IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005 1847
Towards Maximum Achievable Diversity
in Space, Time, and Frequency:
Performance Analysis and Code Design
Weifeng Su, Member, IEEE, Zoltan Safar, Member, IEEE,andK.J.RayLiu,Fellow, IEEE
Abstract—Multiple input multiple output (MIMO) communi-
cation systems with orthogonal frequency division multiplexing
(OFDM) modulation have a great potential to play an impor-
tant role in the design of the next-generation broadband wireless
communication systems. In this paper, we address the problem of
performance analysis and code design for MIMO-OFDM systems
when coding is applied over both spatial, temporal, and frequency
domains. First, we provide an analytical framework for the per-
formance analysis of MIMO-OFDM systems assuming arbitrary
power delay profiles. Our general framework incorporates the
space–time and space–frequency (SF) coding approaches as spe-
cial cases. We also determine the maximum achievable diversity
order, which is found to be the product of the number of transmit
and receive antennas, the number of delay paths, and the rank
of the temporal correlation matrix. Then, we propose two code
design methods that are guaranteed to achieve the maximum
diversity order. The first method is a repetition coding approach
using full-diversity SF codes, and the second method is a block
coding approach that can guarantee both full symbol rate and
full diversity. Simulation results are also presented to support the
theoretical analysis.
Index Terms—Broadband wireless communications, maximum
achievable diversity, MIMO-OFDM systems, multiple antennas,
space–frequency coding, space–time–frequency coding.
I. INTRODUCTION
M
ULTIPLE INPUT MULTIPLE OUTPUT (MIMO) com-
munication systems have a great potential to play an
important role in the design of the next-generation wireless
communication systems due to the advantages that such sys-
tems can offer. By employing multiple transmit and receive
antennas, the adverse effects of the wireless propagation en-
vironment can be significantly reduced. In case of narrow-
band wireless communications, where the fading channel is
frequency nonselective, many modulation and coding methods
Manuscript received August 18, 2003; revised January 29, 2004; accepted
May 9, 2004. The editor coordinating the review of this paper and approving it
for publication is Y.-C. Liang. This work was supported in part by U.S. Army
Research Laboratory under Cooperative Agreement DAAD 190120011.
W. Su is with the Department of Electrical Engineering at the State Univer-
sity of New York (SUNY) at Buffalo, Buffalo, NY 14260 USA (e-mail:
weifeng@eng.buffalo.edu).
Z. Safar is with Modem System Design, Technology Platforms, Nokia
Danmark A/S, Copenhagen, Denmark (e-mail: zoltan.2.safar@nokia.com).
K. J. Ray Liu is with the Department of Electrical and Computer Engi-
neering and Institute for Systems Research University of Maryland, College
Park, MD 20742 USA (e-mail: kjrliu@eng.umd.edu).
Digital Object Identifier 10.1109/TWC.2005.850323
[1]–[9], termed as space–time (ST) codes, have been proposed
to exploit the spatial and temporal diversities available in the
multiantenna channel.
In case of broadband wireless communications, where the
fading channel is frequency selective, orthogonal frequency
division multiplexing (OFDM) modulation can be used to
transform the frequency-selective channel into a set of parallel
frequency flat channels, providing high spectral efficiency and
eliminating the need for high-complexity equalization algo-
rithms. To take advantage of both MIMO systems and OFDM
modulation, MIMO-OFDM systems have been proposed, re-
sulting in two major channel coding approaches for these
systems. The first approach is space–frequency (SF) coding,
where coding is applied within a single OFDM block to ex-
ploit the spatial and frequency diversities. The other approach
is space–time–frequency (STF) coding, where the coding is
applied across multiple OFDM blocks to exploit the spatial,
temporal, and frequency diversities available in frequency-
selective MIMO channels.
Early works on SF coding [10]–[15] used ST codes directly
as SF codes, i.e., previously existing ST codes were used by
replacing the time domain with the frequency domain (OFDM
tones). The performance criteria for SF-coded MIMO-OFDM
systems were derived in [15] and [16], and the maximum
achievable diversity was found to be LM
t
M
r
, where M
t
and
M
r
are the number of transmit and receive antennas, respec-
tively, and L is the number of delay paths in the channel impulse
response. Bölcskei and Paulraj [16] showed that, in general,
systems using ST codes directly as SF codes can achieve only
spatial diversity and are not guaranteed to achieve the full
spatial and frequency diversity LM
t
M
r
. Later, in [17] and [18],
systematic SF code design methods that could guarantee to
achieve the maximum diversity were proposed.
