Full-Diversity and Fast ML Decoding Properties of General Orthogonal Space-Time Block Codes for MIMO-OFDM Systems

Abstract

In this letter we apply the general orthogonal space-time block codes (OSTBC) to MIMO-OFDM systems over frequency-selective fading channels and aim to exploit the potential multipath diversity. By replacing the scalar entry of an OSTBC matrix with the vector of repeated symbols, we obtain a new OSTBC which can achieve both spatial diversity and multipath diversity. Moreover, a fast maximum likelihood (ML) decoding is admitted. Simulation results show that the proposed OSTBC, for two transmit antennas, can obtain a higher diversity gain than the Alamouti code at the same ML decoding complexity

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 20071647
Full-Diversity and Fast ML Decoding Properties of General Orthogonal
Space-Time Block Codes for MIMO-OFDM Systems
Wei Zhang, Member, IEEE, Xiang-Gen Xia, Senior Member, IEEE, and P. C. Ching, Senior Member, IEEE
Abstract—In this letter we apply the general orthogonal
space-time block codes (OSTBC) to MIMO-OFDM systems
over frequency-selective fading channels and aim to exploit the
potential multipath diversity. By replacing the scalar entry of
an OSTBC matrix with the vector of repeated symbols, we
obtain a new OSTBC which can achieve both spatial diversity
and multipath diversity. Moreover, a fast maximum likelihood
(ML) decoding is admitted. Simulation results show that the
proposed OSTBC, for two transmit antennas, can obtain a higher
diversity gain than the Alamouti code at the same ML decoding
complexity.
Index Terms—Diversity technique, multipath channels, space-
time codes, space-time-frequency codes.
I. INTRODUCTION
T
structure of orthogonality guarantees a full-diversity in flat
fading channels and a simple maximum-likelihood (ML) de-
coding algorithm. The first OSTBC design was proposed by
Alamouti in [1] for two transmit antennas and was then
extended by Tarokh et. al. in [2] for any number of transmit
antennas. A class of OSTBC from complex design with the
code rate of 1/2 was also given by Tarokh et. al. in [2].
Later, systematic constructions of complex OSTBC of rates
(k + 1)/(2k) for Mt = 2k − 1 or Mt = 2k transmit
antennas for any positive integer k were proposed in [3]–[5].
The systematic construction of complex OSTBC in [5] is a
closed form design and the coding delay is only half of the
constructions in [3], [4] when Mtis a multiple of 4.
In frequency-selective MIMO channels, the intersymbol
interference induced by multipath fading will severely degrade
the reception performance. To address this issue while ob-
taining the spatial diversity, the previously designed OSTBC
were applied with OFDM [7]. However, when Alamouti
HE orthogonal space-time block codes (OSTBC) design
has recently received recognition because the special
Manuscript received October 7, 2005; revised April 18, 2006; accepted
June 30, 2006. The associate editor coordinating the review of this letter and
approving it for publication was A. Conti. This work was supported in part by
a research grant No. CUHK 4190/03E awarded by the Hong Kong Research
Grant Council, in part by the Air Force Office of Scientific Research (AFOSR)
under Grant No. FA9550-05-1-0161, and by the National Science Foundation
under Grants CCR-0097240 and CCR-0325180. The material in this letter
was presented in part at the IEEE International Conference on Acoustics,
Speech and Signal Processing, Toulouse, France, May 2006.
W. Zhang was with the Department of Electronic Engineering, The
Chinese University of Hong Kong, Shatin, N. T., Hong Kong. He is now
with the Department of Electronic and Computer Engineering, The Hong
Kong University of Science and Technology, Kowloon, Hong Kong (e-mail:
eewzhang@ust.hk).
X.-G. Xia is with the Department of Electrical and Computer Engineering,
University of Delaware, Newark, DE 19716, USA (e-mail: xxia@ee.udel.edu).
P. C. Ching is with the Department of Electronic Engineering, The
Chinese University of Hong Kong, Shatin, N. T., Hong Kong (e-mail:
pcching@ee.cuhk.edu.hk).
Digital Object Identifier 10.1109/TWC.2007.05795.
code was applied to MIMO-OFDM systems in multipath
fading channels, the multipath diversity may not be exploited.
