Kein Folientitel

1
Spatial Temporal Correlation Properties of the
3GPP Spatial Channel Model and the Kronecker MIMO
Channel Model
Dr. Cheng-Xiang Wang
Heriot-Watt University School of Engineering
Physical Sciences Electrical, Electronic and
Computer Engineering Signal Image Processing
Group Phone [phone] Fax
44-131-4514155 E-mail cheng-xiang.wang_at_hw.ac.uk
URL http//www.ece.eps.hw.ac.uk/cxwang
2
Outline

• MIMO MIMO Channel Modeling
• Characteristics of 3GPP SCM KBSM
• Comparisons of 3GPP SCM KBSM
• Conclusions

3
MIMO Communications

• MIMOMultiple Input Multiple Output
• Both the transmitter and receiver are equipped
  with multiple antennas
• Plus additional signal processing which exploits
  the spatial dimension of the mobile radio channel

• Why MIMO?
• Two main benefits Diversity gain
  Multiplexing gain

4
Diversity Gain Multiplexing Gain

• Tradeoff between the diversity gain and the
  multiplexing gain.
• Highest gains are achieved under spatially
  uncorrelated Rayleigh fading channels (multiple
  uncorrelated processes for MIMO channel models).
• In practice, spatial correlation is often
  observed. More accurate MIMO channel models are
  necessary!

5
Classification of MIMO Channel Models

• Deterministic channel models (DCM)
• Stored measured channel impulse responses
• Ray tracing technique
• Stochastic channel models
• Physical models
• Geometric Based Stochastic Models (GBSM)
• One ring, two ring, elliptical,
• Parametric Stochastic Models (PSM)
• 3GPP SCM, COST 259,
• Analytical models
• Correlation Based Stochastic Models (CBSM)
• Kronecker Based Stochastic Model (KBSM)
• Virtual Channel Representation
• Joint Correlation Models (Weichselberger model)

6
Tradeoffs in Channel Modeling

• Tradeoffs in radio channel modeling
• Deterministic approach ?? Stochastic approach
• Physical intuition ?? Analytical traceability

Analytical convenience ! Low complexity ! Low
accuracy! Low adaptability! ( Information
theorists Signal processing people )
High accuracy ! Low adaptability ! High
complexity ! ( Radio propagation people )
Analytical
CBSM
PSM
GBSM
High adaptability ! Moderate complexity ! Enough
accuracy ! ( Radio system engineers )
DCM
Physical
Deterministic
Stochastic
7
Motivations

• Mappings between stochastic channel models

• PSM
• Path angle distribution
• Subpath angle distribution

• GBSM
• Cluster distribution
• Scatterer distribution

• CBSM
• Spatially averaged correlation properties
• Instant correlation coefficient

• Only a few papers addressing the relationship
  between a PSM and a CBSM.
• The spatial temporal correlation properties of
  both types of models still require further
  investigation.
• The interrelation and differences will also be
  studied.

8
Motivations

• 3GPP SCM a practical implementation of PSMs
• The spatial temporal correlation properties are
  implicit. Difficult to connect SCM simulation
  results with theoretical MIMO system analysis.
• The implementation complexity is high it has to
  generate many parameters.
• KBSM a simplified CBSM
• Elegant and concise analytical expressions for
  MIMO channel spatial correlation matrices.
• Less input parameters. Has the KBSM been
  oversimplified?

9
Outline

• MIMO MIMO Channel Modeling
• Characteristics of 3GPP SCM KBSM
• SCM and its spatial-temporal correlation
  properties
• KBSM and its spatial-temporal correlation
  properties
• Comparisons of 3GPP SCM KBSM
• Conclusion

10
SCM (1) Simulation Procedure
11
SCM (2) General Descriptions Definitions

• Channel model description
• The received signal consists of N time-delayed
  multipath replicas of the transmitted signal (N
  6)? Tapped delay line model
• Each resolvable path consists of M irresolvable
  subpaths (M 20). ? superposition of waves ? Sum
  of Sine principle
• Each path is corresponding to a cluster each
  subpath is corresponding to a scatterer within
  the cluster.
• Definition of drop
• A drop is defined as a simulation run for a
  given number of cells/sectors, BSs, and MSs, over
  a specified number of frames.
• During a drop, the channel undergoes fast fading
  due to MS motions.
• Typically, over a series of D drops, the cell
  layout and locations of the BSs are fixed, but
  the location of the MSs are randomly varied at
  the beginning of each drop.

12
SCM (3) Angle Parameters
Angle between the BS-MS LOS and the BS/MS
broadside AoD/AoA of the nth path with respect
to the LOS AoD/AoA Subpath AoD/AoA offset Mean
AoD/AoA
13
SCM (4) Three Level Definitions

• Cluster level
• Fixed BS and MS locations
• Fixed cluster locations
• Random scatterer distributions
• Subpath statistics are emulated by fixed values.