To further improve the performance, one may consider STF
coding across multiple OFDM blocks to exploit all of the avail-
able diversities in the spatial, temporal, and frequency domains.
The STF coding strategy was first proposed in [19] for two
transmit antennas and further developed in [20], [21], and [22]
for multiple transmit antennas. Both [19] and [22] assumed that
the MIMO channel stays constant over multiple OFDM blocks,
and we will show later that in this case, STF coding cannot
provide any additional diversity compared to the SF coding
approach. In [21], an intuitive explanation on the equivalence
between antennas and OFDM tones was presented from
the viewpoint of channel capacity. In [20], the performance
1536-1276/$20.00 © 2005 IEEE
1848 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
Fig. 1. STF-coded MIMO-OFDM system with M
t
transmit and M
r
receive antennas.
criteria for STF codes were derived, and an upper bound on the
maximum achievable diversity order was established. However,
there was no discussion in [20] whether the upper bound can
be achieved or not, and the proposed STF codes were not
guaranteed to achieve the full spatial, temporal, and frequency
diversities.
In this paper, we consider the problem of performance
analysis and full-diversity STF code design for MIMO-OFDM
systems. We provide a general framework, taking into account
coding over the spatial, temporal, and frequency domains.
Our model incorporates the ST and SF coding approaches
as special cases. First, we derive the performance criteria for
STF-coded MIMO-OFDM systems, based on the results of
[18], [23], and [24], and we show that the maximum achievable
diversity order is LM
t
M
r
T , where T is the rank of the
temporal correlation matrix of the channel. Then, we propose
two STF code design methods that are guaranteed to achieve
the maximum achievable diversity order. The first method is
a repetition coding approach, which achieves full diversity
at the price of symbol rate decrease. The advantage of this
approach is that any full-diversity SF code (block or trellis)
can be used to design full-diversity STF codes. The other STF
code design method, a block coding approach, provides both
data rate (full symbol rate) and performance (full diversity).
In this case, the STF codes are constructed using existing
results on signal constellation design for single-antenna fading
channels.
The paper is organized as follows. In Section II, we describe
the MIMO-OFDM system model with an arbitrary power delay
profile. In Section III, we derive the STF code performance
criteria and determine the maximum achievable diversity order.
In Section IV, two STF code design methods are proposed.
The simulation results are presented in Section V, and some
conclusions are drawn in Section VI.
II. S
YSTEM MODEL
We consider an STF-coded MIMO-OFDM system with M
t
transmit antennas, M
r
receive antennas, and N subcarriers, as
shown in Fig. 1. Suppose that the frequency-selective fading
channels between each pair of transmit and receive antennas
have L independent delay paths and the same power delay
profile. The MIMO channel is assumed to be constant over
each OFDM block period, but it may vary from one OFDM
block to another. At the kth OFDM block, the channel im-
pulse response from transmit antenna i to receive antenna j at
time τ can be modeled as
h
k
i,j
(τ)=
L−1
l=0
α
k
i,j
(l)δ(τ − τ
l
) (1)
where τ
l
is the delay and α
k
i,j
(l) is the complex amplitude of the
lth path between transmit antenna i and receive antenna j.The
α
k
i,j
(l)’s are modeled as zero-mean complex Gaussian random
variables with variances E|α
k
i,j
(l)|
2
= δ
2
l
, where E stands for
the expectation. The powers of the L paths are normalized
such that
L−1
l=0
δ
2
l
=1. We assume that the MIMO channel is
spatially uncorrelated, so the channel coefficients α
k
i,j
(l)’s are
independent for different indices (i, j). From (1), the frequency
response of the channel is given by
H
k
i,j
( f)=
L−1
l=0
α
k
i,j
(l)e
−j2πfτ
l
(2)
where j =
√
−1.
