On the MIMO Channel Capacity
Predicted by Kronecker and
Müller Models
Müge KARAMAN ÇOLAKOĞLU
Prof. Dr. Mehmet ŞAFAK
COST 289 4th Workshop, Gothenburg, Sweden
April 11-12, 2007
1
Outline

MIMO Channel Models

Kronecker model

Müller model

Results

Conclusion
2
Kronecker Model-1
Assumptions:
 Flat fading channel
 Only doubly-scattered rays are considered






LOS multipath component ignored
Single scattered signals ignored
Source of fading
 Local scatterers
Number of scatterers  Typically >10
Fading correlations are separated
Tx have no CSI, Rx have CSI
3
Kronecker Model-2
4
Kronecker Model-3
Sensitivity against the model parameters
(wavelength=0.15 m)
5
Müller Model-1
Assumptions:
 Frequency-selective fading channel


Scatterers can be distinguished in time and space
(Different locations and delays)
Only singly-scattered rays are considered



No LOS, no multiple scattering
Tx and Rx at the foci of concentric (equi-delay)
ellipses
Asymptotic in the number of scatterers, and of
the transmit- and receive antennas
6
Müller Model-2
Propagation coefficient
between  th Tx and
Rx antenna
Sl
hv , ,l   A ,l e
v th
j ( ,  ,l  ,v ,l )
 1
Received signal at time
k :
L
y  k    H x  k   1
1
Delay and space coordinates for the Müller model
7
Müller Model-3

Singular values of the random channel matrix H show fewer
fluctuations, become deterministic as its size goes infinity

Approximates finite size matrices

Singular value distributions can be calculated analytically


Only the surviving physical parameters show significant
influence on the singular value distribution and characterise
the MIMO channel
In the asymptotic limit, singular values of H for flat fading and
frequency-selective fading channels are the same
8
Müller Model-4

Surviving physical parameters that dominate
the value of the channel capacity:
System load:
NT

NR
Total richness:
S
 
NR
Attenuation distribution
(assumed):
A ,  1 (for all
,
)
9
Results-1

(m)
dt
(m)
dr
(m)
Rt0
(m)
Rr0
(m)
Dt
(m)
Dr
(m)
R
(m)
0.15
0.15
0.15
50
50
50
50
50000
Parameters used for the Kronecker model
10
Results-2

Effects of the
number of Tx
and Rx antennas
for N T
4
NR
11
Results-3

Effect of the
number of
scatterers for
NT 1

NR 3
S
3
NR
12
Results-4

Effect of the
number of Tx
antennas
13
Results-5

Effect of the
number of
Rx antennas
14
Results-6

Effect of
the number
of
scatterers
15
Conclusion-1
Kronecker model:
 Valid for flat-fading channels
 May be more appropriate for urban channels
 May lead to pessimistic capacity predictions in
suburban areas
 Some measurement results show that model fails
under certain circumstances
16
Conclusion-2
Müller model:



Valid for frequency-selective fading channels
May describe suburban channels more accurately
Simple and characterizes the channel by




the number of Tx antennas
the number of Rx antennas
the chanel richness (usually ignored in other models)
Capacity predictions by the Müller model may be
higher compared with the Kronecker model
17
Thank You...
18
Channel Capacity-1
Finding average capacity-Method 1:
H
 Replace mean value of HH
by deterministic
correlation matrix  Find eigenvalues and the
capacity
 Issue: How to model the correlation matrix ?
(correlation between antenna elements, angular
spread of signals, scattering richness)
 


H 
C  log 2 det  I N R 
HH 
NT

 
19
Channel Capacity-2
Finding average capacity-Method 2:
 Elements of H are zero mean Gaussian random
variables  HH H is central Wishart matrix.
 Joint pdf of the ordered eigenvalues of a complex
Wishart matrix is known.
 Determine the capacity by using joint pdf of the
ordered eigenvalues.
 Issue: Hard to determine the marginal pdf’s
analytically.
20
Kronecker Model-3

Isolates the fading

Simplify the simulation
and the analysis
Underestimates the
channel capacity
(high corelation )

Should be used at low
correlation chanels
Assumes double
scattering from local
scatterers

correlations




More suited for urban
channels
21
Kronecker Model-4
Sensitivity against the model parameters
(wavelength=0.15 m)
22
Müller Model-4




It is assumed that A ,  1,
for all  and
Eigenvalue distribution of the space-time channel
matrix does not changes if the delay times of
particular paths vary.
No need to distinguish between the distributions of
path attenuations conditioned on different delays.
A uniform power delay profile is assumed.

The paths that have same delay are assumed attenuated
23
at the same rate.