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1 Capacity of Correlated MIMO Channels: Channel Power and Multipath Sparsity Akbar Sayeed (joint work with Vasanthan Raghavan, UIUC) Random Matrix Theory and Wireless Communications Workshop Boulder, CO, July Wireless Communications Research Laboratory Department of Electrical and Computer Engineering University of Wisconsin-Madison

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2 Sergio Verdu – Hard Act to Follow! Shannon (belly) Dance! (ISIT 2006, Seattle)

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3 Sergio Verdu – Model Incognito?

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4 Multipath Wireless Channels Multipath signal propagation over spatially distributed paths due to signal scattering from multiple objects –Necessitates statistical channel modeling –Accurate and analytically tractable Understanding the physics! Fading – fluctuations in received signal strength Diversity – statistically independent modes of communication

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5 Antenna Arrays: Multiplexing and Energy Capture Multiplexing – Parallel spatial channels Array aperture: Energy capture Wideband (W): Multi-antenna (N): Dramatic linear increase in capacity with number of antennas

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6 Key Elements of this Work Sparse multipath –i.i.d. model – rich multipath –Seldom true in practice –Physical channels exhibit sparse multipath Modeling of sparse MIMO channels –Virtual channel representation (beamspace) –Physically meaningful channel power normalization –Sparse degrees of freedom –Spatial correlation/coherence The Ideal MIMO Channel –Fastest (sub-linear) capacity scaling with N –Capacity-maximization with SNR for fixed N –Multiplexing gain versus received SNR tradeoff –Simple capacity formula for all SNRs (RMT) Creating the Ideal MIMO Channel in Practice –Reconfigurable antenna arrays –Three canonical configurations: near-optimum performance over entire SNR range –Source-channel matching –New capacity formulation Capacity MUX IDEAL BF C(N) sparse i.i.d. Correlated N

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7 Virtual Channel Modeling Spatial sampling commensurate with signal space resolution Channel statistics induced by the physical scattering environment Abstract statistical models Physical models Virtual Model Tractable Accurate Accurate & tractable Interaction between the signal space and the physical channel

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8 Narrowband MIMO Channel Received signal Transmitted signal Transmit antennas Receive antennas

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9 Uniform Linear Arrays Receive response veector = Rx antenna spacing Transmit steering vector = Tx antenna spacing Spatial sinusoids: angles frequencies d

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10 Physical Model : number of paths : complex path gains : Angles of Arrival (AoAs) : Angles of Departure (AoDs) Non-linear dependence of H on AoAs and AoDs

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11 Virtual Modeling Spatial array resolutions: Physical Model Virtual Model (AS 02) Virtual model is linear -- virtual beam angles are fixed

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12 Antenna Domain and Beamspace Two-dimensional unitary (Fourier) transform Generalization to non-ULAs (Kotecha & AS 04 ; Weichselberger et. al. 04; Tulino, Lozano, Verdu 05;) Unitary (DFT) matrices

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13 Virtual Imaging of Scattering Geometry 2 point scatterers 2 scattering clusters Diagonal scattering Rich scattering

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14 Virtual Path Partitioning Distinct virtual coefficients disjoint sets of paths

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15 Virtual Coefficients are Approximately Independent Channel power matrix: joint angular power profile

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16 Joint and Marginal Statistics Joint distribution of channel power as a function of transmit and receive virtual angles Joint statistics: Marginal statistics: TransmitReceive (diagonal)

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17 Kronecker Product Model Independent transmit and receive statistics Separable angular scattering function (angular power profile) Separable arbitrary kronecker

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18 Communication in Eigen (Beam) Space Multipath Propagation Environment is an image of the far-field of the RX Image of is created in the far-field of TX

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19 Capacity Maximizing Input Optimal input covariance matrix is diagonal in the virtual domain: - Beamforming optimal at low SNR (rank-1 input) - Uniform power input optimal at high SNR (full-rank input) - Uniform power input optimal for regular channels (all SNRs) Veeravalli, Liang, Sayeed (2003); Tulino, Lozano, Verdu (2003); Kotecha and Sayeed (2003)

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20 Degrees of Freedom Dominant (large power) virtual coefficients Statistically independent Degrees of Freedom (DoF) DoFs are ultimately limited by the number of resolvable paths