We consider STF coding across M
t
transmit antennas, N
OFDM subcarriers, and K consecutive OFDM blocks. Each
STF codeword can be expressed as a KN ×M
t
matrix
C =[C
T
1
C
T
2
··· C
T
K
]
T
(3)
where the channel symbol matrix C
k
is given by
C
k
=
c
k
1
(0) c
k
2
(0) ··· c
k
M
t
(0)
c
k
1
(1) c
k
2
(1) ··· c
k
M
t
(1)
.
.
.
.
.
.
.
.
.
.
.
.
c
k
1
(N − 1) c
k
2
(N − 1) ··· c
k
M
t
(N − 1)
(4)
and c
k
i
(n) is the channel symbol transmitted over the nth
subcarrier by transmit antenna i in the kth OFDM block.
The STF code is assumed to satisfy the energy constraint
EC
2
F
= KNM
t
, where C
F
is the Frobenius norm of C.
During the kth OFDM block period, the transmitter applies an
N-point IFFT to each column of the matrix C
k
. After append-
ing a cyclic prefix, the OFDM symbol corresponding to the
ith (i =1, 2,...,M
t
) column of C
k
is transmitted by transmit
antenna i.
SU et al.: DIVERSITY IN SPACE, TIME, AND FREQUENCY: PERFORMANCE ANALYSIS AND CODE DESIGN 1849
At the receiver, after matched filtering, removing the cyclic
prefix, and applying FFT, the received signal at the nth subcar-
rier at receive antenna j in the kth OFDM block is given by
y
k
j
(n)=
ρ
M
t
M
t
i=1
c
k
i
(n)H
k
i, j
(n)+z
k
j
(n) (5)
where
H
k
i,j
(n)=
L−1
l=0
α
k
i,j
(l)e
−j2πn∆fτ
l
(6)
is the channel frequency response at the nth subcarrier between
transmit antenna i and receive antenna j, ∆f =1/T is the sub-
carrier separation in the frequency domain, and T is the OFDM
symbol period. We assume that the channel state information
H
k
i,j
(n) is known at the receiver but not at the transmitter. In
(5), z
k
j
(n) denotes the additive white complex Gaussian noise
with zero mean and unit variance at the nth subcarrier at receive
antenna j in the kth OFDM block. The factor
ρ/M
t
in (5)
ensures that ρ is the average signal-to-noise ratio (SNR) at each
receive antenna.
III. P
ERFORMANCE CRITERIA
In this section, we derive the performance criteria for STF-
coded MIMO-OFDM systems, based on the results of [18],
[23], and [24], and we also determine the maximum achievable
diversity order for such systems.
A. Pairwise Error Probability
Using the notation c
i
[(k −1)N + n]
∆
= c
k
i
(n), H
i,j
[(k −
1)N + n]
∆
= H
k
i,j
(n), y
j
[(k − 1)N + n]
∆
= y
k
j
(n), and z
j
[(k −
1)N + n]
∆
= z
k
j
(n) for 1 ≤ k ≤ K, 0 ≤ n ≤ N − 1, 1 ≤ i ≤
M
t
, and 1 ≤ j ≤ M
r
, the received signal in (5) can be ex-
pressed as
y
j
(m)=
ρ
M
t
M
t
i=1
c
i
(m)H
i,j
(m)+z
j
(m) (7)
for m =0, 1,...,KN − 1. We further rewrite the received
signal in vector form as
Y =
ρ
M
t
DH + Z (8)
where D is a KNM
r
× KNM
t
M
r
matrix constructed from
the STF codeword C in (3) as follows:
D = I
M
r
⊗ [ D
1
D
2
··· D
M
t
] (9)
where ⊗ denotes the tensor product, I
M
r
is the identity matrix
of size M
r
× M
r
, and
D
i
= diag {c
i
(0),c
i
(1),...,c
i
(KN − 1)} (10)
for any i =1, 2,...,M
t
. The channel vector H of size
KNM
t
M
r
× 1 is formatted as in (11) (at the bottom of the
page) where
H
i,j
=[H
i,j
(0) H
i,j
(1) ··· H
i,j
(KN − 1) ]
T
. (12)
The received signal vector Y of size KNM
r
× 1 is given by
(13) (at the bottom of the page) and the noise vector Z, which
has the same form as Y, is given by (14) (at the bottom of the
page).
