Multiple Antennas Channel Characterization and Modeling Mats Bengtsson, Björn Ottersten Channel characterization and modeling 1 September 8, 2005 Signal Processing @ KTH Research Focus Channel modeling Statistical modeling Spatio temporal characteristics Validation against measurement data Transmit strategies Transmit diversity and Space-Time coding Channel knowledge and uncertainty Receiver design Acquisition, synchronization, channel estimation Interference rejection, multi-user detection System issues Spatial multiplexing System capacity and throughput Channel characterization and modeling 2 September 8, 2005
Use of Channel Models Simulation Link performance (transmitter to receiver) System performance (multiple users) Analysis/Understanding Physical propagation Algorithm and system behavior Communication System Design Performance optimization Link design (transmission/modulation, coding, receivers, estimation, equalization, interference suppression, etc.) System level design (resource management, power control, hand over, frequency and coverage planing, etc.) Necessary in frequency duplex communication systems Channel characterization and modeling 3 September 8, 2005 Presentation Outline Algebraic characterizations rank and eigenvalues Stochastic characterizations Geometric characterizations Channel characterization and modeling 4 September 8, 2005
Propagation Environment in Wireless Communications Complex Propagation Environment direct path local scattering (at the terminal) long time delay multipath near field scattering shadowing time varying conditions Channel characterization and modeling 5 September 8, 2005 Direct Path, Line-of-Sight Parameterized Rank One Channel θ x(t) = a(θ)s(t) + n(t) a 1 (θ) a(θ) =. a m (θ) array response assumed known Channel characterization and modeling 6 September 8, 2005
Local Scattering at the Terminal x(t) = i γ i abf(θ i )s(t τ i ) + n(t) small relative time delays = x(t) = hs(t) + n(t) possible to parameterize h causes flat (Rayleigh) fading (fast fading) Channel characterization and modeling 7 September 8, 2005 Long Time Delay Multipath Specular Multipath x(t) = a(θ 1 )s(t)+a(θ 2 )s(t τ)+n(t) τ is on the order of the symbol time T θ2 θ 1 increases the rank of the channel causes frequency selective fading Channel characterization and modeling 8 September 8, 2005
Linear Models r = Hx + n Received signal, channel matrix, transmitted signal, noise Same algebraic model for many different scenarios: Frequency selective scalar channel Spatio temporal model MIMO model... Channel characterization and modeling 9 September 8, 2005 Linear Model, Example Single-Input-Single-Output, frequency selective channel: x(t) = h(t) s(t) + n(t) 2 3 2 3 x(t) h(0) h(l) 0 0 2 3 2 3 x(t 1).. s(t) n(t).... 0 h(0) h(l) s(t 1) n(t 1) 6 = 4. 7 5 6... 7 6 + 4. h(0) h(l) 0 5 4. 7 6 5 4. 7 5 x(t M) 0 0 h(0) h(l) s(t L M) n(t M) x(t) = Hs(t) + n(t) Channel characterization and modeling 10 September 8, 2005
Main Cases of x(t) = Hs(t) + n(t) Low (column) rank: = + Full rank: = + Also full rank: = + Note: Matrix rank more important than matrix dimensions (simplified pictures above)! Channel characterization and modeling 11 September 8, 2005 Example, Space-only Model x 1 (t) x 2 (t) x(t) =. = h s(t) + n(t) = Hs L (t) + n(t). x m (t) s(t) s(t 1) H = [h(1),h(2),...,h(l)] s L (t) =. s(t (L 1)) Channel characterization and modeling 12 September 8, 2005
Single Input Multiple Output Multiple Channels If a signal is observed through at least two channels (FIR) the process is low rank if enough time samples are included. Example, 1 and 2 time samples in x(t) 2 3 2 3 2 3 2 3 4 x 1(t) 5 = 4 h 11 h 12 s(t) 5 4 5 + 4 n 1(t) 5 x 2 (t) h 21 h 22 s(t 1) n 2 (t) 2 3 2 3 2 3 x 1 (t) h 11 h 12 0 2 3 n 1 (t) x 2 (t) h 21 h 22 0 s(t) = 6 6 4 x 1 (t 1) 7 6 5 4 0 h 11 h 12 7 4 s(t 1) 7 5 + n 2 (t) 6 5 4 n 1 (t 1) 7 5 s(t 2) x 2 (t 1) 0 h 21 h 22 n 2 (t 2) Low rank signal model and linear parametrization! Channel characterization and modeling 13 September 8, 2005 Spatial-Temporal Model 2 x M (t) = 6 4 x(t) x(t 1). x(t (M 1)) 3 2 = 6 7 4 5 h 0... 0 h 2 3 n(t) 7 5 s n(t 1) L+M 1(t)+ 6 4. n(t (M 1)) 3 7 5 Low rank signal! x M (t) = Hs L+M 1 (t) + n M (t) Channel characterization and modeling 14 September 8, 2005
