Issues in MIMO Channel Modeling and Simulation

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
Ray-based
Stochastic
Macrocells Microcells Picocells
Time and Frequency Selectivity
Correlations (space, time, and frequency) Fade
Duration
Spectral Sampling Polynomial Embedding
3
Part I

• MIMO Models

4
MIMO Modeling Macrocells

• No scatterer around the elevated base station
  (BS).
• Scatterers around the mobile station (MS) are
  located on a ring.

5
MIMO Modeling Micro Picocells

• Scatterers around the BS and the MS are located
  on separate rings.

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 Clarkes.

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.

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.

9
A 3?4 MIMO Channel

• We use h(t) for the narrowband channels, and
  T(f,t) for wideband channels.

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A 2x2 Channel
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Diffuse and Line-of-Sight Components(Frequency-Fl
at Fading)

• The channel fading gain
• Average power
• Rice factor

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Exact Diffuse Part of the Correlation

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Small Angle Spread at the Base Station

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Wave Scattering around the User

Non-isotropic scattering in a street
Isotropic scattering in an open area
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Users Angle of Arrival von Mises PDF

• is the mean direction of AOA, seen
  by the user
• controls the width of AOA

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The New Space-Time Correlation

• Let

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Special Cases of the Diffuse Part

• Clarkes 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

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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

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Correlation Fitting to Data
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Wideband Macrocell Model
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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/2122 ns
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Comparison with Data
Space-frequency correlation between T11 T21 at
the MS
Space-frequency correlation between T11 T12 at
the BS
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Indoor Model
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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
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Different Types of Correlations
Parallel Corr.
Crossing Corr.
Receive Corr.
Transmit Corr.
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Comparison with Measured Correlation
Parallel Corr.
Crossing Corr.
Tx Corr.
Rx Corr.
Common Transmit Corr.
Common Receive Corr.
Parallel Corr.
Crossing Corr.
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Comparison with Measured Capacity
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Part II

• Fade Duration in
• MIMO Channels

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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

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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/AFD also studied for receive diversity (SIMO)
• How about MIMO channels?!

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Two MIMO Crossing Problems

• Scalar crossing
• There is a scalar process x(t) we count the
  of times it crosses a threshold (a traditional
  crossing problem)

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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)

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Scalar Crossing in MIMO Channels

• Total received SNR

Channel gain from the p-th Tx to the l-th Rx
(a complex Gaussian process)
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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
  ?(t) between two thresholds ?1 and ?2

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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 ?(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!)

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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.

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Example Crossing of Total SNR (cont)
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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
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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

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Vector Crossing in MIMO Channels (cont)

Example MN1
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A Simple Case Study

• Rayleigh fading with no spatial correlation,
• as the temporal correlation
  for each subchannel
• h(t0) ? 1 1 1T, ? gt 0

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Quantitative Analysis of Block Fading

Question For how long, all the subchannels
stay within a hypercube of side 2?, centered at
? 1 1 1T ?
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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

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Part III

• MIMO Simulation

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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)
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The Goal

• Simulation of space-time correlated hij(t)s
• Simulation of m correlated complex Gaussians
  Yk(t) needs cross-correlation and cross-spectrum
  functions

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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

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Spectral Method

• Correlated bandlimited processes
• Spectral Representation Theorem
• Discrete approximation of order q ( of bins in
  )
• 
  

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Sampling Method

• Sampling Theorem
• n determines the window size

max frequency in the spectrum of
, vector of correlated Gaussian variables
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Polynomial Method

• Linear spline approximation over time interval
  a,b
• p is the of subintervals in a,b

, vector of correlated Gaussian variables
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MIMO Channel Model (macrocell)
A. Abdi M. Kaveh, A space-time correlation
model for multielement antenna systems in mobile
fading channels, IEEE JSAC, 2002.
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MIMO Correlation Spectrum

• Closed-form MIMO cross-correlation function

• Closed-form MIMO cross-spectrum function

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Numerical Parameters in Simulation

• 2x2 MIMO channel
• 1000 samples generated for each subchannel,
• over 0,10 seconds
• Max Doppler 0.05 Hz
• Angle spread at the BS 8 deg.
• Isotropic scattering around the MS
• Parallel Tx/Rx arrays
• MS moves towards the BS
• BS element spacing 17 wavelengths
• MS element spacing 0.5 wavelengths
• q60 (spectral), n30 (sampling), p60
  (polynomial)

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Simulated and Theoretical Auto and
Cross-Correlations
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Simulated and Theoretical Distribution and
Level Crossing Rate
PDFs of real and imaginary parts of the first
subchannel,
Level crossing rate of the envelope of
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Theoretical Computational Complexity
Number of Operations Spectral Sampling Polynomial Embedding
Corr. Matrix generation 48,000 3,810,240 3,572,160 983,040
Matrix Decomposition O(3,840) O(16,003,008) O(14,526,784) O(131,072)
White vector size 240 4488 244 8192
Coloring the white vector 960 77504 59536 32768
Calculating the main expression 360,000 252,000 10,000 315,392
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Summary of Part III

• All the four methods provide good results with
  proper choices of parameters (q, n, p)
• Spectral method requires lower computational
  effort (so, it is faster)
• The Matlab file for the Spectral method is
  available on
• http//web.njit.edu/abdi/

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• A Simulation
• Demo