Parameter Estimation of Double Directional Radio Channel Model S-72.4210 Post-Graduate Course in Radio Communications February 28, 2006 Signal Processing Lab./SMARAD, TKK, Espoo, Finland Outline 2 1. Introduction 2. Channel sounding HUT Sounder 3. Double Directional Radio Channel model 4. Parameter estimation Maximum likelihood principle State-space modeling 5. References 6. Homework
Acronyms and abbreviations 3 AWGN DMC EKF i.i.d. IR MIMO PDF PDP RIMAX Rx Tx additive white gaussian noise dense multipath component Extended Kalman Filter independent identically distributed impulse response multiple input multiple output Probability Density Function Power Delay Profile parameter estimation method receiver transmitter f G R (f) G T (f) M R M T M f R s(θ sp ) t x θ dmc θ sp frequency frequency response of the receiver frequency response of the transmitter number of receive antennas number of transmit antennas number of frequency (delay) domain samples covariance matrix observation vector modeling propagation paths time measured observation vector parameters of dense multipath component parameters of concentrated propagation paths Introduction 4 Future wireless MIMO communication systems Exploit the spatial and temporal diversity of the radio channel Require new complex models for simulations Studying and comparing different transceiver structures Models are found through radio channel sounding measurements Measurements are fitted to double directional channel models Signal processing used for parameter estimation Influence of measurement equipment is removed
Channel sounding 5 Sequential channel measurement from between each Tx and Rx ports TKK 5.3 GHz MIMO setup 32 x 32 channels (16 dual polarized elements in arrays at both ends) τ 2 z Length of each impulse response (IR) is 510 samples (120 MHz sampling rate) Observation ( snapshot ) separation 8.7 ms θ 1 φ 1 τ 1 y What sounder produces? Complex array of 32 x 32 x 510 elements for each snapshot x Sounder output (single snapshot) 6 32 x 32 realizations of 510 sample IRs at every 8.7 ms Parameter estimation fits data to a channel model Compresses the channel information using model parameters Remove measurement antenna influence Later the channel model parameters can be plugged into any antenna/transceiver configuration Or parameters can be used to find out model statistics
Double directional channel model 7 Channel frequency response (Fourier transform of IR) at time t constructed with discrete propagation paths H(f,t)=G Rf (f)g Tf (f) p { B R (ϕ R,p,ϑ R,p ) ΓpB T (ϕ T,p,ϑ T,p ) }{{}}{{} C M R 2 C M T 2 T e j2πfτ p } ϕ R,ϕ T azimuthangleat Rx and Tx ϑ R,ϑ T elevationangleat Rx andtx τ time delay of arrival Γ complex path weight matrix G Rf,G Tf frequencyresponseof Rx and Tx [ ] γhh,p γ VH,p C 2 2 Γ p = γ HV,p γ VV,p Sampled double directional channel model 8 In practice discrete samples of H(f,t) are measured Sampled model for the observation consists of two parts: x=s(θsp)+d dmc C M RM T M f 1 Specular propagation paths: Dense multipath component: d dmc N C (0,R(θ dmc )) θsp= {τ, ϕ T, ϑ T, ϕ R, ϑ R, γ} s(θ sp )= ( B RH B TH B f ) γhh + ( B RV B TH B f ) γhv + ( BRH B TV B f ) γvh + ( B RV B TV B f ) γvv where denotes the Khatri-Rao (columnwise Kronecker) product
Illustration of specular paths vs. DMC 9-30 -35-40 Estim ation Residual Specular Paths s(θsp) Estimated DMC + Noise Concentrated Propagation Paths -45 magnitude [db] -50-55 -60-65 -70 Dense Multipath d dmc (distributed diffuse scattering) -75-80 -85 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 norm alized τ Parameter estimation techniques 10 Subspace techniques ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) MUSIC (MUltiple SIgnal Classification) RARE (RAnk Reduction Estimator) Maximum likelihood estimators SAGE (Space-Alternating Generalized Expectation maximization) RIMAX (iterative maximum likelihood) State-Space Methods Extended Kalman Filter
Parameter estimation example: Maximum Likelihood Estimation (1) 11 The Observation x is assumed i.i.d. Gaussian ) x N C (s(θsp) }{{},R(θ }{{ dmc) } mean covariance The Likelihood function, i.e., the pdf of x: l(θ,x) =p ( x θ ) = 1 π M det(r(θ dan )) e (x s(θ sp)) H R 1 (θ dan ) (x s(θ sp )) The maximum likelihood estimates are the parameters θ sp and θ dmc that maximize this function Maximum Likelihood (2) 12 Usually the log-likelihood L(θ,x) is preferred L(θ,x)=ln(l(θ,x))=ln ( 1 π M det(r) ) (x s(θ sp )) H R 1 (x s(θ sp )) Let us assume that R=R(θ dmc ) is known. Then the maximum of L(θ,x) is found by minimizing the last term ˆθ sp,ml =argmin θ sp ( x s(θ sp)) H R 1 (x s(θsp)) The minimum is found by evaluating zeros of the gradient (first order derivatives) of this term
Maximum Likelihood (3) 13 The first order derivatives of are given by the score function: ( x s(θsp )) H R 1 (x s(θ sp ) ) q(x θ,r)=2 R { D H (θ)r 1 (x s(θ)) }, D(θ)= θ Ts(θ) The score function q(x θ,r) has typically several zeros Global search or other initialization (estimates from previous snapshot) required Iterative (e.g. Gauss-Newton or Levenberg-Marquardt) method can be used to reach the optimal parameter estimates Outline of the RIMAX structure [1] 14 1. Read new snapshot x 2. Use previous estimates as initial values 3. Search for new propagation paths 4. Estimate the DMC component 5. Use iterative maximum likelihood method to improve propagation path estimates 6. Check reliability of propagation paths based on estimation error variance 7. Store the estimates and proceed to the next snapshot Channel sounding data Read new Observation x Calculate estimates for the path weights using the structural parameters µ of the previous observation (BLUE, Section 5.1). Improve the parameter estimates of the distributed diffuse components ML-Gauss-Newton Algorithm (Section 6.1.5). Improve the parameter estimates of the propagation paths with the Levenberg-Marquardt algorithm using alternating path group parameter updates (Sections 5.2.4 and 5.2.5). not reached Search for new propagation paths (Section 5.1.5). check convergence reached Check the reliability of the propagation paths. Drop the unreliable paths (Section 5.2.7). Paths dropped? yes no Store the parameter estimates.