Suppose that D and
˜
D are two matrices constructed from two
different codewords C and
˜
C, respectively. Then, the pairwise
error probability between D and
˜
D can be upper bounded
as [23]
P (D →
˜
D) ≤
2r −1
r
r
i=1
γ
i
−1
ρ
M
t
−r
(15)
where r is the rank of (D −
˜
D)R(D −
˜
D)
H
, γ
1
,γ
2
,...,γ
r
are
the nonzero eigenvalues of (D −
˜
D)R(D −
˜
D)
H
, and R =
E{HH
H
} is the correlation matrix of H. The superscript H
stands for the complex conjugate and transpose of a matrix.
Based on the upper bound on the pairwise error probability
in (15), two general STF code performance criteria can be
proposed as follows.
1) Diversity (rank) criterion: The minimum rank of (D −
˜
D)R(D −
˜
D)
H
over all pairs of different codewords C
and
˜
C should be as large as possible.
H =[H
T
1,1
··· H
T
M
t
,1
H
T
1,2
··· H
T
M
t
,2
··· H
T
1,M
r
··· H
T
M
t
,M
r
]
T
(11)
Y =[y
1
(0) ··· y
1
(KN − 1) y
2
(0) ··· y
2
(KN − 1) ··· y
M
r
(0) ··· y
M
r
(KN − 1) ]
T
(13)
Z =[z
1
(0) ··· z
1
(KN − 1) z
2
(0) ··· z
2
(KN − 1) ··· z
M
r
(0) ··· z
M
r
(KN − 1) ]
T
(14)
1850 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
2) Product criterion: The minimum value of the product
r
i=1
γ
i
over all pairs of different codewords C and
˜
C
should be maximized.
B. Performance Criteria and Maximum Achievable Diversity
In case of spatially uncorrelated MIMO channels, i.e., the
channel taps α
k
i,j
(l) are independent for different transmit
antenna index i and receive antenna index j, the correlation
matrix R of size KNM
t
M
r
× KNM
t
M
r
becomes
R = diag (R
1,1
,...,R
M
t
,1
,R
1,2
,...,R
M
t
,2
,
...,R
1,M
r
,...,R
M
t
,M
r
) (16)
where
R
i,j
= E
H
i,j
H
H
i,j
(17)
is the correlation matrix of the channel frequency response
from transmit antenna i to receive antenna j. Using the notation
w =exp(−j2π∆f),from(6),wehave
H
i,j
=(I
K
⊗ W )A
i,j
(18)
where
W =
11··· 1
w
τ
0
w
τ
1
··· w
τ
L−1
.
.
.
.
.
.
.
.
.
.
.
.
w
(N−1)τ
0
w
(N−1)τ
1
··· w
(N−1)τ
L−1
and A
i,j
is defined at the bottom of the page. Substituting ( 18)
into (17), R
i,j
can be calculated as follows:
R
i,j
= E
(I
K
⊗ W )A
i,j
A
H
i,j
(I
K
⊗ W )
H
=(I
K
⊗ W )E
A
i,j
A
H
i,j
(I
K
⊗ W
H
).
With the assumptions that the path gains α
k
i,j
(l) are inde-
pendent for different paths and different pairs of transmit and
receive antennas, and that the second-order statistics of the time
correlation is the same for all transmit and receive antenna
pairs and all paths (i.e., the correlation values do not depend
on i, j, and l), we can define the time correlation at lag m as
r
T
(m)=E{α
k
i,j
(l)α
k+m
∗
i,j
(l)}. Thus, the correlation matrix
E{A
i,j
A
H
i,j
} can be expressed as
E
A
i,j
A
H
i,j
= R
T
⊗ Λ (19)
where Λ = diag {δ
2
0
,δ
2
1
,...,δ
2
L−1
}, and R
T
is the temporal
correlation matrix of size K ×K, whose entry in the pth row
and the qth column is given by r
T
(q − p) for 1 ≤ p, q ≤ K.
We can also define the frequency correlation matrix, R
F
,as
R
F
= E{H
k
i,j
H
k
H
i,j
}, where
H
k
i,j
=
H
k
i,j
(0),...,H
k
i,j
(N − 1)
T
.
