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**Capacity of Correlated MIMO Channels: Channel Power and Multipath Sparsity**

Random Matrix Theory and Wireless Communications Workshop Boulder, CO, July Akbar Sayeed (joint work with Vasanthan Raghavan, UIUC) Part presented at ITW 2004 Wireless Communications Research Laboratory Department of Electrical and Computer Engineering University of Wisconsin-Madison ,

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**Sergio Verdu – Hard Act to Follow!**

Shannon (belly) Dance! (ISIT 2006, Seattle)

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**Sergio Verdu – Model Incognito?**

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**Multipath Wireless Channels**

Fading – fluctuations in received signal strength Diversity – statistically independent modes of communication 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! Flow: Virtual channel governs performance DoF is key Sparse channels: D < N_sp Three key implications: 1) Capacity Scaling with Antennas: best O(sqrt(D(N))) vs O(N) ITW 2004 2) Capacity and Reliability Scaling with Bandwidth (New); Optimal T-W scaling for optimal channel Estimation (whole range of T-W scalings) Rich multipath is a bad assumption as the signal space dimensions increase 3) Maximizing capacity with Reconfigurable Antenna Arrays (ISIT 2006)

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**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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**Key Elements of this Work**

C(N) sparse i.i.d. Correlated N 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

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**Virtual Channel Modeling**

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

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**Narrowband MIMO Channel**

Transmitted signal Received signal Transmit antennas Receive antennas

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**Uniform Linear Arrays Transmit steering vector**

d Transmit steering vector = Tx antenna spacing Relation between theta_T and phi_T. affect of antenna spacing Receive response veector = Rx antenna spacing Spatial sinusoids: angles frequencies

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**Physical Model : complex path gains : number of paths**

: Angles of Arrival (AoA’s) : Angles of Departure (AoD’s) Non-linear dependence of H on AoA’s and AoD’s

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**Virtual Modeling Physical Model Virtual Model**

Spatial array resolutions: Virtual model is linear -- virtual beam angles are fixed (AS ’02)

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**Antenna Domain and Beamspace**

Two-dimensional unitary (Fourier) transform Unitary (DFT) matrices Generalization to non-ULA’s (Kotecha & AS ’04 ; Weichselberger et. al. ’04; Tulino, Lozano, Verdu ’05;)

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**Virtual Imaging of Scattering Geometry**

2 point scatterers 2 scattering clusters Diagonal scattering Rich scattering

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**Virtual Path Partitioning**

Distinct virtual coefficients disjoint sets of paths

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**Virtual Coefficients are Approximately Independent**

Possibly remove Channel power matrix: joint angular power profile

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**Joint and Marginal Statistics**

Joint statistics: Joint distribution of channel power as a function of transmit and receive virtual angles Marginal statistics: Transmit Receive (diagonal)

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**Kronecker Product Model**

Independent transmit and receive statistics Separable angular scattering function (angular power profile) Separable kronecker arbitrary

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**Communication in Eigen (Beam) Space**

Multipath Propagation Environment Image of is created in the far-field of TX is an image of the far-field of the RX

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**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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**Degrees of Freedom Dominant (large power) virtual coefficients**

Statistically independent Degrees of Freedom (DoF) Possibly remove DoF’s are ultimately limited by the number of resolvable paths

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**Channel Power and Degrees of Freedom**

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

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**Prevalent Channel Power Normalization**

The channel power/DoF grow quadratically with N

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**Quadratic Channel Power Scaling?**

is physically impossible indefinitely (received power < transmit power) 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 N

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**Sparse (Resolvable) Multipath**

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

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**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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**(virtual representation)**

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

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**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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**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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**Three Canonical (Regular) Configurations**

Consider p = number of parallel channels (multiplexing gain) q = D/p = DoF’s per parallel channel Received SNR = q/p Beamforming Multiplexing Ideal Draw a picture of morphing: p(N) up q(N) down Multiplexing gain = p increases Received SNR = q/p increases p transmit dimensions; r = max(q , p) receive dimensions

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**Simple Capacity Formula: Multiplexing Gain vs Received-SNR**

Received SNR per parallel channel Multiplexing (MUX) Beamforming (BF) Ideal C_bf = o(C_id), C_mux = o(C_id)

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**Morphing Between the Configurations**

beamforming ideal multiplexing Just say; they get fatter for gamma > 1 all three the same for gamma = 2 say; something about resolutions ; ideal to beam: compressing horizontally; ideal to mux; streching along diagonals . Ideal and bf: iid channels, Mux: D-connected (Liu)

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**Fastest Capacity Scaling: The Ideal Configuration**

BF regime: Ideal regime: Given asymptotics of both BF (high SNR), mul (Low SNR); tight for BF and MUX for any snr. For ideal for high SNR. Combine with earlier slide. MUX regime:

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**Impact of Transmit SNR on Capacity Scaling**

BF: C(N) MUX: BF gamma = 0, ideal; gamma = ½, mux gamma =1 Ideal: N

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**Accuracy of Asymptotic Expressions**

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

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**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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**Capacity Formula Proofs: RMT**

If under broad assumptions on entries of H, the empirical spectral distribution function (Fp) 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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**Beamforming Configuration**

Two cases: In either case, Thus, with

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**Ideal Configuration Case i) Case ii) 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] Result critically depends on the fact that q/p > 0. Else, Grenander and Silverstein not applicable

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**Ideal Configuration Case i) Case ii)**

In either case, empirical density of (normalized by q) converges to Case i) Case ii) Thus, Result critically depends on the fact that q/p > 0. Else, Grenander and Silverstein not applicable

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**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 This method cannot be used for every channel. Need to use regularity, or else we cant obtain closed form capacity limit expressions.

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**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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**Optimum Input Rank versus SNR**

i.i.d. channels: equal power (i.i.d.) input optimal at all SNRs Correlated channels: beamforming (rank-1 input) optimal at low SNR uniform power (full-rank, i.i.d.) input optimal at high SNR 4 Loss of precious channel power! 3 rank 2 1 SNR

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**Ideal Channel: Optimum MG vs SNR tradeoff**

Beamforming: BF Ideal MUX Capacity Ideal: Multiplexing:

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**Impact of Antenna Spacing on Beamstructure**

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**Adaptive-resolution Spatial Signaling**

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

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**Wideband/Low-SNR Capacity Gain**

N-fold increase in capacity (or reduction in ) via BF configuration at low SNR

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**Source-Channel Matching**

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

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**Source-Channel Matching**

Adapting the multiplexing gain p via array configuration: matching the rank of the input to the rank of the effective channel Ideal TX “Square root” rank channel “Square root” rank input RX

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**Source-Channel Matching**

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

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**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 MUX IDEAL BF Optimum number of antennas: N ~ D

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**Summary Sparse multipath Modeling of sparse MIMO channels**

C(N) sparse i.i.d. Correlated N 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

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**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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**Multiplexing Channel:**

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. 2003 Multiplexing Channel: Girko: Theory of random determinants, Springer Publishers, 1st 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. This method cannot be used for every channel. Need to use regularity, or else we cant obtain closed form capacity limit expressions.

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