Previous works show that the maximum achievable diversity
gain over MIMO multipath fading channels is MtMrL, where
Mt is the number of transmit antennas, Mr is the number
of receive antennas and L is the number of taps of the
channel impulse response [8], [9]. To exploit the potential
multipath diversity, a variety of coding schemes have been
proposed, such as space-time trellis coded (STTC) OFDM
in [9]–[12], space-frequency coding (SFC) in [13]–[17], and
space-time-frequency coding (STFC) in [18]–[23]. However,
most of the existing coding methods have to sacrifice low
decoding complexity in order to achieve full diversity. To
achieve the multipath diversity, a sufficiently large effective
length of STTC is necessary thereby leading to an exponential
increase of the STTC complexity. The full-diversity SFC and
STFC mostly rely on some algebraic rotation matrix and do
not have orthogonal structures. Even though some near-ML
decoder, such as sphere decoder [24] can be used, the expected
decoding complexity is still dominated by polynomial terms
of a number of symbols which are jointly detected [25]. A
low-complexity full-diversity STFC was constructed in [20],
but it can only achieve the low decoding complexity over the
paired symbols in the case of L = 2.
In this letter, the general OSTBC are applied to MIMO-
OFDM systems with a special structure of stacking and
repeating. The full-diversity and fast ML decoding properties
of the general OSTBC in flat-fading channels are found still
valid in frequency-selective fading channels and are proved
to hold for any number of transmit antennas and any number
of channel taps. Compared to the SFC design via OSTBC
mapping [14], the OSTBC proposed in this letter occupy sev-
eral OFDM blocks and admit the single-symbol ML decoding.
Simulation results show that the proposed OSTBC can achieve
the multipath diversity, whereas the Alamouti code when
applied to MIMO-OFDM in multipath fading channel cannot
achieve the multipath diversity. For two transmit antennas, the
proposed OSTBC can give a comparable bit-error-rate (BER)
performance with other existing full-diversity STC and SFC at
the same bandwidth efficiency but has a simple ML decoding.
This letter is organized as follows. In Section II, the signal
model and the code design criteria are given. In Section III, a
design of full-diversity OSTBC is proposed for MIMO-OFDM
systems over frequency-selective fading channels. The fast
ML decoding and full-diversity properties are then proved.
In Section IV, simulation results are presented and discussed.
Finally, in Section V, the conclusion is drawn. Throughout
this letter, the following notations will be used. Superscripts
T,
∗andHstand for transpose, conjugate and Hermitian,
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1648IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007
respectively. E[x] represents the expectation of variable x.
Re(x) denotes the real part of x. ||C||Fis the Frobenius norm
of C. ?x? represents the smallest integer larger than x. IN
represents the identity matrix of size N × N. The Kronecker
product is denoted by ⊗. C stands for the complex number
field. j =√−1.
II. SPACE-TIME-FREQUENCY CODED MIMO-OFDM
In this section, the signal model of STFC for MIMO-OFDM
systems is given. Then, the code design criteria are shown.
A. Signal Model
Consider a MIMO-OFDM system with Mttransmit anten-
nas and Mr receive antennas has N OFDM tones (subcar-
riers). A block of Ns information symbols, which are from
discrete alphabet A such as PSK or QAM, are encoded into
a codeword C of size NT × Mt. The codeword C can be
expressed as
⎛
⎝
where T denotes the total number of OFDM blocks involved
in one codeword C, Cu∈ CN×Mt(u = 1,··· ,T) denotes the
signals to be transmitted during the uth OFDM block. The jth
column cu
jth transmit antenna during the uth OFDM block. The code
rate of the STFC is defined as
C =
⎜
C1
...
CT
⎞
⎟
⎠=
⎛
⎝
⎜
c1
...
cT
1
···
...
···
c1
Mt
...
cT
Mt
1
⎞
⎠,
⎟
(1)
jof the matrix Cuwill be fed to the IFFT at the
RSTFC?
Ns
NT.
(2)
When T = 1 in (2), the STFC can be seen as the SFC of size
N ×Mtwith the rate of RSFC= Ns/N. When N = 1 in (2),
the STFC is in fact the STC of size T × Mt with the code
rate RSTC= Ns/T. It has been found that the code rate of
the OSTBC is not more than 3/4 for more than two transmit
antennas [6].
