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Issues in MIMO Channel Modeling and Simulation

Issues in MIMO Channel Modeling and Simulation. Ali Abdi Center for Communications and Signal Processing Research Department of Electrical & Computer Engineering New Jersey Institute of Technology Polytechnic University, December 4, 2003. Overview. MIMO Simulation. MIMO Modeling.

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Issues in MIMO Channel Modeling and Simulation

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  1. Issues in MIMO Channel Modeling and Simulation Ali Abdi Center for Communications and Signal Processing Research Department of Electrical & Computer Engineering New Jersey Institute of Technology Polytechnic University, December 4, 2003

  2. Overview MIMO Simulation MIMO Modeling Narrowband Wideband Stochastic Ray-based Macrocells Microcells Picocells Correlations (space, time, and frequency) Fade Duration Time and Frequency Selectivity Spectral Sampling Polynomial Embedding Polytechnic University, Dec. 4, 2003

  3. Part I MIMO Models Polytechnic University, Dec. 4, 2003

  4. MIMO Modeling: Macrocells • No scatterer around the elevated base station (BS). • Scatterers around the mobile station (MS) are located on a ring. Polytechnic University, Dec. 4, 2003

  5. MIMO Modeling: Micro & Picocells • Scatterers around the BS and the MS are located on separate rings. Polytechnic University, Dec. 4, 2003

  6. Advantage of Ring Models • Simple to analyze  Provide closed-form expressions for space, time, and frequency correlations. • Capture essential characteristics of MIMO channels using few parameters such as mean angle of arrivals/departures, Doppler, angle spreads, etc. • Compatible with previous well-accepted models such as Clarke’s. Polytechnic University, Dec. 4, 2003

  7. Advantage of Ring Models (continued) • Suitable for mobile Tx and Rx scenarios. • Applicable to wideband channels by modifying rings to circular rings. • Example: wideband macrocell model. Polytechnic University, Dec. 4, 2003

  8. MIMO Modeling: Correlations • Closed-form expressions are derived for: • (Narrowband) space-time corr. where is the channel gain between the p-th Tx and l-th Rx antennas. • (Wideband) space-time-frequency corr. , where is the time-varying transfer function between the p-th Tx and l-th Rx antennas. • These correlations are needed to simulate and assess the performance of MIMO systems. Polytechnic University, Dec. 4, 2003

  9. AWGN 1 1 AWGN AWGN + + + AWGN 2 2 Transmitter + Receiver 3 3 4 A 34 MIMO Channel • We use h(t) for the narrowband channels, and T(f,t) for wideband channels. Polytechnic University, Dec. 4, 2003

  10. A 2x2 Channel Polytechnic University, Dec. 4, 2003

  11. Diffuse and Line-of-Sight Components(Frequency-Flat Fading) • The channel fading gain • Average power • Rice factor Polytechnic University, Dec. 4, 2003

  12. Exact Diffuse Part of the Correlation Polytechnic University, Dec. 4, 2003

  13. Small Angle Spread at the Base Station Polytechnic University, Dec. 4, 2003

  14. Wave Scattering around the User Non-isotropic scattering in a street Isotropic scattering in an open area Polytechnic University, Dec. 4, 2003

  15. User’s Angle of Arrival & von Mises PDF • is the mean direction of AOA, seen by the user • controls the width of AOA Polytechnic University, Dec. 4, 2003

  16. The New Space-Time Correlation Let Polytechnic University, Dec. 4, 2003

  17. Special Cases of the Diffuse Part • Clarke’s model: Single Tx-Rx antennas, isotropic scattering • [Abdi99]: Single Tx-Rx antennas, non-isotropic scattering • [Lee70]: Single Tx-multiple Rx antennas, isotropic scattering • [Fulg98]: Multiple Tx-single Rx antennas, isotropic scattering • [Chen00]:Multiple Tx-single Rx antennas, isotropic scattering Polytechnic University, Dec. 4, 2003

