Fire Tom Wada

Professor, Information Engineering, Univ. of the Ryukyus

Chief Scientist at Magna Design Net, Inc

wada@ie.u-ryukyu.ac.jp

http://www.ie.u-ryukyu.ac.jp/~wada/

2013/12/25 1Fire Tom Wada, Univ. of the Ryukyus

Introduction to MIMO OFDM

• REVIEW

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Section 1

SISO Channel

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Base Stat ion

Mobile

Recept ion

Path 2

Path 3

Direct Path

Building

Transmission Antenna

Reception Antenna

Single Input andSingle Output

(SISO)Channel

Fire Tom Wada, Univ. of the Ryukyus

Base Station Receiver

Two path Multi path Channel Example

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Channel Impulse Response = [1, 0.5 , 0, 0]

Two path Multi path Channel Example

2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 5

)3(

)2(

)1(

)0(

)3(000

0)2(00

00)1(0

000)0(

)3(

)2(

)1(

)0(

)3(

)2(

)1(

)0(

1

1

1

1111

4

1

15.000

015.00

0015.0

5.0001

1

1

1

1111

)3(

)2(

)1(

)0(

)3(

)2(

)1(

)0(

***

)3(

)2(

)1(

)0(

963

642

321

963

642

321

X

X

X

X

H

H

H

H

Y

Y

Y

Y

X

X

X

X

Y

Y

Y

Y

X

X

X

X

IFFTChannelFFT

Y

Y

Y

Y

If time domain channel matrix is cyclic, Frequency Domain Channel Matrix is diagonal!

Additive Noise

2013/12/25 6

)3(

)2(

)1(

)0(

)3(

)2(

)1(

)0(

)3(000

0)2(00

00)1(0

000)0(

)3(

)2(

)1(

)0(

)3(

)2(

)1(

)0(

1

1

1

1111

)3(

)2(

)1(

)0(

1

1

1

1111

4

1

15.000

015.00

0015.0

5.0001

1

1

1

1111

)3(

)2(

)1(

)0(

)3(

)2(

)1(

)0(

)3(

)2(

)1(

)0(

***

)3(

)2(

)1(

)0(

963

642

321

963

642

321

963

642

321

N

N

N

N

X

X

X

X

H

H

H

H

Y

Y

Y

Y

noise

noise

noise

noise

X

X

X

X

Y

Y

Y

Y

noise

noise

noise

noise

X

X

X

X

IFFTChannelFFT

Y

Y

Y

Y

Fire Tom Wada, Univ. of the Ryukyus

How to recover sending signal from receiver signal.- EQUALIZE -

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)3(

)2(

)1(

)0(

)3(

1000

0)2(

100

00)1(

10

000)0(

1

)3(

)2(

)1(

)0(

)3(

)2(

)1(

)0(

)3(000

0)2(00

00)1(0

000)0(

)3(

)2(

)1(

)0(

***

)3(

)2(

)1(

)0(

Y

Y

Y

Y

H

H

H

H

X

X

X

X

Then

X

X

X

X

H

H

H

H

X

X

X

X

IFFTChannelFFT

Y

Y

Y

Y

NoiseIgnore　

Summary of Matrix model of OFDM

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Transmission Antenna

Reception Antenna

Channel

)3(

)2(

)1(

)0(

X

X

X

X

IFFT

Channel

Cyclic

FFT

)3(

1000

0)2(

100

00)1(

10

000)0(

1

H

H

H

H

)3(

)2(

)1(

)0(

X

X

X

X

)3(

)2(

)1(

)0(

)3(

1000

0)2(

100

00)1(

10

000)0(

1

)3(

)2(

)1(

)0(

X

X

X

X

IFFTChannel

CyclicFFT

H

H

H

H

X

X

X

X

Important Mathematics

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Cyclic Matrix can be diagonalized by FFT and IFFT.

XH is Hermitian of X, that is, complex conjugate and transpose.

)),1(),0((

)3(000

0)2(00

00)1(0

000)0(

)),1(),0((

)3(000

0)2(00

00)1(0

000)0(

HHdiag

H

H

H

H

UMatrix

CyclicU

UIFFT

H

FFT

Since

HHdiag

H

H

H

H

IFFTMatrix

CyclicFFT

H

Unitary Matrix Unitary Matrix U can satisfy following property.

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IUUUU HH

)(0

)(1

,

1

ji

ji

then

vectorverticaliswhen

j

H

i

N

i

ee

eeU

e

　　　　

U

H

H

H

H

UMatrix

CyclicH

)3(000

0)2(00

00)1(0

000)0(

Eigen value of Channel Cyclic Matrix is Channel Transfer Function as (H(0), H(1), H(2), … ).

• MIMO OFDM Channel Modeling

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Section 2

SISO Channel OFDM makes Multi-path channel simple complex h(k) for freq=k.