Channel Modeling Time Space To increase spectral efficiency (suppress interference), the channel rank is critical. In the spatial domain, the signal is in general low rank! x(t) = Hs(t) +n(t) }{{} low rank This is not important if the goal is primarily spatial diversity gain. If the goal is system capacity increase, this is critical. Why? The channel rank is related to interference supression capabilities (spatial degrees of freedom) and receiver complexity. Channel characterization and modeling 15 September 8, 2005 Rank Characterizations Many different criteria have been proposed as functions of the singular values of H, related to: Channel capacity Diversity order Matrix condition number Interference rejection capability... Channel characterization and modeling 16 September 8, 2005
Example, GSM Field Trial Data Experimental data collected by Ericsson Radio Systems Two transmitters in an urban environment (Düsseldorf, Germany) about 1.5 km from the base station, non line of sight. Dual polarized antenna element Antenna mounted about 40 meters above ground Rank one channel (most of the time) SNR approximately 23 db burst Channel characterization and modeling 17 September 8, 2005 Channel Rank Profile GSM, 4 and 8 Branches 0 5 10 Rank 1 Rank 2 Rank 3 4 branches, single polarization. Suburban 1 0 5 10 Rank 1 Rank 2 Rank 3 Rank 4 2x4 dual polarized branches. Suburban 1 Relative model error, db 15 20 25 30 Relative model error, db 15 20 25 35 30 40 35 45 0 0.2 0.4 0.6 0.8 1 Probability 40 0 0.2 0.4 0.6 0.8 1 Probability Low rank approximations of the GSM channel at 1800 MHz, suburban environment. Results from 10000 bursts. Channel characterization and modeling 18 September 8, 2005
Stochastic Channel Characterization Continuous time model: K H(t,τ) = A k (t)a Rx (θ k (t))a Tx,* (α k (t))δ(τ τ k (t)) k=1 N Discrete time model: H(t,τ) = H n (t)δ(τ nt), where n=1 K H n (t) A k (t)a Rx (θ k )a Tx,* (α k ) sinc(τ n /T n) k=1 (corresponding to sinc shaped transmit/receive filters) Channel characterization and modeling 19 September 8, 2005 Stochastic Channel Characterization, Common assumptions: cont. Slow time variations in A k (t), θ k (t), α k (t) large scale fading, Shadow fading Small movements = small variations in τ k (t) = phase variations between the terms = small scale fading. Many terms contribute to each tap, H n (t) H n (t) Gaussian distributed, H n (t) fully desribed by 2 nd order statistics. The small scale fading is Rayleigh fading! Each tap is stochastically independent from the others. Channel characterization and modeling 20 September 8, 2005
Characteristics of the Stochastic Channel Models Physical parameters + give insight and understanding + extrapolation, can change carrier frequency, band width, antenna characteristics (configuration, aperture) difficult (impossible) to find compact representation in micro-cellular environments, often not identifiable Non-physical parameters + compact representation + useful for simulation, analysis and design cannot separate channel from measurement equipment limited insight Channel characterization and modeling 21 September 8, 2005 Second Order Characterization Full Spatial characterization: R H,n = E[vec{ H n }vec{ H n } ] N Tx N Rx N Tx N Rx Hermitian matrix (N Tx N Rx ) 2 real valued parameters! [ Generating channels: Hn = unvec R 1/2 H,n ], g g CN(0,I NTx N Rx ) Reduced number of parameters? Geometric model with physical parameters (main directions, spread angles,...) Impose algebraic structure Common choice: the Kronecker model. Channel characterization and modeling 22 September 8, 2005