  18. The Channel Measurements • Locations: suburban and urban areas • Data format: 12 pairs of narrowband inphase and quadrature components • Length of each record: 47 m or 7 s • Speed of the mobile receiver: fixed at 6.7 m/s • Carrier frequency: 910.25 MHz • Nominal power of the transmitter: 0.2 W • Sampling frequency: 35156.25 Hz Polytechnic University, Dec. 4, 2003

  19. Correlation Fitting to Data Polytechnic University, Dec. 4, 2003

  20. Wideband Macrocell Model Polytechnic University, Dec. 4, 2003

  21. Outdoor Wideband Data • Comparing different characteristics of the circular ring model with the data reported in Pedersen et al. "A Stochastic Model of …,” IEEE Trans. Vehic. Technol., vol. 49, pp. 437-447, 2000. • Environment: Typical urban • Location: Aarhus, Denmark • Carrier Frequency: fC =1.8 GHz • BS/user Separation: 300-3000m • BS antenna height: 32m (12 m above the average rooftop level) • No line-of-sight between the BS and user • Sampling Interval: TS =TC/2=122 ns Polytechnic University, Dec. 4, 2003

  22. Power delay profile Delay distribution Angle distribution at BS Comparison with Data Space-frequency correlation between T11 & T21 at the MS Space-frequency correlation between T11 & T12 at the BS Polytechnic University, Dec. 4, 2003

  23. Indoor Model Polytechnic University, Dec. 4, 2003

  24. Indoor Narrowband Data • Comparing different types of correlations of the two-ring model with the data collected at Brigham Young University, 2000-2001. • Location:Fourth floor of a five story engineering building on Brigham Young University campus • Carrier Frequency: fC =2.45 GHz • Data Format:Narrowband (25 KHz) 10 x10 channel matrix. 124 such matrices • collected over 80 ms. Multiple such channel matrices obtained for several Rx • and Tx locations in each room • Antenna Spacing:/4 at both the Tx and Rx sides • No line-of-sight between the BS and user Polytechnic University, Dec. 4, 2003

  25. Different Types of Correlations Parallel Corr. Crossing Corr. Receive Corr. Transmit Corr. Polytechnic University, Dec. 4, 2003

  26. Comparison with Measured Correlation Parallel Corr. Crossing Corr. Tx Corr. Rx Corr. Common Transmit Corr. Common Receive Corr. Parallel Corr. Crossing Corr. Polytechnic University, Dec. 4, 2003

  27. Comparison with Measured Capacity Polytechnic University, Dec. 4, 2003

  28. Part II Fade Duration in MIMO Channels Polytechnic University, Dec. 4, 2003

  29. Outline • Motivation • Two Crossing Problems in MIMO Systems • Scalar Crossing • Theory and a numerical example • Applications (adaptive modulation and Markov modeling) • Vector Crossing • Theory • Application (block fading model in MIMO channels) • Summary Polytechnic University, Dec. 4, 2003

  30. Motivation • LCR (Level Crossing Rate) & AFD (Average Fade Duration) extensively studied in SISO channels: • Interleaver optimization • Adaptive modulation • Outage analysis in multiuser systems • Throughput estimation of protocols • Markov modeling of fading channels • LCR/AFDalso studied for receive diversity (SIMO) • How about MIMO channels?! Polytechnic University, Dec. 4, 2003

  31. x(t) t t0 Two MIMO Crossing Problems • Scalar crossing: There is a scalar processx(t) & we count the # of times it crosses a threshold (a traditional crossing problem) Polytechnic University, Dec. 4, 2003

  32. y(t) t0 x(t) Two MIMO Crossing Problems (cont) • Vector crossing: There is a vector process w(t) = [x(t) y(t)]T & we count the # of times it crosses a multidimensional surface (needs a multidimensional approach) Polytechnic University, Dec. 4, 2003

  33. AWGN AWGN AWGN 1 1 + + + AWGN 2 2 Transmitter + Receiver 3 3 4 Scalar Crossing in MIMO Channels • Total received SNR : Channel gain from the p-th Tx to the l-th Rx (a complex Gaussian process) Polytechnic University, Dec. 4, 2003

