Xiang-Gen Xia

University of Delaware

Newark, DE 19716

xxia@ee.udel.edu http://www.ee.udel.edu/~xxia

Space-Time Codes, Orthogonal

Designs, and Compositions of

Quadratic Forms

Alamouti code from 2 by 2 orthogonal design

Alamouti Code for 2 Transmit Antennas

(1998)

** 12

21

xx

xx

Information bits

are mapped to

complex symbols

x1 and x2

encoder

** 12

21

xx

xx

-x2* x1

x1* x2

It is an option in 3G

Alamouti Scheme: Fast ML

Decoding and Full Diversity

Signal Model:

Y=CA+W，

where

is a signal constellation, for example

S21

12

21,:

**xx

xx

xxC C

},1{ jS

S

Alamouti Code: Fast ML Decoding

ML decoding is to minimize

Orthogonality:

for any values x1 and x2.

The cross term x1x2 can be canceled and x1 and x2 can be separated：

x1 and x2 can be decoded separately:

The decoding complexity is reduced from to

}{}{}{

)}(){(|||| 2

CCAAtrYCACAYtrYYtr

CAYCAYtrCAY

HHHHHH

HF

22

22

1 )|||(| IxxCCH

)()(|||| 2211

2 xfxfCAY F

)(minand)(minmin 2211),( 21

221

xfxfSxSxSxx

2|| S

||2 S i.e., complex symbol-wise decoding

For any two different matrices

Their difference matrix is also orthogonal

Because of the orthogonality, B has full rank

Alamouti Code: Full Rank Property

),(),(~

),(~~

,),(

2121

*

1

*

2

21

21*

1

*

2

21

21

yyxxCC

yy

yyyyCC

xx

xxxxCC

),(*)(*)(

)~

,( 2211

1122

2211yxyxC

yxyx

yxyxCCB

22

222

11 )|||(|)~

,())~

,(( IyxyxCCBCCB H

General Size

CHC

Y - CA

H

Y - CA

For L=2 transmit antennas:

k=p=2

Rate R=k/p=1

12

21

2xx

xxL

Its proof is given in next slide.

Proof of rate : pkeiR .,.,1

.so,spaceldimensionain

vectorstindependenlinearlymostatare

Theret.independenlinearlyare1size

of,,...,2,1,)(vectorsthatprovesThis

.,...,2,1,00)()(

Thus.)()(

,ofityorthogonalthetoDue.ofcolumns

firsttheare)(where,)()(

:)(,ofcolumnfirst

the.matricesconstantrealreal

are,,...,2,1,where,Let

1

11

1

2

11

1

1

11

1

1

pkp

p

p

kiA

kixCC

xCC

CA

AxAC

CC

Considernp

kiAxAC

i

i

T

k

i

i

T

i

i

k

i

ii

i

k

i

ii

For 8 transmit antennas:

– k=p=8

– Rate R=k/p=1

12345678

21436587

34127856

43218765

56781234

65872143

78563412

87654321

8

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

L

• A1, A2,…, An of size and B1, B2,…, Bk of size are two

families of Hurwitz matrices if and only if the following two C are real

orthogonal designs where

and Two different representations

• There are n square Hurwitz matrices A1, A2,…, An of size by

using Clifford algebra with k=p rate=1

• There are p Hurwitz matrices B1, B2,…, Bk of size with

kp np

kk xBxBC 11

xx nAAC ,,1 T

kxx ],,[ 1 x

pp

)( pn np

)( pn

The basic problem for real orthogonal designs or real space-time

block codes for PAM signals is solved.

12

2

12

21

1

xx

xx

xx

xx

Amicable family: family of matrices of the

same size forms an

amicable family if

is a

complex orthogonal design iff

{Ai,Bi,i=1,…,k} is an Amicable design.

ts BBAA ,,,,, 11

tjsiABBAiii

tji

sji

BBBB

AAAAii

tjsiIBBAAi

i

H

jj

H

i

i

H

jj

H

i

i

H

jj

H

i

j

H

ji

H

i

1,1,)(

,1

,1

,0

,0)(

1,1,)(

k

i

iiiik xBxAxxC1

1 )Im()Re(),,( i

However the other representation

does not work!