Assume that the cyclic prefix is larger than the maximum
delay of the multipath fading channels and that the perfect
OFDM timing and frequency synchronization are available
at the receivers. We can write the received signals in the
frequency domain at the ith receive antenna during the uth
OFDM block as
?ρ
j=1
where diag(·) denotes a diagonal matrix by putting the vector
on the main diagonal of the matrix, Hi,j∈ CN×1denotes the
frequency response of the channel between the jth transmit
antenna and the ith receive antenna, nu
noise and its kth entry nu
Gaussian with zero mean and unit variance and assumed to
be statistically independent with respect to k, i and u. The
normalization factor ρ ensures that ρ is the signal-to-noise
ratio (SNR) at each receive antenna, independently of Mt.
With some matrix permutations, (3) can be rewritten as [23]
?ρ
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Yu
i=
Mt
Mt
?
diag(cu
j)Hi,j+ nu
i,
(3)
i∈ CN×1denotes
i(k) is a circular symmetric, complex
Yu
i=
MtXuhu
i+ nu
i,
(4)
for u = 1,··· ,T and i = 1,··· ,Mr, where
Xu
=
D0Cu
?
with
Dl= diag?
and
˜hu
hu
?
···
···
DL−1Cu?,
(˜hu
(5)
hu
i
=
(˜hu
i,0)T
i,L−1)T
?T
,
(6)
1
e−j2πτl/Ts
···
e−j2π(N−1)τl/Ts?
i,Mt(l)
i,l=?
i,1(l)
···
hu
?T,
for l = 0,··· ,L − 1. The entry hu
neous channel coefficient of the lth tap of the channel between
the jth transmit antenna and the ith receive antenna during the
uth OFDM block. The identical channel power-delay profile is
assumed for all MIMO channels, i.e., E[hu
for any l,i,j and u. τl is the delay of the lth path and Ts
represents the effective duration of one OFDM block. The
MIMO channels are further assumed to be quasi-static during
at least T OFDM blocks, then it has h1
we focus on the received signals in T OFDM blocks only,
without loss of generality the superscript u of hu
omitted for notational brevity.
Stacking all the received signals in (4) into a vector, we
obtain
Y =
MtXh + n
where
?
h
=
hT
1
···
n
=
(n1
?
G =?
B. Code Design Criteria
Let C andˆC be two different STFC of size NT × Mt
related to X andˆX. The design criteria of STFC over quasi-
static frequency-selective channels are summarized as [23]:
• Diversity (rank) criterion: The minimum rank r of (X−
ˆX)Λ(X−ˆX)Hover all pairs of different codeword C and
ˆC should be as large as possible, where Λ = E[hhH];
• Product (determinant) criterion: The minimum value of
the product?r
the nonzero eigenvalues of (X −ˆX)Λ(X −ˆX)H.
Let Υ = diag( δ2
0
δ2
1
···
From (6) and (9), we get Λ = IMr⊗ Q. Using the property
of rank(AAH) = rank(A), we obtain
?
≤
In order to achieve the maximum diversity gain MtMrL
over frequency-selectivefading channels, STC, SFC and STFC
i,j(l) denotes the instanta-
i,j(l)(hu
i,j(l))∗] = δ2
l
i= ··· = hT
i. Because
ican be
?ρ
Y
X
=
=
(Y1
1)T···(YT
IMr⊗ G
?
1)T···(Y1
Mr)T···(YT
Mr)T?T(7)
(8)
hT
Mr
1)T···(n1
XT
T
?T
(9)
?
1)T···(nT
XT
1
Mr)T···(nT
?T. From (1) and (5), we can
(IT⊗ DL−1)C
Mr)T?T
(10)
with G =
rewrite G ∈ CTN×MtLas:
(IT⊗ D0)C
···
···
?.
(11)
i=1λiover all pairs of different codeword
C andˆC should be maximized, where λ1,··· ,λr are
δ2
L−1) and Q = Υ ⊗ IMt.
r
=
Mr· min
MtMrL.
rank
?
(G −ˆG)Q
1
2
??
(12)

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 20071649
have been proposed in MIMO-OFDM systems but most of
them have a high ML decoding complexity [9]–[23]. Recently,
a low-complexity full-diversity STFC has been proposed [20],
but the single complex symbol decoding exists only in the case
of L = 2. In the following, a new OSTBC is constructed which
can achieve the full-diversity in multipath fading channels
while admitting a single-symbol ML decoding for any value
of L.
