Page 1

3244

IEICE TRANS. FUNDAMENTALS, VOL.E88–A, NO.11 NOVEMBER 2005

LETTER

Design and Performance Analysis of Quasi-Orthogonal Space-Time

Block Codes for Four Antennae

Yan ZHAO†∗a), Nonmember and Chen HE†, Member

SUMMARY

orthogonal space-time block codes for four antennae is presented. Compar-

ison with the design method proposed by Jafarkhani, this method enlarges

the number of quasi-orthogonal space-time block codes. The performance

of these codes is also analyzed and the simulation results show that it is

similar to even better than that of the codes proposed by Jafarkhani.

key words:

space-time block codes, quasi-orthogonal space-time block

codes, full rate

In this letter, a novel general design method of quasi-

1. Introduction

Space-time coding is a kind of new coding and signal pro-

cessing technique in wireless communication. It uses multi-

transmit and multi-receive antennae, which can greatly im-

prove the capacity of the wireless communication system.

Comparison with space-time trellis codes, space-time block

codes have the feature of simple decoding, and many re-

searchershavedone alot ofworkinthis field. Alamouti pro-

posed a space-time code for two transmit antennae and two

symbol periods, which have a symbol transmission rate of 1

and an optimal linear processing decoder in [1]. The gener-

alization of the Alamouti scheme to more than two transmit

antennae was done by Tarokh, and space-time block codes

from orthogonal design were given in [2],[3]. Liang proved

that there exist no rate-one space-time block codes from

generalized complex linear processing orthogonal designs

for more thantwo transmitantennaeusinganycomplexcon-

stellation in [4]. Heath and Paulraj presented a space-time

code design, using the linear dispersion code framework,

for MIMO Rayleigh fading channels, which provided code

that had the same ergodic capacity performance as spatial

multiplexing but allowed for improved diversity advantage

in [5]. Jafarkhani proposed a quasi-orthogonal space-time

block code (QOSTBC) with full rate, which sacrifice the or-

thogonality and have a relatively simple decoding method

in [6]. Hou analyzed the matrices of several QOSTBC and

compared the performance of these codes in [7].

In this letter, a novel general design method of

QOSTBC for four transmit antennae is proposed. Here we

can educe a general class of QOSTBC, and some existed

Manuscript received December 27, 2004.

Manuscript revised April 18, 2005.

Final manuscript received July 15, 2005.

†The authors are with the Department of Electronic Engineer-

ing, Shanghai Jiaotong University, P.R. China.

∗Presently, with the Department of Electronic and Information

Engineering, Shanghai University of Electric Power, P.R. China.

a)E-mail: zhaoyan79@smmail.cn

DOI: 10.1093/ietfec/e88–a.11.3244

codes can be treated as the special case. Comparison with

the method proposed by Jafarkhani, this method can greatly

increase the usable codes of QOSTBC for four antennae.

The simulation results show that the BER performance of

the codes designed by this new method is similar to that of

Jafarkhani’s.

Notation: Bold upper (lower) letters denote matrices

(vectors); (.)∗, (.)Tand (.)Hdenote conjugate, transpose, and

Hermitian transpose respectively; I denotes the identity ma-

trix; det[·]denotesthedeterminantofamatrix; Re{·}denotes

the real part of a complex number.

2. The System Model of Space-Time Codes

Consideringawirelesscommunicationsystem, thetransmit-

ter has N antennae and the receiver has M antennae. At time

t, the symbol si

tenna isadjustedtothe average energy Es. Inthesametrans-

mission interval, the N symbols are transmitted simultane-

ously from N transmit antennae. Then at time t the receive

signal rj

t,t = 1,...,l,i = 1,...N at the ith transmit an-

t, at the jth receive antenna is given by

rj

t=

N

?

i=1

hi,jsi

t+ nj

t

(1)

Where hi,j,i = 1,...,N, j = 1,..., M are the fading coef-

ficients from the ith transmit antenna to the jth receive an-

tenna and nj

at time t. We assume that hi,jare independent samples of a

zero-meancomplexGaussianrandomvariablewithvariance

0.5 per dimension and nj

pendent samples of zero-mean complex Gaussian random

variable with variance N0/2 per dimension. In addition, it

is assumed that the fading coefficients are constant during a

frame and change independently from frame to frame. As-

suming perfect channel state information is available at the

receiver, and then it will compute the decision matrix (2)

over all code words s1

in favor of the code word that minimizes the sum.

tis the additive noise for the jth receive antenna

t,t = 1,...,l, j = 1,..., M are inde-

1s2

1...sN

1s1

2s2

2...sN

2...s1

ls2

l...sN

land decides

l ?