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)()()(

)3(

)2(

)1(

)0(

)3(000

0)2(00

00)1(0

000)0(

)3(

)2(

)1(

)0(

***

)3(

)2(

)1(

)0(

kXkhkY

X

X

X

X

h

h

h

h

X

X

X

X

IFFTChannelFFT

Y

Y

Y

Y

Transmission Antenna

Reception Antenna

SISO ChannelIFFT FFT

)(kX )(kY

bjakh )(

MIMO Channel- Nr X Nt SISO Channels for Freq=k -

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NtNtNrNtNrNr

Nt

Nt

Nr X

X

X

X

X

X

hhh

hhh

hhh

Y

Y

Y

2

1

2

1

21

22221

11211

2

1

0

00

0

0

H

)(1 kX)(11 kh

)(2 kX

)(3 kX

)(kX Nt

)(1 kY

)(2 kY

)(3 kY

)(kYNr

)(21 kh

)(32 kh

)(khNrNt

H

Singular value decomposition of Nr x Nt Matrix H Nr x Nt matrix H can be decomposed as below using

Nr x Nr Unitary matrix V and Nt x Nt Unitary matrix U.

Σ is Nr x Nt diagonal matrix.

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NtNtNrNr

K

H

0000

0000

0000

2

1

UVH

.,1HH

K andofvalueeigenis HHHH

SVD Example by Matlab(1) H =

1 2 3

2 4 5

>> [U,S,V] = svd(H)

U =

-0.4863 -0.8738

-0.8738 0.4863

S =

7.6756 0 0

0 0.2913 0

V =

-0.2910 0.3396 -0.8944

-0.5821 0.6791 0.4472

-0.7593 -0.6508 -0.0000

H =

1 2

2 4

3 5

>> [U,S,V] = svd(H)

U =

-0.2910 -0.3396 -0.8944

-0.5821 -0.6791 0.4472

-0.7593 0.6508 -0.0000

S =

7.6756 0

0 0.2913

0 0

V =

-0.4863 0.8738

-0.8738 -0.4863

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SVD Example by Matlab(2) H =

1.0000 + 1.0000i 2.0000 + 1.0000i

1.0000 - 3.0000i 3.0000 - 1.0000i

>> [U,S,V] = svd(H)

U =

-0.4616 - 0.0659i -0.4907 + 0.7361i

-0.3956 + 0.7913i -0.2863 - 0.3680i

S =

5.0000 0

0 1.4142

V =

-0.6594 0.7518

-0.5934 + 0.4616i -0.5205 + 0.4048i

>> U*S*V'

ans =

1.0000 + 1.0000i 2.0000 + 1.0000i

1.0000 - 3.0000i 3.0000 - 1.0000i

>> U*S*V'

ans =

1.0000 + 1.0000i 2.0000 + 1.0000i

1.0000 - 3.0000i 3.0000 - 1.0000i

>> U'*U

ans =

1.0000 0.0000 - 0.0000i

0.0000 + 0.0000i 1.0000

>> U*U'

ans =

1.0000 0.0000 - 0.0000i

0.0000 + 0.0000i 1.0000

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MIMO communication

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)(1 kX

)(2 kX

)(3 kX

)(kX Nt

)(1 kY

)(2 kY

)(3 kY

)(kYNr

NtNr X

X

X

Y

Y

Y

2

1

2

1

H

HMIMO

Channel

Introduce pre-processing and post-processing

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)('1 kX

)('2 kX

)('3 kX

)(' kX Nt

)('1 kY

)('2 kY

)('3 kY

)(' kYNr

'

'

'

'

'

'

2

1

2

1

Nt

H

Nr X

X

X

Y

Y

Y

HUV

HMIMO

Channel

NtxNt

UNrxNr

HV

There are K(=rank(H)) independent channel

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0'

0'

''

''

'

'

'

000

000

00

00

00

'

'

'

1

111

2

11

2

1

Nr

K

KKK

Nt

K

Nr

Y

Y

XY

XY

X

X

X

Y

Y

Y

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

2

1

2

1

2

1

2

1

2

1

2

1

NtNr

Nt

HH

Nr

Nt

H

Nr

X

X

X

Y

Y

Y

X

X

X

Y

Y

Y

X

X

X

Y

Y

Y

UUVV

HUV

SVD-MIMO system

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)('1 kX

)('2 kX

)('3 kX

)(' kX Nt

)('1 kY

)('2 kY

)('3 kY

)(' kYNr

H=VΣUH

MIMOChannel

NtxNt

UNrxNr

HV

1

K

)('1 kX

)('2 kX

)('3 kX

)(' kX Nt

)('1 kY

)('2 kY

)('3 kY

)(' kYNr

Put them altogetherMIMO-OFDM system Space Division Multiplexing by MIMO (K stream)

Orthogonal Frequency Division Multplxing (OFDM)

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)1:0('1 NX

)1:0('2 NX

)1:0('1 NY

)1:0('2 NY

IFFT FFT

IFFT

IFFT

IFFT

FFT

FFT

FFT

NtxK

NtxK

NtxK

NtxK

NtxK

NtxK

NtxK

KxNr

U HV

Summary This presentation shows matrix based modeling for both

OFDM and MIMO and there are many similarity in mathematics.

1. OFDM realizes many parallel communication channels in frequency domain.

2. OFDM converts multi-path channel to simple one tap channel such as h(k)=a+bj for Frequency=k.

3. Then OFDM-based MIMO system can focus on simple channel matrix.

4. By singular value decomposition (SVD), MIMO channel matrix H can be decomposed to V*Σ*UH.

5. Non-zero elements of Σ (rank of H) indicates parallel communication channel in space.

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