III. AN OSTBC FOR ACHIEVING FULL DIVERSITY AND
FAST ML DECODING
In this section, we present a new design of the OSTBC
for MIMO-OFDM systems and show its properties of full-
diversity and fast ML decoding.
A. Code Structure
Before presenting our code structure, let us have a brief
review of the general OSTBC developed in [2]. Let¯GMtbe
a Q × Mt(Q ≥ Mt) matrix with entries of complex linear
combinations of elements from the set
{±s1,±s∗
1,±s2,±s∗
2,··· ,±sNu,±s∗
Nu}.
It is a general OSTBC design if the following orthogonality
holds for any complex values of sj?,(j?= 1,2,··· ,Nu):
¯GH
For Mttransmit antennas, each column of the codeword¯GMt
is transmitted through different transmit antennas in Q time
slots. The rate of the general OSTBC¯GMtis defined as
ROSTBC? Nu/Q. In [5], the rate for closed form designs of
complex general OSTBC is given by
ROSTBC= (?Mt/2? + 1)/(2 · ?Mt/2?).
Based on the general OSTBC design, for example, in
[3]–[5], we develop an OSTBC tailored for MIMO-OFDM
systems with Mttransmit antennas. The code structure is of
the following form:
C =√γGMt⊗ 1Γ×1,
where the scalar√γ ensures that the OSTBC obeys the
energy constraint, i.e., ||C||2
length-Γ column vector with all ones. The matrix GMtof
size
¯GMtbut an information symbol, sj?, in ¯GMtis replaced
by a vector of information symbols, Sj?, where Sj?
[ sj?(1)
···
the discrete alphabet such as PSK or QAM constellations
normalized into the unity power and Γ = 2?log2L??for any
1 ≤ L?≤ L. Clearly, Γ ≥ L?and
Comparing (13) with (1), we get T = Q.
The orthogonality of (13) can be verified by
CHC
=
γ(GMt⊗ 1Γ×1)H(GMt⊗ 1Γ×1)
=
γΓ(GH
⎛
j?=1
⎛
j?=1
Mt¯GMt= (|s1|2+ |s2|2+ ··· + |sNu|2)IMt.
(13)
F = QNMt, and 1Γ×1 is a
NQ
Γ× Mthas the same structure as the OSTBC matrix
=
sj?(N
Γ) ]T∈ A
N
Γ×1,j?= 1,··· ,Nu. A is
N
Γis a positive integer.
MtGMt)
Nu
?
Nu
?
=
γΓ
⎝
⎝
SH
j?Sj?
⎞
⎠IMt
|sj?(n)|2
=
γΓ
N/Γ
?
n=1
⎞
⎠IMt.
(14)
TABLE I
PROPOSED OSTBC TRANSMISSION FOR TWO TRANSMIT ANTENNAS
(C1= S1⊗ 1Γ, C2= S2⊗ 1Γ)
Time-Frequency \ Space
OFDM block 1
OFDM block 2
Antenna 1
C1
−C∗
Antenna 2
C2
C∗
21
TABLE II
FULL-DIVERSITY SFC [14] TRANSMISSION FOR TWO TRANSMIT
ANTENNAS (Mt= 2,L = 2,N = 4)
Frequency \ SpaceAntenna 1 Antenna 2
OFDM block 1
¯G2
...
¯G2
?
?
?
?
?
?
?
?
?
L
The important properties of the proposed OSTBC are shown
in the following theorem.
Theorem 1: Suppose that a MIMO-OFDM system with Mt
transmit antennas and Mr receive antennas has N OFDM
tones and that the MIMO channels are spatially uncorrelated
and each channel experiences the frequency-selective fading
characterized by L independent taps of the channel impulse
response, in which the maximum path delay is less than the
length of the cyclic prefix of OFDM and the path gains are
constant over Q OFDM blocks. For any 1 ≤ L?≤ L, the
OSTBC as described in (13) can
1) achieve a code rate as
where Γ = 2?log2L??;
2) have a fast ML decoding algorithm with single-symbol
decoding;
3) achieve a diversity gain of MtMrL?. If L?= L, then
the full-diversity will be achieved.
