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On the MIMO Channel Capacity Predicted by

Kronecker and Müller Models

Müge KARAMAN ÇOLAKOĞLU

Prof. Dr. Mehmet ŞAFAK

COST 289 4th Workshop, Gothenburg, SwedenApril 11-12, 2007

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Outline

MIMO Channel Models Kronecker model Müller model

Results Conclusion

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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

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Kronecker Model-2

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Kronecker Model-3

Sensitivity against the model parameters(wavelength=0.15 m)

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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

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Müller Model-2

Delay and space coordinates for the Müller model

Propagation coefficient between th Tx and th Rx antenna

v

l

lvl

Sj

llv eAh1

)(,,,

,,,,

1

1L

k k

y H x

Received signal at time :k

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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

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Müller Model-4 Surviving physical parameters that dominate

the value of the channel capacity:

R

T

N

NSystem load:

Total richness:RN

S

Attenuation distribution (assumed):

, 1A (for all ),

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Results-1

Parameters used for the Kronecker model

(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

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Results-2 Effects of the

number of Tx and Rx antennas for

4R

T

N

N

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Results-3 Effect of the

number of scatterers for

3

1

R

T

N

N 3RN

S

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Results-4 Effect of the

number of Tx antennas

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Results-5 Effect of the

number of Rx antennas

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Results-6 Effect of

the number of scatterers

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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

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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

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Thank You...

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Channel Capacity-1

Finding average capacity-Method 1: Replace mean value of 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)

HHH

H

TN N

CR

HHI

detlog2

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Channel Capacity-2

Finding average capacity-Method 2: Elements of are zero mean Gaussian

random variables 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.

HHHH

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Kronecker Model-3

Isolates the fading

correlations

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

More suited for urban channels

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Kronecker Model-4

Sensitivity against the model parameters(wavelength=0.15 m)

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Müller Model-4

It is assumed that 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

at the same rate.

, 1,A for all and