Space-Time Block Codes

t=1

M

?

j=1

|rj

t−

N

?

i=1

hi,jsi

t|2

(2)

3.

3.1Orthogonal Space-Time Block Codes

Orthogonal space-time block codes (OSTBC) denote that

Copyright c ? 2005 The Institute of Electronics, Information and Communication Engineers

Page 2

LETTER

3245

the coding matrix S satisfies the condition SHS = αI, where

α is a constant. Several OSTBCs have been designed in [2].

The often-quoted OSTBC is the code proposed by Alamouti

which meets the condition SHS = αI, where α = |s1|2+|s2|2.

?

As the orthogonality of these codes is necessary, the

usable codes are limited seriously till now.

S =

s1

−s∗

s2

s∗

21

?

(3)

3.2Quasi-Orthogonal Space-Time Block Codes

As a complex orthogonal design providing full rate for

space-time block codes is not possible for more than two

transmit antennae, some researchers have proposed a new

code, which is quasi-orthogonal and can provide full rate. A

kind of QOSTBC mentioned in [6] is based on the Alamouti

scheme mentioned above.

Define Sij =

−s∗

QOSTBC for N=4:

When vi,i = 1,...,4 denote the column vectors of S, it can

be easily found that:

?

si

sj

s∗

ji

?

, then get the following

S =

?

S12

−S∗

S34

S∗

1234

?

=

s1

−s∗

−s∗

s4

s2

s∗

−s∗

−s3

s3

−s∗

s∗

1

−s2

s4

s∗

s∗

s1

2143

342

(4)

< v1,v2>=< v1,v3>=< v2,v4>=< v3,v4>= 0,

< v1,v4>= β,< v2,v3>= −β

(5)

Where < vi,vj>=

vj. Thus

4?

α

−β

0

l=1(vi)l(vj)∗

lis the inner product of viand

SHS =

vH

1

vH

2

vH

3

vH

4

α

0

0

β

?

v1

v2

v3

v4

?

=

00

β

0

0

α

−β

α

0

(6)

Where α =

4?

i=1|si|2, β = (s1s∗

Obviously, it can be found that the columns of OS-

TBC’s coding matrix S are orthogonal to each other, while

the columns of QOSTBC’s coding matrix S are not orthog-

onal to each other. As we can see from the formulae (5) and

(6) that the first and second columns of QOSTBC’s coding

matrix S are only related to the fourth and third columns,

so we name this kind of codes quasi-orthogonal space-time

block codes.

4+ s4s∗

1) − (s2s∗

3+ s3s∗

2).

4. The General Design Method of QOSTBC

As QOSTBCs based on the Alamouti scheme are few, a

novelgeneraldesignmethodofQOSTBC is proposedinthis

section.

Assuming a system has four transmit antennae and the

user’s data is defined as s = [s1, s2, s3, s4]T. Then a design

of QOSTBC for four transmit antennae is to obtain a 4 ×

4 quasi-orthogonal matrix (defined as S) with elements of

±si,±s∗

conditions:

(1) An arbitrary symbol in s is assigned only once to a

given antenna.

(2) An arbitrary symbol in s is not transmitted by sev-

eral antennae at the same time.

i,i = 1,...,4. S needs to satisfy the following three

(3) The matrix S should satisfy SHS =

α 0 β 0

0 α 0 β

β 0 α 0

0 β 0 α

00

αβ

βα

00

, or SHS =

α

β

0

0

β

α

0

0

0

0

α

β

0

0

β

α

or SHS =

i, i = 1,...,4, we

α

0

0

β

β

0

0

α

where α,β = ±γ are constants.

Conditions (1) and(2) ensurethat the code has full rate.

When a,b,c,d represent ±sior ±s∗

can get 24 kinds of QOSTBCs that meet the condition (1)

and (2) by changing the collocation of the row vectors of

A1,A2,A3,A4in (7).

A3=

cadb

dcca

A1=

a

b

c

d

a

b

b

a

d

c

b

d

c

d

a

b

c

a

d

c

b

a

d

c

,

A2=

a

b

c

d

a

b

c

d

b

c

d

a

b

a

d

c

c

d

a

b

c

d

b

a

d

a

b

c

d

c

a

b

,

,

A4=

(7)

It can be easily found that however the content of ev-

ery element in A2,A3,A4is adjusted, these matrices cannot

satisfy condition (3). So there are only six basic forms of

QOSTBC, whichcansatisfyallthe three conditions, marked

by S1∼ S6.