• Link level
• Fixed BS and MS locations
• Random cluster distributions
• Random scatterer distributions

• System level
• Random BS and MS locations
• Random cluster distributions
• Random scatterer distributions

14
SCM (5) Channel Coefficient Formulation

• The NLOS channel coefficient between the sth BS
  antenna element and the uth MS antenna element of
  the nth path (cluster) is given by

Power of the nth path Log-normal
shadowing Number of subpaths per path Subpath
AoD Subpath AoA BS antenna array gains MS
antenna array gains
Relative BS antenna distance (meter) Relative
MS antenna distance (meter) Subpath random
path MS speed MS direction Wave number
M
k
15
SCM (6) Spatial Temporal Correlation Function

• General formula of the spatial temporal
  correlation function

• Spatial temporal correlation function at the
  cluster level (no expectation)

• A deterministic function of

• The spatial-temporal separability is not a
  property of the SCM at the cluster level .

16
SCM (7) Spatial Correlation Function 1

• Spatial correlation function general formula

• Spatial correlation function at the BS

• Spatial correlation function at the MS

• Spatial correlation functions at the cluster
  level are functions of

• Spatial separability is not a property of the SCM
  at the cluster level.

17
SCM (8) Spatial Correlation Function 2

• Ideal spatial correlation function when

Joint PDF of
and

• Ideal spatial correlation function at the BS / MS

PDF of

• Still, spatial separability is not a property of
  the SCM at the cluster level

18
SCM (9) Temporal Correlation Function

• Temporal correlation function general formula

• Ideal temporal correlation function when

• Take the expectation over which is uniformly
  distributed over

• Spatial-temporal separability is a SCM property
  at the system level

19
Outline

• MIMO MIMO Channel Modeling
• Characteristics of 3GPP SCM KBSM
• SCM and its spatial-temporal correlation
  properties
• KBSM and its spatial-temporal correlation
  properties
• Comparisons of 3GPP SCM KBSM
• Conclusion

20
KBSM (1) Channel Coefficient Formulation

• Basic assumptions
• The channel coefficients of a narrowband MIMO
  channel are complex Gaussian distributed with
  average power.
• The scattering environment around each end is
  independent of each other ? spatial separability.
• Spatial characteristics and temporal
  characteristics are separable ? spatial temporal
  separability.

Tx array

• Channel Coefficient Formulation

MS array correlation matrix
Rx array
BS array correlation matrix
Complex Gaussian process with desirable
temporal characteristics
21
KBSM (2) Spatial Correlation Function 1

• Spatial correlation function at the BS

PDF of

• Spatial correlation function at the MS

PDF of

• General spatial correlation function

• Spatial separability is one of the basic
  properties of the KBSM.

22
KBSM (3) Spatial Correlation Function 2

• Channel correlation matrix

Kronecker product

• Spatial filtering structure of KBSM

Spatially Uncorrelated Gaussian Processes
Spatially Correlated Gaussian Processes
Spatial filtering
23
KBSM (4) Temporal Correlation Function
Spatial Temporal Correlation Function

• Temporal correlation function (U shape Doppler
  filtering)

• Doppler filtering structure of a KBSM

Correlated Gaussian Processes
Uncorrelated Gaussian Processes

• Spatial temporal correlation function

• The temporal correlation property of a KBSM is
  static .
• Spatial temporal separability is one of the basic
  properties of KBSM.

24
Outline

• MIMO MIMO Channel Modeling
• Characteristics of 3GPP SCM KBSM
• Comparisons of 3GPP SCM KBSM
• Theorectical Comparison
• Simulation Based Comparison
• Conclusion

25
Fundamental Differences Equivalent Conditions

• Fundamental differences between SCM KBSM
• Num. of
  subpaths AoA-AoD Correlation
• SCM Finite
  (20)
  Correlated
• KBSM Infinite (Gaussian
  process) Independent

• Equivalent Conditions
• The number M of subpaths in each path for the SCM
  tends to infinity.
• Two links share the same antenna element at one
  end, i.e., the spatial correlations at either the
  MS or the BS.
• The same set of angle parameters are used.
• The PAS function are the same.

• Equivalent conditions will be used for the
  calibration of two models.