Use a different representation

where

T

kk

nk

xxxx

AAxxC

))Im(),Re(,),Im(),(Re(where

,,,),,(

11

11

x

xx

xxxx nn BABA ,,11

T

kxx ),,( 1 x

Two Questions

Can a non-square p by n complex

orthogonal design have rate 1, i.e., k=p,

when n>2? If not, what is the bound?

How to construct rate over ½ complex

orthogonal designs?

Rate Upper Bounds for Complex Orthogonal

Designs

Liang-Xia’03 showed that their symbol rates, k/p, is strictly less than 1 for more than 2 transmit antennas.

H.Wang-Xia’03 showed that their symbol rates, k/p, can not be above ¾ when n>2, and conjectured that their symbol rates are upper bounded by

Su-Xia’03 first showed that ¾ holds for n>2 when no linear processing is allowed.

Liang’03 showed that this conjecture holds when no linear processing is allowed.

H.Wang-Xia’03showed that for a p by n generalized complex orthogonal design, the rate is upper bounded by 4/5 when n>2.

22

12

n

n

p

k

For 3 and 4

transmit antennas

Rates 4

3

p

k

• (Wang-Xia’03) The above rate upper bounds hold for a

finite QAM (excluding PSK or PAM) signal constellation.

• (Liang 2003, Su-Xia-Liu 2004, Lu-Fu-Xia 2005)

constructed complex orthogonal designs with the above rates

and the constructions by Lu-Fu-Xia 2005 have closed-

forms.

Closed Form for COD Self-Similarity

Construct COD Bn+2, Bn+1 from Bn

Construction Unites for n=2k-1

nn d is variablescomplex nonzero of number COD, np : nB

nn BB as riablescomplex va ofset same vector,1p : n

nn BB as riablescomplex va ofset same vector,1q :ˆn1,

nm,n

n,n

d v

BQQ

variablescomplex nonzero of number

COD, nq : nm,nm, 0

m,n

n,n

Qa

BQQ

s riablescomplex va ofset same

vector,1q : 0n1,-mnm,

m,n

n,n

Qa

BQQ

s riablescomplex va ofset same

ˆˆ COD, 1q :ˆ0n1,mnm,

A Theorem (Lu-Fu-Xia’05)

Orthogonality among Units

)()1()(

)()(

iBjB

jBiB

n

k

n

nn12 where

)(ˆ)(ˆ

)()(k- n

iBjB

jBiB

nn

nn

)()(

)()(

,1

,1

iBjQ

jQiB

nn

nn

)(ˆ)(ˆ

)()(

,,

,,

iQjQ

jQiQ

nmnm

nmnm

)(ˆ)(

)()(

,,1

,1,

iQjQ

jQiQ

nmnm

nmnm

)(ˆ)(

)()(

,1

,1

iBjQ

jQiB

nn

nn

)(ˆ)(

)()(ˆ

,2

,2

iBjQ

jQiB

nn

nn

)(ˆ)(

)()(ˆ

,1,1

,1,1

iQjQ

jQiQ

nmnm

nmnm

All the above matrices are COD

Orthogonal Property among Units

Bn(i) has the same structure of Bn, but the indices of nonzero complex

variables in Bn(i) are from (i-1)dn+1 to idn, where dn is the number of

nonzero complex variables in Bn.