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21 Channel Power and Degrees of Freedom D(N) = number of dominant non-vanishing virtual coefficients = Degrees of Freedom (DoF) in the channel The D non-vanishing virtual coefficients are O(1) Simplifying assumption: Assume equal number of transmit and receive antennas – Channel power:

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22 Prevalent Channel Power Normalization The channel power/DoF grow quadratically with N

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23 Quadratic Channel Power Scaling? is physically impossible indefinitely (received power < transmit power) N Total TX power Total RX power Quadratic growth in channel power Linear growth in total received power Linear capacity scaling Increasing power coupling between the TX and RX due to increasing array apertures

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24 Sparse (Resolvable) Multipath Degrees of Freedom (D) Rich (linear) Sparse (sub-linear) Channel Dimension Sub-quadratic power scaling dictates sparsity of DoF

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25 Capacity Scaling: Sparse MIMO Channels For a given channel power/DoF scaling law what is the fastest achievable capacity scaling? New scaling result: coherent capacity cannot scale faster than and this scaling rate is achievable (Ideal channel) (AS, Raghavan, Kotecha ITW 2004)

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26 MIMO Capacity Scaling C(N) Correlated channels (kronecker model) Chua et. al. 02 i.i.d. model Telatar 95 Foschini 96 physical channels (virtual representation) Liu et. al. 03 AS et. al. 04

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27 Sparse Virtual Channels Sub-quadratic power scaling dictates sparse virtual channels: Capacity scaling depends on the spatial distribution of the D(N) channel DoF in the possible channel dimensions

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28 Simple Model for Sparse MIMO Channels 0/1 mask matrix with D non-zero entries Sparsity in virtual (beam) domain correlation/coherence in the antenna (spatial) domain

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29 Three Canonical (Regular) Configurations Beamforming p = number of parallel channels (multiplexing gain) q = D/p = DoFs per parallel channel Ideal Multiplexing Consider p transmit dimensions; r = max(q, p) receive dimensions Multiplexing gain = p increases Received SNR = q/p increases Received SNR = q/p

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30 Simple Capacity Formula: Multiplexing Gain vs Received-SNR Received SNR per parallel channel Beamforming (BF) Ideal Multiplexing (MUX)

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31 Morphing Between the Configurations beamforming ideal multiplexing

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32 Fastest Capacity Scaling: The Ideal Configuration BF regime: Ideal regime: MUX regime:

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33 Impact of Transmit SNR on Capacity Scaling BF: MUX: Ideal: C(N) N

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34 Accuracy of Asymptotic Expressions C(N) N BF and MUX tight at all SNRs Ideal tight in the low- or high-SNR regimes

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35 Capacity Formula Proofs: RMT If H is r x p, coherent ergodic is given by If, in addition, H is regular

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36 Capacity Formula Proofs: RMT If under broad assumptions on entries of H, the empirical spectral distribution function (F p ) of (normalized by p) converges to a deterministic limit (F) Limit capacity computation Approach 1: Sometimes this limit can be characterized explicitly Approach 2: Often, the limit can only be characterized implicitly via the Stieltjes transform. The limit capacity formula is the solution to a set of recursive equations

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37 Beamforming Configuration Two cases: In either case, Thus, with

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38 Ideal Configuration Two cases: Case i) reduces to a q x p i.i.d. channel Case ii) reduces to a q-connected p-dimensional channel [LRS 2003] Case i)Case ii)

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39 Ideal Configuration In either case, empirical density of (normalized by q) converges to Case i) Case ii) Thus, Case i)Case ii)

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40 Multiplexing Configuration Previous result due to [Grenander & Silverstein 1977] not applicable Two cases: In either case, empirical density is unknown The implicit characterization of [Tulino, Lozano & Verdu 2005] based on results due to [Girko] can be easily extended here Exploiting the regular nature of H

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41 The Ideal MIMO Channel: Fixed N 0/1 mask matrix with D non-zero entries Spatial distribution of the D channel DoF in the possible dimensions (resolution bins) that yields the highest capacity

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42 Optimum Input Rank versus SNR Correlated channels: - beamforming (rank-1 input) optimal at low SNR - uniform power (full-rank, i.i.d.) input optimal at high SNR SNR rank i.i.d. channels: equal power (i.i.d.) input optimal at all SNRs Loss of precious channel power!