The Kronecker Model Definitions: Receive side correlation matrix: R Rx H,n = 1 N Tx E[ H n H n ] Transmit side correlation matrix: R Tx H,n = 1 N Rx E[( H n H n ) T ] Model (tap-wise Kronecker): R H,n R Tx H,n RRx H,n Generating channels: Hn (R Rx H,n )1/2 G(R Rx H,n )T/2, vec[g] CN(0,I NTx N Rx ) Interpretation: The total correlation between two links is the product of the correlation seen from the transmit and receive sides, respectively. Number of parameters: N 2 Rx + N2 Tx real valued parameters. Channel characterization and modeling 23 September 8, 2005 Experimental Validation Solve (its an eigenvalue problem!): Define the relative error as min X,Y R H X Y F Ψ(A,B) = A B F A F Results from experimental data (indoor, non line of sight) 2x2 3x3 Ψ(ˆR H,X Y) 0.76 % 4.52 % Ψ(ˆR H, ˆR Tx H ˆR Rx H ) 0.86 % 4.79 % Ψ(ˆR Tx H,X) 0.40 % 1.74 % Ψ(ˆR Rx H,Y) 0.03 % 1.60 % Channel characterization and modeling 24 September 8, 2005
Comments on the Kronecker Model Necessary condition: Spatial stationarity small movements of a transmit antenna does not affect the receive side correlation and vice versa. Sufficient condition: Spatial properties seen from transmit and receive sides are statistically independent (more precise statement later). Common criticism: Cannot describe scattering that is localized both in DoA and DoD. Overestimates the capacity (at least in certain scenarios). Channel characterization and modeling 25 September 8, 2005 Further Comments Easy to find scenarios where the model does not hold. Only relevant for small antenna apertures. Allows for analytical results. Possible extensions: Add line-of-sight component for Rice fading Use as base functions in more complex model: H n = k γ k(r Rx H,n (k))1/2 G k (R Rx H,n (k))t/2 (one interpretation of the Weichselberger model). Channel characterization and modeling 26 September 8, 2005
Tap-wise vs. Narrowband Kronecker Narrowband channel: R narrowband N n=1 Rwideband n 80 60 45 80 60 50 40 40 40 45 20 20 0 35 0 40 20 30 20 40 60 25 40 60 35 80 20 80 60 40 20 0 20 40 60 80 2D beamforming diagram from a tap-wise Kronecker model, R wideband n = R Tx n R Rx n. 80 30 80 60 40 20 0 20 40 60 80 2D beamforming diagram for a narrowband Kronecker model, R narrowband = R Tx R Rx. Tap-wise Kronecker narrowband Kronecker Channel characterization and modeling 27 September 8, 2005 Geometric Characterizations Example, Direct Path, Line-of-Sight Parameterized Rank One Channel θ x(t) = a(θ)s(t) + n(t) a 1 (θ) a(θ) =. a m (θ) array response assumed known Channel characterization and modeling 28 September 8, 2005
Local Scattering Angular Spread What is it? Characterization? Estimation? Channel characterization and modeling 29 September 8, 2005 Example: Flat fading MISO (Same ideas apply to frequency selective, Doppler and MIMO Models) L x(t) = α n a(θ 0 + θ n ) s(t) + n(t) n=1 } {{ } h Assumption: The L specular paths are stochastically independent and equally distributed. Full stochastic characterization: Joint PDF of α n, θ n : f α, θ(α, θ) Channel characterization and modeling 30 September 8, 2005
How to Describe the Scattering Shape? Marginal PDF of the azimuth deviations: f θ( θ) = f α, θ(α, θ)dα (Marginal) Power Azimuth Spectrum (PAS): P( θ) = L α 2 f α, θ(α, θ)dα Conditional Power Azimuth Spectrum: [ ] P conditional ( θ) = LE α 2 θ = L α f 2 α, θ (α, θ) dα = P( θ) f θ( θ) f θ( θ) Angular Spread: σ 2 θ = θ 2 P( θ)d θ = θ 2 f θ( θ)p conditional ( θ)d θ Channel characterization and modeling 31 September 8, 2005 Second Order Characterization Spatial Covariance Matrix: R h = E[hh ] = a(θ 0 + θ n )a(θ 0 + θ n ) P( θ)d θ = a(θ 0 + θ n )a(θ 0 + θ n ) f θ( θ)p conditional ( θ)d θ If L large: Central Limit Theorem = h Gaussian = Completely characterized by R h = completely characterized by P( θ) = f θ( θ)p conditional ( θ). Conclusion: P conditional ( θ) and f θ( θ) not unique, only their product. Channel characterization and modeling 32 September 8, 2005
