The ergodic high SNR capacity of the Introduction - PowerPoint PPT Presentation

ITW Jeju, South Korea, 2015 1 / 18 Ramy Gohary and Halim Yanikomeroglu The ergodic high SNR capacity of the Introduction spatially-correlated non-coherent MIMO System Model channel within an SNR-independent gap The right singular vectors of X Asymptotic high SNR Ramy Gohary and Halim Yanikomeroglu 1 non-coherent capacity Bounds Main Result Systems and Computer Engineering Dept., Carleton University, Canada Conclusions October 2015 1Work supported by Huawei Inc. and Ontario Research Funding

ITW Jeju, South Korea, Introduction 2015 2 / 18 Ramy Gohary and Halim Yanikomeroglu • Non-coherent MIMO communication system: No Introduction channel state information (CSI) is available at either the System Model Tx or Rx. The right singular vectors of X • The analysis of non-coherent systems accounts for the Asymptotic communication resources expended to acquire high SNR accurate CSI. non-coherent capacity • Training cost tolerable in static and slow fading Bounds Main Result scenarios, but not in fast and block fading ones. Conclusions • In fast fading, more beneficial to use signalling strategies that do not require Rx to know CSI. (Hochwald et al. ’00, Zheng et al. ’02)

ITW Jeju, South Korea, Previous Work 2015 3 / 18 Ramy Gohary • For spatially-white channel, input matrices that achieve and Halim Yanikomeroglu capacity have the following structures: • At any SNR and any coherence interval, the product of Introduction an isotropically distributed unitary component and a System Model diagonal component with non-negative entries. The right singular (Hochwald et al. ’00) vectors of X • At high SNRs and coherence interval greater than a Asymptotic high SNR threshold, τ , isotropically distributed unitary on the non-coherent Grassmann manifold. (Zheng et al. ’02) capacity Bounds • At high SNRs and coherence interval less than τ , the Main Result product of an isotropically distributed unitary component Conclusions and a diagonal component with random entries distributed as the square root of the eigenvalues of a beta matrix. (Yang et al. ’13) • At low SNRs, only one entry of the diagonal component is potentially non-zero. (Srinivasan et al. ’09) • What about spatially-correlated channels?

ITW Jeju, South Korea, Spatial correlation 2015 4 / 18 Ramy Gohary • Spatial correlation arises due to proximity of physical and Halim Yanikomeroglu antennas, especially in prospective massive MIMO systems. Introduction System Model • Correlation nonnegligible, even when spacing exceeds The right multiple wavelengths. singular vectors of X • Kronecker model: left and right multiplication of the Asymptotic spatially-white channel matrix with Tx and Rx high SNR non-coherent correlation matrices. capacity Bounds • Correlation matrices, vary much more slowly than Main Result instantaneous channel parameters. Can be estimated Conclusions accurately and assumed known. (Yu et al. ’04) • Correlation: significant impact on signalling methodology and achievable rate. • Kronecker correlation noncoherent model considered in (Jafar et al. ’05) at any SNR.

ITW Jeju, South Korea, Non-coherent communication 2015 5 / 18 on spatially correlated channels: Ramy Gohary and Halim Yanikomeroglu What is not known? Introduction System Model The right singular vectors of X Asymptotic • No closed-form expressions for capacity, or bounds high SNR non-coherent thereof. capacity Bounds • No constructive signalling strategy to approach Main Result capacity. Conclusions

ITW Jeju, South Korea, This Work 2015 6 / 18 Ramy Gohary and Halim Yanikomeroglu • Derive an expression for the ergodic high SNR non-coherent capacity for block Rayleigh fading Introduction channels with Kronecker correlation. System Model The right • Expression accurate within an SNR-independent gap singular vectors of X and an error that decays as 1 / SNR. Asymptotic • Derive an upper bound on the gap to the actual high SNR non-coherent capacity. Gap decreases monotonically with logarithm capacity Bounds of condition number of Tx correlation. Main Result • Show that input signals that achieve capacity lower Conclusions bound can be expressed as product of isotropically distributed random Grassmannian component and deterministic component comprising eigenvectors and inverse of eigenvalues of Tx correlation matrix.

ITW Jeju, South Korea, System Model 2015 7 / 18 Ramy Gohary and Halim Yanikomeroglu • Frequency-flat block Rayleigh fading channel with equal Introduction number of Tx and Rx antennas, M . System Model • Correlated signals emitted from Tx and correlated The right singular signals impinging on Rx. Channel vectors of X Asymptotic H = A 1 / 2 H w B 1 / 2 , high SNR non-coherent capacity where A and B are Tx and Rx pd correlation matrices, Bounds Main Result and H w random with zero-mean unit-variance i. i. d. Conclusions circularly-symmetric complex Gaussian entries. • We assume A and B are full rank and Tr A = Tr B = 1. • Block fading model with coherence time T .