  34. γ(t) γ2 Probability γ1 t Incrossing Rate More on Total SNR in MIMO • Total MIMO SNR is useful for Markov modeling, channel characterization, and system design • Need to calculate ASD (Average Stay Duration) of γ(t) between two thresholds γ1 and γ2 Polytechnic University, Dec. 4, 2003

  35. Incrossing Rate of Total SNR • Space-time correlated Rayleigh fading  correlated zero-mean complex Gaussian hlp(t)’s • We have used this paper for incrossing of γ(t): A. M. Hasofer, “The upcrossing rate of a class of stochastic processes,” in Studies in Probability and Statistics. E. J. Williams, Ed., 1974. • For an MN channel, a (2MN-1)-fold integral needs to be solved (very time consuming) • We are developing a simpler technique (not done yet!) Polytechnic University, Dec. 4, 2003

  36. Example: Crossing of Total SNR • MIMO channel (macrocell) model, taken from: • A. Abdi and M. Kaveh, “A space-time correlation model for multielement antenna systems in mobile fading channels,” IEEE JSAC, 2002. Polytechnic University, Dec. 4, 2003

  37. Spatial correlation 0.99 0.88 21 channel Doppler = 20 Hz 0 Example: Crossing of Total SNR(cont) Polytechnic University, Dec. 4, 2003

  38. γ(t) t MIMO Application of Scalar Crossing Markov modeling: Use ICR of SNR to determine the transition probability from one state to another Adaptive Modulation: Use the ASD of SNR in each region, to choose proper power/rate adaptation policy Polytechnic University, Dec. 4, 2003

  39. Vector Crossing in MIMO Channels • Joint dynamic behavior of all the subchannels is of interest • In an MN channel, there are 2MN real space-time correlated processes. Put them into the vector h(t) • Need to calculate the ASD (Average Stay Duration) of the vector process h(t) within a hypercube Polytechnic University, Dec. 4, 2003

  40. Probability Outcrossing Rate Vector Crossing in MIMO Channels (cont) Example: M=N=1 Polytechnic University, Dec. 4, 2003

  41. A Simple Case Study • Rayleigh fading with no spatial correlation, as the temporal correlation for each subchannel • h(t0) =  [1 1 … 1]T,  > 0 Polytechnic University, Dec. 4, 2003

  42. Quantitative Analysis of Block Fading Question: For how long, all the subchannels stay within a hypercube of side 2, centered at  [1 1 … 1]T ? Polytechnic University, Dec. 4, 2003

  43. Summary of Part II • Two different types of crossing in MIMO channels • Scalar crossing is related to the total SNR • Adaptive modulation and Markov modeling in MIMO channels entail a scalar crossing • Vector crossing considers the joint variations of all the subchannels • The MIMO block fading model was analyzed using the vector crossing approach Polytechnic University, Dec. 4, 2003

  44. Part III MIMO Simulation Polytechnic University, Dec. 4, 2003

  45. AWGN AWGN AWGN + + + AWGN + The MIMO Channel • Propagation medium: Frequency-flat and time-varying multipath Rayleigh channel 1 1 2 2 Transmitter Receiver 3 3 4 Channel gains (zero-mean complex Gaussian processes) Polytechnic University, Dec. 4, 2003

  46. The Goal • Simulation of space-time correlatedhij(t)’s • Simulation of m correlated complex Gaussians Yk(t) needs cross-correlation and cross-spectrum functions Polytechnic University, Dec. 4, 2003

  47. Four Simulation Techniques • Spectral Representation Method needs MIMO cross-spectra • Sampling Theorem Method • Random Polynomial Method • Circulant Embedding Method the last three need MIMO cross-correlations Polytechnic University, Dec. 4, 2003

  48. Spectral Method • Correlated bandlimited processes: • Spectral Representation Theorem: • Discrete approximation of order q(# of bins in ): Polytechnic University, Dec. 4, 2003

  49. Sampling Method • Sampling Theorem: • n determines the window size max frequency in the spectrum of , vector of correlated Gaussian variables Polytechnic University, Dec. 4, 2003

  50. Polynomial Method • Linear spline approximation over time interval [a,b] • p is the # of subintervals in [a,b] , vector of correlated Gaussian variables Polytechnic University, Dec. 4, 2003

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