Example of succesful parameter estimation [1] 15 PDP [db] PDP [db] -80-82 -84-86 -88-90 -92-94 -96-98 -100 0 1000 2000 3000 4000 Time delay [ns] 5000 6000-80 -82-84 -86-88 -90-92 -94-96 -98-100 0 1000 2000 3000 4000 Time delay [ns] 5000 6000 Example for the PDP of a measured impulse response (blue) and of the estimated concentrated propagation paths (red). Example for the PDP of the remainder (blue) of a measured impulse response after removing the estimated concentrated propagation paths. Red line is the estimated PDP of the DMC. PDP after whitening [db] 15 10 5 0-5 -10-15 0 1000 2000 3000 4000 Time delay [ns] 5000 6000 Example for the PDP of the remainder of a measured impulse response after removing the estimated concentrated propagation paths and whitening (removing the DMC). Illustration of estimation results 16 Panoramic (full 360 ) view at courtyard of Technical University of Ilmenau Rx at the middle of the courtyard (at point where the photo has been taken) Tx going around the courtyard
Alternative approach: Tracking of the propagation path parameters 17 Propagation path parameter estimation as a multi-target tracking problem Number of (reliable) paths P represent multiple targets Large number of parameters for each target τ 2 z θ 1 τ 1 Linear vs. Nonlinear motion model Nonlinear Measurement model φ 1 y Modeling the noise process x State-space methods 18 State transition (possibly nonlinear): x k+1 =f k (x k,q k ) Measurement equation (nonlinear): y k =h(x k,r k ) Extended Kalman Filter (EKF) Measured PDP over time compared to PDP of EKF estimates. EKF assumes Gaussian distribution Linearizes state transition and measurement equations through Taylor series approximation Tracks the parameters over time using recursive filtering (Kalman filters are popular in radar applications) Low computational complexity compared with iterative maximum likelihood Initialization using e.g. RIMAX Reliable tracking requires some statistics of the behavior of the parameters
RIMAX vs. EKF 19 EKF is computationally lighter than RIMAX Time per snapshot(s) 10 2 10 1 Simulation results show how EKF filters the parameters resulting in lower estimation error variance Rx Azimuth ϕr(degrees) -117.5-118 -118.5-119 -119.5-120 -120.5-121 -121.5-122 500 1000 1500 2000 2500 Snapshot index RIMAX EKF Original ML-ss EKF(improved Q) 10 20 30 40 50 60 70 80 90 Snapshot index Conclusions 20 Parameter estimation fits measured data to a channel model Compresses the channel information to model parameters Removes measurement antenna influence Later the channel model parameters can be used for 1. Statistical analysis of the channel parameters 2. Simulations with arbitrary antenna/transceiver configurations Most popular classes of parameter estimation techniques are subspace and maximum likelihood State-space methods are under research for revealing and utilizing the time-dependt properties of the radio propagation environments
References 21 [1] A. Richter, Estimation of radio channel parameters: Models and algorithms, Ph. D. dissertation, Technische Universität Ilmenau, Germany, 2005, [Online]. Available: www.db-thueringen.de [2] A. Richter, M. Enescu, V. Koivunen, State-Space Approach to Propagation Path Parameter Estimation and Tracking, in Proc. 6th IEEE Workshop on Signal Processing Advances in Wireless Communications, New York City, June 2005. [3] J. Salmi, Statistical Modeling and Tracking of the Dynamic Behavior of Radio Channels, Master s Thesis, Helsinki University of Technology, Espoo, Finland, June 2005. [4] V-M. Kolmonen, J. Kivinen, L. Vuokko, P. Vainikainen, 5.3 GHz MIMO radio channel sounder, in Proc. 22nd Instrumentation and Measurement Technology Conference, IMTC 05, Ottawa, Ontario, Canada, May 16-19 2005, pp.1883-1888. Homework 22 Maximum likelihood estimation of mean and variance Consider a discrete-time received signal r(k)=µ+w(k), k=0,1,...,n 1 where µ is a constant mean and w(k) ~ N(0,σ 2 ) is AWGN with variance σ 2. The PDF (likelihood) of the observation vector r is thus given by p( r µ,σ 2 ) = 1 ( 1 N 1 2πσ 2 ) Ne 2σ 2 k=0 (r(k) µ)2 Derive the maximum likelihood estimates for both the mean µ and the variance σ 2. HINT: Differentiate the log-likelihood function with respect to both parameters and set the derivatives to zero.