Then, R
F
= W ΛW
H
. As a result, we arrive at
R
i,j
=(I
K
⊗ W )(R
T
⊗ Λ)(I
K
⊗ W
H
)
= R
T
⊗ (W ΛW
H
)=R
T
⊗ R
F
(20)
yielding
R = I
M
t
M
r
⊗ (R
T
⊗ R
F
). (21)
Finally, combining (4), (9), (10), and (21), the expression for
(D −
˜
D)R(D −
˜
D)
H
in (15) can be rewritten as
(D −
˜
D)R(D −
˜
D)
H
= I
M
r
⊗
M
t
i=1
(D
i
−
˜
D
i
)(R
T
⊗ R
F
)(D
i
−
˜
D
i
)
H
= I
M
r
⊗
(C −
˜
C)(C −
˜
C)
H
◦ (R
T
⊗ R
F
)
(22)
where ◦ denotes the Hadamard product.
1
Denote
∆
∆
=(C −
˜
C)(C −
˜
C)
H
(23)
and R
∆
= R
T
⊗ R
F
. Then, substituting (22) into (15), the pair-
wise error probability between C and
˜
C can be upper bound-
ed as
P (C →
˜
C) ≤
2νM
r
− 1
νM
r
ν
i=1
λ
i
−M
r
ρ
M
t
−νM
r
(24)
where ν is the rank of ∆ ◦ R, and λ
1
,λ
2
,...,λ
ν
are the
nonzero eigenvalues of ∆ ◦R. The minimum value of the
product
ν
i=1
λ
i
over all pairs of distinct signals C and
˜
C is
termed as coding advantage, denoted by
ζ
STF
= min
C=
˜
C
ν
i=1
λ
i
. (25)
As a consequence, we can formulate the performance criteria
for STF codes as follows.
1) Diversity (rank) criterion: The minimum rank of ∆ ◦ R
over all pairs of distinct codewords C and
˜
C should be as
large as possible.
2) Product criterion: The coding advantage or the minimum
value of the product
ν
i=1
λ
i
over all pairs of distinct
signals C and
˜
C should also be maximized.
1
Suppose that A = {a
i, j
} and B = {b
i, j
} are two matrices of size
m × n. The Hadamard product of A and B is defined as A ◦ B =
{a
i, j
b
i, j
}
1≤i≤m, 1≤ j≤n
.
A
i,j
=[α
1
i,j
(0) α
1
i,j
(1) ··· α
1
i,j
(L − 1) ··· α
K
i,j
(0) α
K
i,j
(1) ··· α
K
i,j
(L − 1) ]
T
SU et al.: DIVERSITY IN SPACE, TIME, AND FREQUENCY: PERFORMANCE ANALYSIS AND CODE DESIGN 1851
If the minimum rank of ∆ ◦R is ν for any pair of distinct
STF codewords C and
˜
C, we say that the STF code achieves a
diversity order of νM
r
. For a fixed number of OFDM blocks K,
transmit antennas M
t
, and correlation matrices R
T
and R
F
,the
maximum achievable diversity or full diversity is defined as the
maximum diversity order that can be achieved by STF codes of
size KN ×M
t
.
According to the rank inequalities on Hadamard products and
tensor products [38], we have
rank(∆ ◦ R) ≤ rank(∆)rank(R
T
)rank(R
F
).
Since the rank of ∆ is at most M
t
and the rank of R
F
is at most
L, we obtain
rank(∆ ◦ R) ≤ min {LM
t
rank(R
T
),KN}. (26)
Thus, the maximum achievable diversity is at most
min {LM
t
M
r
rank(R
T
),KNM
r
}, in agreement with the
results of [20]. However, there is no discussion in [20] whether
this upper bound can be achieved or not. In the following
sections, we show that this upper bound can indeed be
achieved. We can also observe that if the channel stays constant
over multiple OFDM blocks (rank(R
T
)=1), the maximum
achievable diversity is only min{LM
t
M
r
,KNM
r
}.Inthis
case, STF coding cannot provide additional diversity advantage
compared to the SF coding approach.
