ANOVA with more experimental designs Xuhua Xia [email protected] .
Space-Time Codes, Orthogonal Designs, and Compositions of ...xxia/ELEG812/Lecture4_6_11_2011.pdf ·...
Transcript of Space-Time Codes, Orthogonal Designs, and Compositions of ...xxia/ELEG812/Lecture4_6_11_2011.pdf ·...
Xiang-Gen Xia
University of Delaware
Newark, DE 19716
[email protected] 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
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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!