UEC Tokyo

Multi-Keyhole Modelfor MIMO Radio-Relay Systems

EuCAP 2007 Nov. 12-16, 2007

Y. KARASAWA, M. Tsuruta and T. Taniguchi

Advanced Wireless Communication research Center (AWCC)

Univ. Electro-Communications (UEC Tokyo)

UEC Tokyo

Outline of Presentation

1. Radio-relay system for MIMO service area expansion

2. Multi-keyhole model for the system performance evaluation

3. Empirical formula for PDF of the largest eigenvalue

4. Empirical formula for PDFs of all eigenvalues

5. Evaluation of the estimation accuracy

6. Conclusions

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Motivation

Why do we need MIMO channel models?High data rate transmission (WLAN, WiMAX)

Service expansion to isolated areasMIMO repeater system

(MIMO radio-relay system)Ad Hoc Network MIMO repeater

PC 1PC 2 PC 3

Internet

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MIMOrepeater

AP

MIMO repeater system

Extension of high capacity link (= multi-stream transmission) to isolated areas

Equivalent Multi-keyholePropagation Channel

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Keyhole environment

Multi-keyhole environmentMulti-antenna relay

Single-antenna relayM x 1 x N

M x K x N

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M K N

1

2

3

1

2

3

１

K

１

( ) rvrptr nnsHGHr ++=

tre KHHH 1

=

s Ht nrpHr nrv

r

Received signal

Eqivalent channel (CSI)when the effect of thermal noise powerin RS is negligible.

Channel Expression of Radio-Relay Systems

HeeHHR ≡

Hereafter, we discussPDFs of eigenvaluesfor

Accesspoint

Userterminal

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Eigenvalues in MIMO repeater system

2×1×2 2×2×2 2×3×2

1st Eigenvalue

2nd Eigenvalue

2×2

keyhole Multi-keyhole idealChannel model

Rayleigh fading environment with i.i.d.

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PDF of the largest eigenvaluein single keyhole environment

( )

( ) ( ) ( )1

121

120 21

22

1)(

λλ

λλ

NM

MN

MN

KMN

duu

pupu

p

−

−+

∞

ΓΓ=

⎟⎠⎞

⎜⎝⎛= ∫

[ ] [ ]tM

ttTrN

rr

tre

hhhhhh 1121112111 LL=

= HHH

)(1HeeTrace HH=λM N1

Ht Hr

Channel response matrix

The p.d.f. of eigenvaluecentral chi-square

distributed with 2Ndegrees of freedom

central chi-square distributed with 2Mdegrees of freedom

(Kv: v-th order modified Bessel function of the second kind)

p2N

p2M

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Approximated PDF of the largest eigenvalue

M+L N+L1

Before the transformation

After the transformation

linked to the theory of space diversity at transmission and reception sides, respectively

Converting the number of repeater antennas to 1

introduction of the number of effective increment antennas L

It results to p.d.f. of keyhole

M NK

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Introduction of the number of effective increment antennas L

( ) ( )( )βα 1,, −= KNMKNML6681.0,4343.0 == βα

the number of effective increment antennas L

M NK

Before the transformation

M+L N+L1

After the transformation

It is the resonable approximation.

100

101

100 101 102 103

2X2 3X3 4X4 5X5 6X6 近似式

L(M,N,K)

M(K-1)N

Approximation

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Approximation equation of p.d.f. of largest eigenvalue for MxKxN case

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

Λ+Γ+ΛΓ

⎟⎠⎞

⎜⎝⎛Λ= −

−++

1

12

2

1

1 22

)( λλ

λ NM

LMN

KLMLN

p

Λ: Power adjustment factor (average value of eigenvalue)

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Comparison between simulated values and calculated values ~the largest eigenvalue~

0.001

0.01

0.1

1

0.1 1 10

Cumulativ

e pro

babilit

y

Eigenvalue

N,M=2

N,M=3

K=2K=3K=4

The number K of repeater antennas

K=2K=3K=4

Simulation

Approximation

0.001

0.01

0.1

1

0.1 1 10 100

Cumulative probability

Eigenvalue

N=2,M=4

The number K of repeater antennasSimulation

K=2K=3K=4K=5

K=2K=3K=4K=5

Approximation

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PDF of each Eigenvalue in i.i.d. Environment (approximated)

M N

Gamma distributionwith diversity order of

λ1: : M x N

λ2: : (M-1) x (N-1)

λ3: : (M-2) x (N-2)

M x N

(Nt-1) x (Nr-1)

(Nt-2) x (Nr-2)

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Approximate Formulas of PDFs for All Eigenvalues

M NK

Mi+Li Ni+Li1

MiNiKi

Mi=M-i+1Ki=K-i+1Ni=N-i+1

Li=L-i+1

Similar way when obtaining the largest eigenvalue for (M-i+1) x (K-i+1) x (N-i+1)

Equivalent transformation: Reduce each antenna number by i-1

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

Λ+Γ+ΓΛ

⎟⎟⎠

⎞⎜⎜⎝

⎛Λ

= −

−++

i

iNM

iiiii

LMN

i

i

i ii

iii

KLMLN

p λλ

λ 22

)(

12

2

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Comparison between simulated values and calculated values ~all eigenvalues~

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 100

Cumulative probability

Eigenvalue

K=1K=2K=3

The number K of repeater antennas

K=1K=2K=3

Simulation

Approximation

λ1λ2

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 100

Cumulative probability

Eigenvalue

K=1K=2K=3

The number K of repeater antennas

K=1K=2K=3

Simulation

Approximation

2×K×2 2×K×4λ1

λ2

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Conclusion

MIMO radio-relay system (or MIMO repeater system) which can expand service area to isolated areas is introduced.

For the system designing, channel model is important. A channel model named multi-keyhole model for this purpose is presented.

An empirical calculation method for PDFs of eigenvaluesin the channel is developed.

The proposed scheme realizes very accurate estimate of the PDFs.

We are ready to evaluate digital transmission characteristics of radio-relay systems using the proposed channel model.