123

*

2

*

13

*

3

*

12

*

3

*

21

0

0

0

0

xxx

xxx

xxx

xxx

B4 with d4 =3 B4( i )

1)1(32)1(33)1(3

*

2)1(3

*

1)1(33)1(3

*

3)1(3

*

1)1(32)1(3

*

3)1(3

*

2)1(31)1(3

0

0

0

0

iii

iii

iii

iii

xxx

xxx

xxx

xxx

Inductive Construction for n+1 and

n+2

)1()1()2(

)2()1(1

n

k

n

nn

nBB

BBB

)2(ˆ)3(ˆ)4(

)1()1()4()3(

)4()1()1()2(

)3()2()1(

,1

,1

,1

2

nnn

n

k

nn

nn

k

n

nnn

n

BBQ

BQB

QBB

BBB

B

)1(ˆ

)2(

)3(

)4()1( ,1

2

n

n

n

n

k

n

B

B

B

Q

B

)4(ˆ

)3(ˆ

)2(ˆ

)1()1(

ˆ

,1

2

n

n

n

n

k

n

Q

B

B

B

B

)2(ˆ)3(ˆ)4(

)1(ˆ)4()3(

)4()1(ˆ)2(

)3()2()1(

,,,1

,1,1,

,1,1,

,,,1

2,

nmnmnm

nmnmnm

nmnmnm

nmnmnm

nm

QQQ

QQQ

QQQ

QQQ

Q

)4(

)3(

)2(

)1(

,1

,

,

,1

2,

nm

nm

nm

nm

nm

Q

Q

Q

Q

Q

)4(ˆ

)3(ˆ

)2(ˆ

)1(ˆ

ˆ

,1

,

,

,1

2,

nm

nm

nm

nm

nm

Q

Q

Q

Q

Q

Orthogonality for New Units

)()1()(

)()(

iBjB

jBiB

n

k

n

nn

COD

)()1()(

)()(

22

22

iBjB

jBiB

n

k

n

nn

?

)()1()(

)()(

22

22

iBjB

jBiB

n

k

n

nn

)1(ˆ)1()6(ˆ)7(ˆ)8(

)2()1()5()1()8()7(

)3()1()8()5()1()6(

)4()7()6()5(

)5(ˆ)2(ˆ)3(ˆ)4(

)6()1()1()4()3(

)7()4()1()1()2(

)8()1()3()2()1(

1

,1

,1

1

,1

,1

,1

,1

,1

,1

n

k

nnn

n

k

n

k

nn

n

k

nn

k

n

nnnn

nnnn

nn

k

nn

nnn

k

n

n

k

nnn

BBBQ

BBQB

BQBB

QBBB

BBBQ

BBQB

BQBB

QBBB

COD

Rate Formula

nn

nmnmnmnm

nnn

nn

nmnmnmnm

nnn

dv

mvvvv

vvv

pq

mqqqq

qqq

,0

,1,,12,

,1,02,0

,0

,1,,12,

,1,02,0

0,2

3

0,2

3

)!1()!(

)!12(

)!()!1(

])1(

[)!2(

12,

12,

mkmk

kv

mkmk

k

mmkk

q

km

km

k

k

q

vR

k

k

k2

1

12,0

12,0

12

Design Examples

)2(ˆ)3(ˆ)4(

)1()1()4()3(

)4()1()1()2(

)3()2()1(

111,1

11,11

1,111

111

3

BBQ

BQB

QBB

BBB

Bk

k

11 xB

23

*

13

*

12

*

3

*

21

0

0

0

xx

xx

xx

xxx

*

11 xB 11ˆ xB 01,1 Q 01,1 Q 1,1Q̂

*

12

*

21

11

11

2)1()1()2(

)2()1(

xx

xx

BB

BBB

k

Design Examples

156

246

345

654

423

513

612

321

33

33

4

0

0

0

0

0

0

0

0

)1()1()2(

)2()1(

xxx

xxx

xxx

xxx

xxx

xxx

xxx

xxx

BB

BBB

k

COD for 4 antennas

p = 6, d = 8, R = 3/4

)2(ˆ)3(ˆ)4(

)1()1()4()3(

)4()1()1()2(

)3()2()1(

333,1

33,13

3,133

333

5

BBQ

BQB

QBB

BBB

Bk

k

COD for 5 antennas

p = 15, d = 10, R = 2/3

33 than ' COD size (half)smaller one exists thereodd, isk if 1,-2k n nn BB

)1(ˆ)2(ˆ)3(ˆ)4(

)2()1()4()3(

)3()4()1()2(

)4()3()2()1(

'