Proof of Theorem 1.-1)
Using (1), we can observe that the total number of informa-
tion symbols embedded in one codeword of (13) transmitted
through N subcarriers in Mt transmit antennas during Q
OFDM blocks is Ns = NuN/Γ. Thus the code rate of the
proposed STBC in (13) is
NQ=1
Γ
Q
The Proof of Theorem 1.-2) and the Proof of Theorem
1.-3) will be elaborated in Section III-B and Section III-C,
respectively.
An example of the proposed OSTBC for two transmit
antennas is given by the following 2N × 2 matrix:
C =
−S∗
where S∗
vector S1 and S2, respectively. The transmission scheme is
shown in Table I. It can be observed that the Alamouti coded
OFDM in [7] is a special case of (16) when Γ = 1. It
should be mentioned that the idea of repeating was previously
applied in the design of SFC in [14]. For comparison, the SFC
transmission scheme of [14] is shown in Table II. Because
the SFC in [14] is constructed by mapping the Alamouti
1
Γof that of general OSTBC,
R =
Ns
Nu
=1
ΓROSTBC.
(15)
?
S1
S2
S∗
21
?
⊗ 1Γ×1,
(16)
1and S∗
2represent the conjugate of all elements of
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1650IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007
codes along subcarriers during one OFDM block only, it
cannot admit the single-symbol ML decoding. However, for
our proposed OSTBC, the single-symbol ML decoding can be
achieved and will be shown later.
B. Fast ML Decoding
Combining (13) with (3), we can write the received signals
at the ith receive antenna during T (T = Q) OFDM blocks
as
⎡
⎣
where Φ ∈ CNQ×NMtis of the same form as the general
OSTBC¯G ∈ CQ×Mtbut with the N ×N block matrix entries
Λj? =√γdiag(Sj? ⊗ 1Γ×1)
instead of the scalar entry sj? in¯G, for j?= 1,··· ,Nu. Let
H =?
With the notations (7) and (10), we can obtain
?ρ
where Φ?= IMr⊗ Φ.
Using
⎢
Y1
...
YT
i
i
⎤
⎥
⎦ =
?ρ
MtΦ
⎡
⎣
⎢
Hi,1
...
Hi,Mt
⎤
⎥
⎦+
⎡
⎣
⎢
n1
...
nT
i
i
⎤
⎦
⎥
(17)
HT
1,1···HT
1,Mt···HT
Mr,1···HT
Mr,Mt
?T.
Y =
MtΦ?H + n,
(18)
(Φ?)HΦ?
=
IMrMt⊗ (
Nu
?
j?=1
|Λj?|2)
⎛
j?=1
=
γIMrMt⊗ diag
⎝
Nu
?
|Sj?|2⊗ 1Γ×1
⎞
⎠(19)
the ML decoding at the receive can be simplified as
????Y −
= argmin
S
ρ
MtHH(Φ?)HΦ?H
ˆS
=argmin
S
?ρ
MtΦ?H
?ρ
????
????
2
F
????YHY − 2
⎧
⎩
MtRe?YHΦ?H?
+
=argmin
sj?(n)∈S
⎨
Nu
?
j?=1
N/Γ
?
n=1
fj?,n(sj?(n))
⎫
⎬
⎭,
(20)
where S is the vector with entries sj?(n) for all j?and n and
fj?,nis the quadratic form of sj?(n). It can be seen that the
ML detection can be done separately from each sj?(n). This
leads to the fast ML decoding.
If the constant modulus modulation with unit-energy is
used, such as PSK, (19) will be equal to (Φ?)HΦ?
γNuIMrMtN. The ML detection of PSK should be equivalent
to
ˆS = argmax
S
=
Re?YHΦ?H?,
which is a linear processing.
C. Full-Diversity
From (12) it is known that the achievable diversity gain
of the proposed OSTBC depends on the minimum rank of
(G−ˆG)Q
G (andˆG accordingly) is defined in (11).
For any˘C = C−ˆC ?= 0, there exists at least one nonzero
˘ sj?(n0) = sj?(n0) − ˆ sj?(n0) for 1 ≤ n0≤N
Nu. Let σ(n0) denote the smallest row index of sj?(n0) in C
and
bm,u= (u − 1)N + σ(n0) + (m − 1)ν,
m = 1,··· ,Γ; u = 1,··· ,T,
where ν is the separation factor representing the distance
between any two repeated symbols during one OFDM block.