Why Consider P conditional ( θ)? Typical simulation procedure: 1. For each specular path, find random θ n from f θ( θ). 2. Given θ n, determine average power of α n : P conditional ( θ). 3. Find random gain α n with that average power. Special Case 1: (Uniformly distributed directions) Generate θ n from a uniform distribution (f θ( θ) =const.). Use only P conditional ( θ) P( θ) to determine the shape. Special Case 2: (Same average power per path) Use P conditional ( θ) =const. Use only f θ( θ) P( θ) to determine the shape. Channel characterization and modeling 33 September 8, 2005 Idealized Examples 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 Power 1 Power 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 15 20 25 30 35 40 45 DoA 0 15 20 25 30 35 40 45 DoA Case 1: Uniformly distributed directions Case 2: Same average power per path Note: Same PAS and same channel covariance! Channel characterization and modeling 34 September 8, 2005
Estimation Attempt True model: Single realization of a diffusely scattered cluster. main direction: θ, spread angle: σ θ. False assumption: One (or several) point source(s). Direction: θ, power P. Estimation Strategy: {ˆθ, ˆP } = estdoa({x(t)} N t=1) Estimation algorithm: MUSIC, ESPRIT, ML, SAGE, WSF,... Plot distribution of the estimates! Channel characterization and modeling 35 September 8, 2005 Numerical Example Single cluster, 24 rays. Uniform spread in azimuth and Doppler. 8 element ULA. N = 4096 snapshots for each channel realization. ML estimation. Channel characterization and modeling 36 September 8, 2005
Azimuth PDF 0.09 0.08 0.07 Dashed: true Histogram: empirical Solid: theoretical PDF 0.06 0.05 0.04 0.03 0.02 0.01 0 2 4 6 8 10 12 14 16 18 Estimated azimuth angle ˆθ [Degrees] Channel characterization and modeling 37 September 8, 2005 Doppler PDF 0.06 Dashed: true Histogram: empirical Solid: theoretical PDF 0.04 0.02 0 50 40 30 20 10 0 10 20 30 40 50 Estimated Doppler velocity ˆν [m/s] Channel characterization and modeling 38 September 8, 2005
Power Azimuth Spectrum 2 Simulated Theoretical 0 Dashed: theoretical Solid: empirical PAS [db] 2 4 6 8 10 10 8 6 4 2 0 2 4 6 8 10 Estimated azimuth deviation θ Channel characterization and modeling 39 September 8, 2005 Theoretical Expression for the Empirical Distributions Empirical Azimuth PDF: ( θ = ˆθ θ) f θ( θ) = 1 σ θ ( 2 θ2 σ 2 θ 1 + 1 ) 3/2 Empirical Power Azimuth Spectrum: (γ e 0.5772) E[10log 10 ( ˆ P S ) θ] ( 10log 10 e 2 γ e ) ( θ2 10log10 (4 σ 2 θ ) ) + 1 Channel characterization and modeling 40 September 8, 2005
Comments Separate estimation of Azimuth PDF and Marginal Power Azimuth Spectrum is an ill-posed problem. At least for Rayleigh channels. Estimation of the PAS is difficult (if not impossible). Trying to resolve irresolvable clusters may lead to wrong conclusions. Empirical shape functions are independent of the true shape. Our analysis is generic, holds for all DOA direction estimation algorithms when spread is small. Channel characterization and modeling 41 September 8, 2005 Angular Distribution from Physical Model Power However, angular spread can be estimated consistently! This is sufficient for modeling, simulation, and design purposes. σ θ Angle Channel characterization and modeling 42 September 8, 2005
Corresponding MIMO Model L H = α n a Rx (θ n )a Tx,* (φ n ) n=1 θ n : Direction of Arrival (DoA) φ n : Direction of Departure (DoD) Full stochastic characterization: Joint PDF, f α,θ,φ (α,θ,φ) Joint (Marginal) PAS: P(θ,φ) = L α 2 f α,θ,φ (α,θ,φ)dα Receive Side PAS: P Rx (θ) = L α Rx 2 f α,θ,φ (α,θ,φ)dα dφ Transmit Side PAS: P Tx (φ) = L α Tx 2 f α,θ,φ (α,θ,φ)dα dθ Kronecker holds if P(θ,φ) P Rx (θ)p Tx (φ). Channel characterization and modeling 43 September 8, 2005