ITW Jeju, South Korea, System Model (cont’d) 2015 8 / 18 Ramy Gohary and Halim Yanikomeroglu • The received signal matrix can be expressed as Introduction Y = X A 1 / 2 H w B 1 / 2 + V , System Model The right singular where X ∈ C T × M is Tx signal, and V ∈ C T × M additive vectors of X noise; the entries of V are i. i. d. standard complex Asymptotic high SNR Gaussian random variables. non-coherent capacity Bounds • Tx power constraint: Main Result E { Tr ( XX † ) } ≤ TP . Conclusions • The matrices A and B are known but H w is not.

ITW Jeju, The right singular vectors of X South Korea, 2015 9 / 18 Ramy Gohary and Halim Yanikomeroglu • Conditioned on X , Y is Gaussian and Introduction � − 1 vec ( Y ) � B ⊗ X A X † + I MT � − vec † ( Y ) System Model � exp p ( Y | X ) = . The right B ⊗ X A X † + I MT π TM det singular � � vectors of X Asymptotic high SNR • For deterministic Φ , p (Φ Y | Φ X ) = p ( Y | X ) , yielding non-coherent capacity optimal Bounds X = Q X D U † Main Result A , where Conclusions • Q X isotropically distributed unitary matrix; • D random diagonal with non-negative entries; and • U A is the matrix containing the eigenvectors of A .

ITW Jeju, Conditional Entropy h ( Y | X ) South Korea, 2015 10 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction 1 C ( P ) = h ( Y ) − h ( Y | X ) � � max . System Model T p ( X ) , E { Tr ( XX † ) }≤ TP The right singular Our goal is to evaluate C ( P ) as P → ∞ . vectors of X • Evaluating h ( Y | X ) straightforward Asymptotic high SNR non-coherent capacity h ( Y | X ) = MT log π e Bounds Main Result + M log det AB + E { log det D 2 } + O ( 1 / P ) . Conclusions • Approximation valid when D full rank and its entries scale with P .

ITW Jeju, Nonconditional Entropy h ( Y ) South Korea, 2015 11 / 18 Ramy Gohary and Halim Yanikomeroglu • Computing entropy of signal component plus noise component formidable task. Introduction System Model • At high SNR, write The right singular h ( Y ) = h ( X A 1 / 2 H w B 1 / 2 ) + O ( 1 / P ) vectors of X Asymptotic = h ( Q X D Λ 1 / 2 A H w ) + T log det B + O ( 1 / P ) . high SNR non-coherent capacity Bounds Main Result • Λ A diagonal matrix of eigenvalues. Conclusions • The matrix Q X ∈ C T × M , T ≥ M . • An expression for h ( Q X D Λ 1 / 2 A H w ) can be obtained by transforming from Cartesian to QR coordinates (Zheng et al. ’02).

ITW Jeju, South Korea, 2015 12 / 18 Ramy Gohary and Halim • Coordinate change yields Yanikomeroglu h ( Q X D Λ 1 / 2 A H w ) = h ( Ψ D Λ 1 / 2 Introduction A H w ) System Model + log | G M ( C T ) | + ( T − M ) E { log det H † w D 2 Λ A H w } . The right singular vectors of X Asymptotic • G M ( C T ) is the Grassmann manifold; and high SNR non-coherent • Ψ ∈ C M × M isotropically distributed. capacity Bounds • Computing h ( Ψ D Λ 1 / 2 A H w ) is the difficult part. Main Result • Without spatial correlation Λ A = I M and optimal D = I M . Conclusions • For case with spatial correlation, we develop bounds.

ITW Jeju, South Korea, Upper Bound on Capacity 2015 13 / 18 Ramy Gohary • Gaussian distribution maximizes entropy yields and Halim Yanikomeroglu A H w ) ≤ M 2 log π eT h ( Ψ D Λ 1 / 2 M λ A 1 P . (1) Introduction System Model The right singular • Bound not achievable unless A = 1 M I M and vectors of X E { D 2 } = PT M I M . Asymptotic high SNR non-coherent • Upper bound on capacity: capacity Bounds 1 − M log TP 1 − 2 M Main Result � � � � C ( P ) ≤ M log det A π eM + T T Conclusions + M 2 T log λ A 1 + 1 T log | G M ( C T ) | 1 − M � � E { log det B H w H † w } + O ( 1 / P ) . + T

ITW Jeju, South Korea, Lower Bound on Capacity 2015 14 / 18 • Restricting D to particular distribution yields lower Ramy Gohary and Halim bound on capacity. Yanikomeroglu • Set D to deterministic Introduction TP � System Model D = Λ − 1 / 2 A Tr Λ − 1 The right A singular vectors of X • Choice ensures Ψ D Λ 1 / 2 A H w Gaussian, i.i.d. entries Asymptotic high SNR A H w ) = M 2 log π eTP non-coherent h ( Ψ D Λ 1 / 2 capacity . Bounds Tr Λ − 1 A Main Result Conclusions • Lower bound on capacity: 1 − M TP + 1 T log | G M ( C T ) | � � C ( P ) ≥ M log T π e Tr Λ − 1 A 1 − M � � E { log det B H w H † w } + O ( 1 / P ) . + T