If the symbols are repeated on neighboring subcarriers then
ν = 1. For the proposed OSTBC in (13), it has ν = 1 and
σ(n0) = (n0− 1)Γ + 1, for 1 ≤ n0≤N
Furthermore we define
I =?
and
G?=?
for 1 ≤ L?≤ L and Γ = 2?log2L??. It is easily seen that G?
is in fact the first MtL?columns of the matrix G in (11).
Then we construct a matrix B ∈ CTΓ×MtL?by extracting the
I(p)th row of G?∈ CTN×MtL?as the pth row of B for all p
and 1 ≤ p ≤ TΓ. In fact, B is a sub-matrix of G?after some
rows selection. Consequently, we can see the relationship as:
1
2 for any different C andˆC given by (13), where
Γand 1 ≤ j?≤
(21)
Γ.
b1,1
···
b1,T
···
bΓ,1
···
bΓ,T
?.
?,
(22)
(IT⊗ D0)C
···
(IT⊗ DL?−1)C
(23)
G
CTN×MtL
Moreover, the matrix˘B = B−ˆB is given by [Appendix I]
˘B =√γ(V ⊗ IT)?IL? ⊗¯G?,
˘ s1(n0), ˘ s2(n0),··· , ˘ sNu(n0) and the Γ×L?(Γ ≥ L?) matrix
V is shown in (31).
For the OSTBC proposed in (13), it has bm,1−b1,1= m−1,
for m = 1,··· ,Γ. Then we can write the first L?rows of V
as
˜V = V1V2,
−→
G?
CTN×MtL?
−→
B
CTΓ×MtL?
(24)
where¯G is the OSTBC matrix of size T ×Mtwith variables
(25)
where the diagonal matrix V2is
V2= diag?
and the Vandermonde matrix V1is
⎛
⎜
ξ(L?−1)τ0
Obviously det(V2) ?= 0 and
det(V1) =
ξ(b1,1−1)τ0
···
ξ(b1,1−1)τL?−1?
1
ξτL?−1
...
ξ(L?−1)τL?−1
(26)
V1=
⎜
⎜
⎝
1
ξτ0
...
1
ξτ1
...
ξ(L?−1)τ1
···
···
...
···
⎞
⎟
⎟
⎟
⎠. (27)
?
0≤q<p≤L?−1
(ξτp− ξτq).
(28)
Since τL−1≥ τL?−1≥ τp> τq≥ τ0for L?− 1 ≥ p > q ≥ 0
and τL−1< Ts, it has ξτp?= ξτqand det(V1) ?= 0. Therefore,
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 20071651
TABLE III
PERFORMANCE COMPARISON AMONG VARIOUS CODING SCHEMES FOR
2 × 1 OFDM SYSTEM IN 2-RAY FADING CHANNEL AT 1 BIT/S/HZ.
SimulatedSymbolModulation
CodesRate
STC [7]1BPSK
STC [14]
0.5
QPSK
SFC [15]1BPSK
OSTBC (13)
0.5
QPSK
Diversity
Gain
2
4
4
4
No. of symbols
in joint decoding
1
2
4
1
the square matrix˜V is of full rank. Then we can obtain that
rank(V) = rank(˜V) = L?.
Let˜Q be the diagonal matrix of size MtL?× MtL?with
˜Q =˜ Υ⊗IMtand˜Υ = diag( δ2
we have
0
···
δ2
L?−1). Using (24),
˜Q
H
2 ˘BH˘B˜Q
1
2= γ(˜Υ
H
2VHV˜ Υ
1
2) ⊗ (¯GH¯G).
(29)
The rank is given by
rank
?˜Q
H
2 ˘BH˘B˜Q
1
2
?
= rank
?˜Υ
H
2VHV˜Υ
1
2
?
· rank?¯GH¯G?
=
L?Mt,
(30)
for 1 ≤ L?≤ L.
Because T ≥ Mtand Γ ≥ L?, from (30) we can see that
(B −ˆB)˜Q
B −ˆB is a sub-matrix of G?−ˆG?with some selected rows,
we know that (G?−ˆG?)˜Q
rank. As seen from (23), the matrix G?is a sub-matrix of G
with the first MtL?columns of G. Then the minimum rank of
(G −ˆG)Q
obtain that the proposed OSTBC achieves the diversity gain
MtMrL?over frequency-selective fading channels. If L?= L,
the OSTBC achieves the full-diversity MtMrL.