S3=

cdab

badc

S1=

a

b

c

d

a

d

b

a

d

c

b

c

c

d

a

b

c

b

d

c

b

a

d

a

,

S2=

a

b

d

c

a

d

b

c

a

c

b

d

b

a

c

d

b

c

a

d

b

d

a

c

c

d

b

a

c

b

d

a

c

a

d

b

d

c

a

b

d

a

c

b

d

b

c

a

,

,

S4=

,

S5=

a

c

d

b

b

d

c

a

c

a

b

d

d

b

a

c

,

S6=

.

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3246

IEICE TRANS. FUNDAMENTALS, VOL.E88–A, NO.11 NOVEMBER 2005

Itcanbeeasilyfoundthatthe codingmatrices proposed

in [6] and [7] are all special examples of S1and S2. S3∼ S6

are the four groups of QOSTBC for four transmit antennae

designed by our new method. After getting the six basic

matrixes, you can adjust the content of every element in the

matrixes S1∼ S6to meet the condition (3). Consequently,

the number of usable coding matrices of QOSTBC is greatly

increased. The period of QOSTBC derived from S1 ∼ S6

can be divided into two steps:

1. Decide the position of conjugate in the matrix. It can

be found that two arbitrary rows or columns can be chosen

as elements’ conjugate.

2. Adjust the position of minus sign in the matrix to

meet quasi-orthogonal condition.

The following six matrices are special examples of

S1 ∼ S6named Mi,i = 1,...,6. M1is the existed coding

matrix proposed by Jafarkhani.

M3=

s3 s∗

4

s1 s∗

2

s2 −s∗

similar quasi-orthogonality to and the same decoding com-

plexity as Jafarkhani’s.

M1=

s1

−s∗

−s∗

s4 −s3 −s2 s1

s∗

s4 s3 −s2 −s1

s∗

1

The new QOSTBCs designed by the new method have

s2

s∗

s3 s4

4s∗

s∗

1

2

3−s∗

1−s∗

4

3

s∗

2

M2=

s∗

s∗

s∗

s∗

s1 s2

s∗

4

s2 −s1 s4 −s3

s∗

1

2−s1 s∗

4

s3 −s∗

3−s4 −s∗

s2

s∗

3

4−s3

2−s1

1s2

s3 s4

3−s∗

s4

s1 s∗

s4 −s∗

2

s3 s∗

3s2 −s∗

4

1

1s4 −s∗

s3 s4

4−s∗

s∗

3

M4=

s∗

2−s∗

1

3−s∗

s1 s2 s3 s4

s∗

s2 s1 s4 s3

s∗

4−s∗

1s∗

2

M5=

s1 s2

3−s∗

1s∗

2

2−s∗

4−s∗

3

M6=

3s∗

4−s∗

1−s∗

2

4s∗

3−s∗

2−s∗

1

5.Performance Analysis

When analyzing the performance of space-time codes, we

always use pairwise error probability (PEP). Assuming that

11

...

1

s

2

s

2

...

s

2

...

s

l

s

l

...

s

l

has ideal channel state information, the probability of trans-

mitting S and deciding in favor of S

approximated by

?

Where d2?

S =

s1

s1

...

s1

1

s2

s2

...

s2

1

...

...

...

...

sN

1

sN

2

...

sN

l

22

ll

is transmitted,

S

?=

s

?1

s

?2

s

?N

?1

?2

?N

...

...

...

?1

?2

?N

is received, and the receiver

?at the decoder is well

P

S → S

?/

?

hi,j

??

??

≤ exp

?

−d2(S,S

?)Es

4N0

?

?

(8)

S,S

=

M ?

j=1

l?

t=1

??????

N ?

i=1hi,j

si

t− s

?i

t

???????

2

=

M ?

variance per dimension, Apq=

p,q ≤ N.

Assuming the error matrix is

j=1ΩjAΩH

j, Ωj =

?

h1,j

···

hN,j

?

, N0/2 is the noise

l?

t=1(sp

t− s

?p

t)(sq

t− s

?q

t)∗,1 ≤

B

?

S,S

??

= S − S

?=

s1

...

sN

˜ s1

...

˜ sN

1− s

?1

1

···

...

···

s1

...

sN

l− s

?1

l

1− s

?N

1

l− s

?N

l

=

˜ s1

...

˜ sN

1

···

...

···

l

1

l

??

(9)

Then get:

A

?

S,S

??

= BH?

S,S

??

B

?