,1

,1

,1

,1

3

nnnn

nnnn

nnnn

nnnn

n

BBBQ

BBQB

BQBB

QBBB

B

332

1' nn pp

Smaller Size COD for n=4l

Smaller Size COD for n=4l

123

213

312

321

1111,1

111,11

11,111

1,1111

4

0

0

0

0

)1(ˆ)2(ˆ)3(ˆ)4(

)2()1()4()3(

)3()4()1()2(

)4()3()2()1(

'

xxx

xxx

xxx

xxx

BBBQ

BBQB

BQBB

QBBB

B

COD from our design for 4 antennas: d = 3, p = 4, R = 3/4

----- coincides with the existing one

Liang’s and Su-Xia-Liu’s: d = 6, p = 8, R = 3/4

• A design example for n=8

transmit antennas.

• In this case, d=35, p=56.

• Rate =d/p=5/8

• This construction is inductive

for all n with closed-forms

• Liang’s and Su-Xia-Liu’s:

d = 70, p = 112, R = 5/8

• These constructions do

not have closed-forms

and computer-aid or

manual help is needed

COD Construction Comparison

d p d p Rate=d/p

1 1 1 1 1 1

2 2 2 2 2 1

3 3 4 3 4 3 / 4

4 6 8 3 4 3 / 4

5 1 0 1 5 1 0 1 5 4 / 6

6 2 0 3 0 2 0 3 0 4 / 6

7 3 5 5 6 3 5 5 6 5 / 8

8 7 0 1 1 2 3 5 5 6 5 / 8

9 1 2 6 2 1 0 1 2 6 2 1 0 6 / 1 0

1 0 2 5 2 4 2 0 2 5 2 4 2 0 6 / 1 0

1 1 4 6 2 7 9 2 4 6 2 7 9 2 7 / 1 2

1 2 9 2 4 1 5 8 4 4 6 2 7 9 2 7 / 1 2

1 3 1 7 1 6 3 0 0 3 1 7 1 6 3 0 0 3 8 / 1 4

1 4 3 4 3 2 6 0 0 6 3 4 3 2 6 0 0 6 8 / 1 4

Liang &Su-Xia-Liu Lu-Fu-Xian

Conclusion

Orthogonal designs have been applied in wireless

communications

Rate upper bounds have been presented for COD

(with linear processing)

A closed-form construction for COD of rate

reaching the conjectured optimal rate has been

presented

The rate upper bound is still open

What is a smallest p, i.e., the smallest number of

rows in a p by n COD, is still open

Some Papers to Read S. M. Alamouti, A simple transmit diversity technique for wireless

communications, IEEE J. Select. Areas Commun., Oct. 1998.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block codes from

orthogonal designs, IEEE Trans. on Information Theory, July 1999.

W. Su and X.-G. Xia, On space-time block codes from complex orthogonal

designs , Wireless Personal Communications , April, 2003.

X.-B. Liang and X.-G. Xia, On the Nonexistence of Rate-One Generalized

Complex Orthogonal Designs, IEEE Trans. on Information Theory, Nov.

2003.

H. Wang and X.-G. Xia, Upper Bounds of Rates of Complex Orthogonal

Space-Time Block Codes, IEEE Trans. on Information Theory , Oct. 2003.

longer version .

X.-B. Liang, Orthogonal designs with maximal rates, IEEE Trans. on

Information Theory, Oct. 2003.

W. Su, X.-G. Xia, and K. J. R. Liu, A Systematic Design of High-Rate

Complex Orthogonal Space-Time Block Codes, IEEE Communications

Letters, June 2004.

K. Lu, S. Fu, and X.-G. Xia, Closed Form Designs of Complex Orthogonal

Space-Time Block Codes of Rates (k+1)/(2k) for 2k-1 or 2k Transmit

Antennas, IEEE Trans. on Information Theory, Dec. 2005.

Thank You!