1
2 ∈ CTΓ×MtL?is of full rank. Considering that
1
2 ∈ CTN×MtL?is also of full
1
2 ∈ CTN×MtLis MtL?. Therefore, from (12) we
IV. SIMULATION RESULTS
To investigate the performance of the proposed OSTBC
over frequency-selective fading channels, we perform the
simulation experiments and compare with other existing STC
and SFC for MIMO-OFDM systems. In the simulation, we
use a 2 × 1 system with 64 OFDM tones. The bandwidth of
OFDM system is 20 MHz and the length of the cyclic prefix
is 16, i.e., 0.8 μs. Hence the duration of one OFDM symbol
(cyclic prefix excluded) is Ts = 3.2 μs. A two-ray channel
model is considered with the equal gain and the second path
delay as 0.5 μs.
Four coding schemes combined with OFDM are simulated.
They are STC without repeating, i.e., Alamouti coded OFDM
[7], STC via repeating [14], SFC via algebraic rotation [15]
and the proposed OSTBC shown in (13) with L?= L = 2. To
fix the bandwidth efficiency at 1 bit/s/Hz, different modulation
schemes are used as shown in Table III. When the channel
power-delay profile is known at transmitter, the subcarriers
permutation can be applied to optimize the diversity product
[15]. The proposed OSTBC for QPSK modulation is simulated
with two schemes: without permutation (ν = 1) and with
the optimum permutation (νopt = 16), where ν denotes the
subcarrier spacing between any two repeated symbols in one
2468 1012 1416 1820
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
SNR (dB)
BER
BPSK, Alamouti−OFDM
QPSK, STC, repeat 2 times
BPSK, SFC, without permutation (ν=1)
BPSK, SFC, with permutation (νopt=16)
QPSK, proposed STBC, without permutation, (ν=1)
QPSK, proposed STBC, with permuation, (νopt=16)
Fig. 1.
STC via repeating [14], space-frequency coding (SFC) via algebraic rotation
[15], and the proposed OSTBC for 2 × 1 OFDM system in 2-ray channel
model (N = 64, BW=20 MHz, τ = [0,0.5]μs) at 1 bit/s/Hz.
Performance comparison among the Alamouti coded OFDM [7],
OFDM block and νoptis calculated by optimizing the diversity
product [15, Eq. (4.22)]. The same permutations are also
applied for the SFC [15] but BPSK is used to keep at 1
bit/s/Hz.
From Fig. 1 it can be seen that the STC without repeating
[7] marked by “∗” gives the worst BER performance among
all the simulated coding schemes. This is due to the fact
that the space diversity gain 2 is achieved only. To achieve
the maximum diversity gain MtL = 4 in the simulated
system, the STC was repeated twice in [14]. It can also be
observed that the STC via mapping [14], the SFC (ν = 1)
[15], and the proposed OSTBC (ν = 1) can achieve the
comparable performance with a deeper slope than the STC
without repeating. With optimum permutation (νopt = 16),
the BER performance can be further improved for the SFC
and the proposed OSTBC, and the improvement is more than
2 dB in SNR at the BER of 10−4. Furthermore, the proposed
OSTBC has a slight BER advantage over the SFC, not more
than 1 dB. The advantage is due to the better intrinsic diversity
product given by the proposed OSTBC where the signal space
is not rotated compared with the SFC in [15].
The ML decoding complexity can be evaluated by the
number of symbols involved in the joint detection. In Table III,
it is shown that only the Alamouti coded OFDM in [7]
and our proposed OSTBC have single-symbol ML decoding
complexity. Moreover, the proposed OSTBC can achieve the
maximum multipath diversity gain but has a loss of full symbol
rate. The Alamouti coded OFDM in [7] can achieve the full-
rate but cannot exploit the multipath diversity. To further
investigate the BER performance comparison between the two
coding schemes with low decoding complexity, we perform
simulations under a 6-ray COST207 channel model (L = 6)
[26], where the maximum path delay is 5μs. In the simulation,
we use N = 128 and the 5 MHz bandwidth. To keep at
1 bit/s/Hz bandwidth efficiency, BPSK is used for Alamouti
coded OFDM [7] and QPSK is for our proposed OSTBC in
(13) with L?= 2. The simulation results are illustrated in
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