S,S

(10)

Especially, PEP of Rayleigh fading channel in [3] is

≤

i=1

Where r is the rank of A, λiare the eigenvalues of A. Thus

a diversity of rM is achieved.

The rank deficiency problem in the above codes has

been solved by a constellation rotation scheme in [8]. In

that scheme, the first two symbols in the quasi-orthogonal

design are drawn from a constellation Φ, while the other two

symbols are drawn from a rotated constellation ejωΦ. The

optimal rotation angles ω can be obtained for some com-

monly used signal constellations. In Table 1, the optimal

rotation angles ω for several widely used constellations are

given [9]. When the constellation is QPSK, the optimal ro-

tation angle is π/4.

When A is full rank, the diversity product λ can be de-

fined as:

P

?

S → S

??

≤

N

?

j=1

?

1 +

Es

4N0λi

?

−M

r ?

λi

−M ?Es

4N0

?−rM

(11)

λ = min

{S?S?}|det[A]|1/2N= min

= min

{S?S?}|det[BHB]|1/2N

?)H(S − S

{S?S?}|det[(S − S

?)]|1/2N

(12)

Where N is the number of transmit antennae. Then get that:

λ2?−NM?Es

So the measurement of PEP can be obtained by calculating

the diversity product λ. For simplicity here we only calcu-

late the diversity product of M2. The diversity products of

P

?

S → S

??

≤

?

4N0

?−NM

=

?

λ2Es

4N0

?−NM

(13)

Table 1

Optimal rotating angle of QOSTBCs.

PSK

48

π/2

π/4

π/8

QAM

Constellation

Rotation Angle

28

π/4

16

π/4

32

π/4

64

π/4

Page 4

LETTER

3247

M3∼ M6are similar to M2. The diversity products of Ja-

farkhani code (viz. M1) and M2are

λM1= min

{S?S?}|det[BH

???????????

M1BM1]|1/8

α?

0

0

α?

0

−β?

β?

0

= min

{S?S?}

det

0

−β?

α?

0

β?

0

0

α?

???????????

1/8

= min

{S?S?}|(α?2− β?2)2|1/8

(14)

Where α?=

4?

{S?S?}|det[BH

???????????

i=1|˜ si|2, β?= (˜ s1˜ s∗

4+ ˜ s4˜ s∗

1) − (˜ s2˜ s∗

3+ ˜ s3˜ s∗

2).

λM2= min

M2BM2]|1/8

= min

{S?S?}

det

α

0

−β

0

0

α

0

β

β

0

α

0

0

−β

0

α

???????????

1/8

= min

{S?S?}|(α2+ β2)2|1/8

(15)

Where α =

4?

i=1|˜ si|2, β = (˜ s1˜ s∗

Comparing the formulae (14) and (15), we can see that

M2has similar diversityproducts to Jafarkhani’s coding ma-

trix, so these new QOSTBC have similar PEP performance

to Jafarkhani’s. Because the formula (13) is only an upper

bound of PEP, the diversity products’ comparison can only

reflect the similarity of QOSTBC. So the BER formula of

QOSTBC needs more researches. And now we can only

compare the BER performance of QOSTBCs through simu-

lation.

3− ˜ s3˜ s∗

1) + (˜ s2˜ s∗

4− ˜ s4˜ s∗

2).

6.Simulation Results

In this section, we provide simulation results of our pro-

posed codes. In all simulations, we consider only one re-

ceive antenna for simplicity. The signal modulation mode is

QPSK. It is assumed that the channel is flat Rayleigh fading

channel and the receiver has perfect channel state informa-

tion.

From Fig.1, it can be found that the six QOSTBCs’

BER are nearly the same in low SNR. With the increase of

SNR, the six QOSTBCs’ BER are appreciably different be-

cause of the differences in the coding matrices. At the same

time, we can see that the BER to SNR curve of M2and M4

are similar to that of Jafarkhani’s coding matrix (M1) and

the BER performance of M3and M6are not as good as that

of M1, while the BER of M5is better than that of M1.

7. Conclusions

In this letter, a novel general design method of quasi-

orthogonal space-time block codes for four transmit anten-

nae is proposed. Comparison with the existed method, this

Fig.1

Bit error rate vs. SNR for QOSTBC.

method can greatly increase the usable codes of QOSTBC

for four transmit antennae. Simulation results show that the

BER performance of the codes designed by our new method

is similar to or better than that of existed codes.

Acknowledgement

This research is supported by Natural Science Foundation

of China (under Grant No. 60272082, No. 60372076) and

Shanghai Research Center for Wireless Technologies (under

Grant No. 03DZ15010).

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