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Fire Tom Wada Professor, Information Engineering, Univ. of the Ryukyus Chief Scientist at Magna Design Net, Inc [email protected] http://www.ie.u-ryukyu.ac.jp/~wada/ 2016/2/1 1 Fire Tom Wada, Univ. of the Ryukyus Introduction to MIMO OFDM

Introduction to MIMO OFDMMIMOwithOFDM).pdf · Introduction to MIMO OFDM ... OFDM and MIMO and there are many similarity in mathematics. 1. OFDM realizes many parallel communication

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Fire Tom Wada

Professor, Information Engineering, Univ. of the Ryukyus

Chief Scientist at Magna Design Net, Inc

[email protected]

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

2016/2/1 1Fire Tom Wada, Univ. of the Ryukyus

Introduction to MIMO OFDM

• REVIEW

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 2

Section 1

SISO Channel

2016/2/1 3

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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 4

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

Two path Multi path Channel Example

2016/2/1 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

2016/2/1 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 -

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 7

)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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 8

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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 9

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.

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 10

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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 11

Section 2

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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 12

)()()(

)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 -

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 13

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.

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 14

NtNtNrNr

K

H

0000

0000

0000

2

1

UVH

.,1

HH

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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 15

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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 16

MIMO communication

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 17

)(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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 18

)('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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 19

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

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 20

)('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)

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 21

)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.

2016/2/1 Fire Tom Wada, Univ. of the Ryukyus 22