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43 Ideal Channel: Optimum MG vs SNR tradeoff Capacity Beamforming: Ideal: Multiplexing: BF Ideal MUX

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44 Impact of Antenna Spacing on Beamstructure

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45 Adaptive-resolution Spatial Signaling Ideal Medium resolution TX and RX Multiplexing gain and spatial coherence: Fewer independent streams with wider beamwidths at lower SNRs. High resolution TX and RX Multiplexing Beamforming Low resolutionTX and High-Res. RX

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46 Wideband/Low-SNR Capacity Gain N-fold increase in capacity (or reduction in ) via BF configuration at low SNR

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47 Source-Channel Matching Adapting the multiplexing gain p via array configuration: rank of the inputrank of the effective channel matching the rank of the input to the rank of the effective channel Multiplexing Full-rank channel Full-rank input RX TX

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48 Source-Channel Matching Ideal Square root rank channel Square root rank input RX TX Adapting the multiplexing gain p via array configuration: rank of the inputrank of the effective channel matching the rank of the input to the rank of the effective channel

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49 Source-Channel Matching Beamforming Rank-1 channel Rank-1 input TX RX Adapting the multiplexing gain p via array configuration: rank of the inputrank of the effective channel matching the rank of the input to the rank of the effective channel

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50 New Capacity Formulation for Reconfigurable MIMO Channels To achieve O(N) MIMO capacity gain at all SNRs Optimal channel configuration realizable with reconfigurable antenna arrays Optimum number of antennas: N ~ D Capacity MUX IDEAL BF

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51 Summary Sparse multipath –i.i.d. model – rich multipath –Seldom true in practice –Physical channels exhibit sparse multipath Modeling of sparse MIMO channels –Virtual channel representation (beamspace) –Physically meaningful channel power normalization –Sparse degrees of freedom –Spatial correlation/coherence The Ideal MIMO Channel –Fastest (sub-linear) capacity scaling with N –Capacity-maximization with SNR for fixed N –Multiplexing gain versus received SNR tradeoff –Simple capacity formula for all SNRs (RMT) Creating the Ideal MIMO Channel in Practice –Reconfigurable antenna arrays –Three canonical configurations: near-optimum performance over entire SNR range –Source-Channel Matching –New capacity formulation Capacity MUX IDEAL BF C(N) sparse i.i.d. Correlated N

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52 Extensions: Implications of Sparsity Relaxing the 0-1 sparsity model Non-uniform sparsity Wideband MIMO channels/doubly-selective MIMO channels Space-time coding Reliability (error exponents) Impact of TX CSI (full or partial) Channel estimation (compressed sensing) and feedback Network implications (learning the network CSI)

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53 REFERENCES Beamforming Channel: Bai & Yin: Convergence to the semicircle law, Annals Prob., vol. 16, pp , 1988 Ideal Channel: Marcenko & Pastur: Distribution of eigenvalues for some sets of random matrices, Math- USSR-Sb., vol. 1, pp , 1967 Bai: Methodologies in spectral analysis of large dimensional random matrices: A review, Statistica Sinica, vol. 9, pp , 1999 Silverstein & Bai: On the empirical distribution of eigenvalues of a class of large dimensional random matrices, Journal of Multivariate Analysis, vol. 54, no. 2, pp , 1995 Grenander & Silverstein: Spectral analysis of networks with random topologies, SIAM Journal on Appl. Math., vol. 32, pp , 1977 Liu, Raghavan & Sayeed, Capacity and spectral efficiency of wideband correlated MIMO channels, IEEE Trans. Inform. Theory, vol. 49, no. 10, pp , Oct Multiplexing Channel: Girko: Theory of random determinants, Springer Publishers, 1 st edn, 1990 Tulino, Lozano & Verdu: Impact of antenna correlation on the capacity of multiantenna channels, IEEE Trans. Inform. Theory, vol. 51, no. 7, pp , July 2005 Multi-antenna Capacity of Sparse Multipath Channels, V. Raghavan and A. Sayeed.

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