𝑌 2 , … , 𝑌 (𝑁) ] 𝑈 • Polynomial Expansion detectors , Moshavi et al. ‘96: 𝑌 = 𝑛 ෩ 𝑫 • Multistage Wiener filters (MSWF), Goldstein et al. ‘98 : 𝑌 = ሚ 𝐷 𝑙𝑙 In DASs, MSWF and Polynomial Expansion Detectors are equivalent with −1 𝑛 ෩ 𝑈 𝑈 𝒕 𝑁 𝑛 ෩ 𝑫 𝑻 𝑁 𝑫 𝒕 𝑁 𝑛 ෩ SINR 𝑁 = 𝑫 −1 𝑛 ෩ 𝑈 𝑈 1−𝒕 𝑁 𝑛 ෩ 𝑫 𝑻 𝑁 𝑫 𝒕 𝑁 𝑛 ෩ 𝑫 R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi - Stage Linear Detection in CF Massive MIMO” 12