26
PAS Fitting 1 (Model Calibration)

• In the SCM, subpath statistics are emulated by
  specifying fixed value subpath AoA/AoD offsets

• In the KBSM, subpath statistics are described by
  the Power Azimuth Spectrum (PAS) functions of
  subpaths
• ---- There are three candidate PASs
  Uniform, Gaussian, Laplacian

• Model calibration is to find a PAS function for
  KBSM which fits best its spatial correlation
  properties to those of the SCM

• Equivalent conditions are applied for model
  calibration
• Interpolate the subpath AoA/AoD offsets 100 times
  to approximate infinite M? Interpolated SCM
• Spatial correlation at the BS and MS
• Calculate the spatial correlations as functions
  of mean AoA/AoD and normalized antenna spacing
• Find the best fit PAS function using least square

27
PAS Fitting 2 Interception 1

• Spatial correlation as functions of mean AoA/AoD
  with fixed normalized antenna spacing equal to 2

28
PAS Fitting 3 Interception 2

• Spatial correlation as functions of normalized
  antenna spacing with fixed mean AoA/AoD equals to
  60 degree

29
PAS Fitting 4 Observations

• In all cases, Gaussian PAS fit best to the
  interpolated SCM In all cases, Gaussian PAS fit
  best to the SCM
• ! ! Gaussian is not the 3GPP SCM choice for
  link level calibrations.
• The spatial correlation functions obtained from
  the SCM have unstable fluctuations around the
  ideal value approximated by the interpolated SCM
• Caused by insufficient number of M
• We refer as implementation loss
• Implementation loss become significant when the
  AS is as high as 35 degree
• The original SCM have relatively poor performance
  compared with the ideal MIMO channel model in
  terms of the spatial correlation function.
• We suggest to increase the number of subpaths in
  the SCM to increase the simulation accuracy.

30
Theoretical SCM Cluster Level

• The cluster level spatial correlation function of
  the SCM as a function of the mean AoA and mean AoD

31
Theoretical KBSM Cluster Level

• The cluster level spatial correlation function of
  the KBSM as a function of the mean AoA and mean
  AoD

32
Outline

• MIMO MIMO Channel Modeling
• Characteristics of 3GPP SCM KBSM
• Comparisons of 3GPP SCM KBSM
• Theorectical Comparison
• Simulation Based Comparison
• Conclusion

33
Comparison Approach Simulation Set Up

• For fair comparison, we use exactly the same
  parameters for the SCM and KBSM. ? KBSM is
  embedded into the SCM to share the SCM parameter
  generator
• We consider a 2 by 2 MIMO link
• Long sequence of the SCM channel coefficients are
  generated and its spatial/temporal correlation
  functions are calculated numerically
• Spatial/temporal correlation functions for the
  KBSM are calculated according to the
  corresponding parameters
• For spatial correlation, we compared the first
  column of the 4 by 4 MIMO channel correlation
  matrix
• The temporal correlation is evaluated at the
  system level

34
Spatial Correlation Cluster Level Simulation

• The cluster level comparison of instant spatial
  correlation functions of the SCM and KBSM

35
Spatial Correlation Link Level Simulation 1

• Spatial correlation functions of the SCM and KBSM
  averaged at the link level, as functions of the
  normalized MS antenna spacing

36
Spatial Correlation Link Level Simulation 2

• Spatial correlation functions of the SCM and KBSM
  averaged at the link level, as functions of the
  normalized BS antenna spacing

37
Spatial Correlation Link Level Conclusions

• At the link level, cluster level effects of
  implementation loss and the imperfect PAS
  fitting tend to be averaged out.
• At the link level, the SCM has the same property
  of the spatial separability as the KBSM.

38
Spatial Correlation System Level Simulation 1

• Spatial correlation functions of the SCM and KBSM
  averaged at the system level, as functions of the
  normalized BS antenna spacing

39
Spatial Correlation System Level Simulation 2

• Spatial correlation functions of the SCM and KBSM
  averaged at the system level, as functions of the
  normalized MS antenna spacing

40
Spatial Correlation System Level Conclusions

• At the system level, cluster level effects of
  implementation loss and the imperfect PAS
  fitting tend to be average out
• At the system level, the SCM has the same
  property of the spatial separability as the KBSM
• At system level, the spatial correlation function
  approximates the well-known Bessel function given
  as

41
Temporal Correlation System Level Simulations

• Temporal correlations functions of the SCM and
  KBSM at the system level

42
Temporal Correlation Conclusions

• At the system level, the SCM has the same spatial
  temporal separability as the KBSM.
• At the system level, both models tend to have the
  identical temporal correlation function.
• The temporal properties of the KBSM remain
  static, whereas vary largely for SCM from
  individual runs.

43
Conclusions

• The SCM KBSM properties can be evaluated at
  three levels the cluster level, link level and
  system level.
• At all the three levels, the KBSM has both the
  spatial separability and spatial temporal
  separability.
• The SCM has the spatial temporal separability
  only at the system level, and has the spatial
  separability at the link and system levels.
• The KBSM with the Gaussian PAS is found to fit
  best the SCM spatial correlation properties.
• The KBSM is restricted to model only the
  averaging effects of spatial correlation and
  temporal correlation.
• The SCM provides more insights of the variations
  of different MIMO links.

44
Thank You !