Note that the proposed analytical framework includes ST
and SF codes as special cases. If we consider only one subcar-
rier (N =1)and one delay path (L =1), the channel becomes
a single-carrier time-correlated flat fading MIMO channel. The
correlation matrix R simplifies to R = R
T
, and the code de-
sign problem reduces to that of ST code design, as described
in [24]. In the case of coding over a single OFDM block
(K =1), the correlation matrix R becomes R = R
F
, and the
code design problem simplifies to that of SF codes, as discussed
in [18].
IV. F
ULL-DIVERSITY STF CODE DESIGN METHODS
We propose two STF code design methods to achieve the
maximum achievable diversity order min {LM
t
M
r
rank(R
T
),
KNM
r
} in this section. Without loss of generality, we assume
that the number of subcarriers N is not less than LM
t
,sothe
maximum achievable diversity order is LM
t
M
r
rank(R
T
).
A. Repetition-Coded STF Code Design
In [18], we proposed a systematic approach to design full-
diversity SF codes. Suppose that C
SF
is a full-diversity SF code
of size N × M
t
. We now construct a full-diversity STF code
C
STF
by repeating C
SF
K times (over K OFDM blocks) as
follows:
C
STF
= 1
k×1
⊗ C
SF
(27)
where 1
k×1
is an all one matrix of size k × 1.Let
∆
STF
=(C
STF
−
˜
C
STF
)(C
STF
−
˜
C
STF
)
H
and
∆
SF
=(C
SF
−
˜
C
SF
)(C
SF
−
˜
C
SF
)
H
.
Then, we have
∆
STF
=
1
k×1
⊗ (C
SF
−
˜
C
SF
)
1
1×k
⊗ (C
SF
−
˜
C
SF
)
H
= 1
k×k
⊗ ∆
SF
.
Thus,
∆
STF
◦ R =(1
k×k
⊗ ∆
SF
) ◦ (R
T
⊗ R
F
)
= R
T
⊗ (∆
SF
◦ R
F
).
Since the SF code C
SF
achieves full diversity in each OFDM
block, the rank of ∆
SF
◦ R
F
is LM
t
. Therefore, the rank of
∆
STF
◦ R is LM
t
rank(R
T
),soC
STF
in (27) is guaranteed to
achieve a diversity order of LM
t
M
r
rank(R
T
).
We observe that the maximum achievable diversity depends
on the rank of the temporal correlation matrix R
T
.Ifthe
fading channels are constant during K OFDM blocks, i.e.,
rank(R
T
)=1, the maximum achievable diversity order for
STF codes (coding across several OFDM blocks) is the same as
that for SF codes (coding within one OFDM block). Moreover,
if the channel changes independently in time, i.e., R
T
= I
K
,
the repetition structure of STF code C
STF
in (27) is sufficient,
but not necessary to achieve the full diversity. In this case
∆ ◦ R = diag (∆
1
◦ R
F
, ∆
2
◦ R
F
,...,∆
K
◦ R
F
)
where ∆
k
=(C
k
−
˜
C
k
)(C
k
−
˜
C
k
)
H
for 1 ≤ k ≤ K. Thus, in
this case, the necessary and sufficient condition to achieve full-
diversity KLM
t
M
r
is that each matrix ∆
k
◦ R
F
be of rank
LM
t
over all pairs of distinct codewords simultaneously for all
1 ≤ k ≤ K.
The proposed repetition-coded STF code design ensures full
diversity at the price of symbol rate decrease by a factor of 1/K
(over K OFDM blocks) compared to the symbol rate of the
underlying SF code. The advantage of this approach is that any
full-diversity SF code (block or trellis) can be used to design
full-diversity STF codes.
B. Full-Rate STF Code Design
We can also design a class of STF codes that can achieve
a diversity order of ΓM
t
M
r
rank(R
T
) for any fixed integer
Γ(1≤ Γ ≤ L) by extending the full-rate full-diversity SF code
construction method (coding over one OFDM block, i.e., the
K =1case) proposed in [25].
We consider an STF code structure consisting of STF code-
words C of size KN by M
t
C =[C
T
1
C
T
2
··· C
T
K
]
T
(28)
where
C
k
=
G
T
k,1
G
T
k,2
··· G
T
k,P
0
T
N−P ΓM
t
T
(29)
