370 IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 9, SEPTEMBER 2002
Space-Time-Frequency (STF) Coding for
MIMO-OFDM Systems
Andreas F. Molisch, Senior Member, IEEE, Moe Z. Win, Senior Member, IEEE, and Jack H. Winters, Fellow, IEEE
Abstract—We consider the capacity of multiple-input–mul-
tiple-output (MIMO) systems that use OFDM as the modulation
format. We point out a basic equivalence between antennas and
OFDM-tones. This similarity immediately allows us to essentially
reuse all space-time codes designed for flat-fading channels in
MIMO-OFDM systems operating in frequency-selective channels.
An optimum code would thus code across all antennas and tones
(as well as time) simultaneously. Since this can become very
complex, we propose a method for grouping antennas and codes
in such a way that the inherent diversity is retained, while the
complexity is greatly reduced. Capacity computations between
the full-complexity and the reduced-complexity systems illustrate
this tradeoff.
Index Terms—MIMO, OFDM, space-time coding.
I. INTRODUCTION
W
IRELESS SYSTEMS with multiple antennas at
the transmitter and receiver (multiple-input–mul-
tiple-output, MIMO) have much larger capacity in fading
channels than standard wireless systems [1], [2]. The appro-
priate use of space-time (ST) processing [3] and ST codes
[4] allows us to achieve, or at least approach, these capacities
in practical systems. For frequency-selective channels, a
combination of MIMO with OFDM (orthogonal frequency
division multiplexing) is promising [5], [6]. The simplest way
to perform ST coding in a MIMO-OFDM system would be to
apply the ST-codes for the frequency-flat channels to each tone
separately. However, a recent paper [7] has pointed out that
this is suboptimum, as the inherent frequency diversity of the
frequency-selective channel is not exploited. It was also stated
that construction of codes that code across tones would be
difficult. In this paper, we show by a very simple observation
how we can extend ST code design rules to frequency selective
channels without sacrificing performance. We will also develop
further simplifications and illustrate the performance-com-
plexity tradeoff.
II. A
NTENNAS AND TONES—UNIFIED PERSPECTIVE
In a conventional OFDM system [see Fig. 1(a)], i.e., without
exploitation of the frequency diversity, the data streams for the
OFDM tones enter separate ST coders whose outputs are then
forwarded to the different antennas. The tones at each antenna
are inverse Fourier-transformed and the resulting time signal is
Manuscript received October 12, 2001. The associate editor coordinating the
review of this letter and approving it for publication was Dr. J. Ritcey.
The authors were with the Wireless Systems Research Department,
AT&T Labs—Research, Middletown, NJ 07748-4801 USA (e-mail: An-
dreas.Molisch@ieee.org; win@ieee.org; jack.winters@ieee.org).
Publisher Item Identifier 10.1109/LCOMM.2002.802047.
(a)
(b)
Fig. 1. System model for: (a) separate coding for each tone (upper figure) and
(b) joint space/time/frequence coding (lower figure). IFFT blocks not shown for
simplicity.
upconverted to the carrier frequency and transmitted across the
mobile radio channel.
As pointed out in [7], coding across the tones is required
to exploit the inherent frequency diversity in a time-dispersive
channel. A full-complexity coder thus must use the symbols
from all tones as input, and distribute them to all tones on all
antennas jointly [Fig. 1(b)]. Assuming
transmit antennas
and
tones, the size of the coder is thus .
1
Systematic
methods for designing these codes thus seem difficult to derive.
2
From the antennas, the signal is sent through the mobile radio
channel, which is assumed to be constant within one OFDM
block. The fading of the signals at thedifferent antenna elements
is assumed to be identically distributed, but not necessarily in-
dependent [9], [10]. The channel matrix can thus be written as
(1)
where
denotes the transfer function at frequency
between the antenna pair . The are zero-mean,
1
In a slight abuse of notation, we call the number of outputs of a (vector)
coder the “size” of the coder. The total number of possible signal constellations
per timestep is
L
, where
L
is the alphabet size of each antenna and tone.
2
We note that existing system proposals use a rudimentary form of coding
across the tones. Any standard coder (e.g., Reed–Solomon code) that is applied
to the data stream before serial/parallel conversion distributes the redundancy
across tones and thus exploits frequency diversity in some way [8]. However,
it seems difficult to formulate analytical criteria and systematic construction
methods for ST codes from that point of view.
1089-7798/02$17.00 © 2002 IEEE
MOLISCH et al.: STF CODING FOR MIMO-OFDM SYSTEMS 371
circularly symmetric complex Gaussian random variables. The
channel also adds white Gaussian noise, which is assumed to
be independent at all
receiver antenna elements.
Let us now consider the similarity between antennas and
tones.
3
We will start out with the simplest imaginable system,
to illustrate the basic similarity between tones and antennas,
and then step by step relax the assumptions and show where
the practical differences lie.
Let us first consider an FDM system with tones that have
a narrow bandwidth, yet are widely spaced apart. Similarly,
the antennas are assumed to be spaced wide enough apart to
exhibit independent fading. Finally, we will assume that the
-th transmit antenna can only communicate with the -th re-
ceive antenna, so that there is no crosstalk between the antennas.
Thus, we have
parallel (isolated) communications chan-
nels. It does not matter whether they are distinguished by dif-
ferent tones or different antennas—in any case, they are inde-
pendently fading, and experience no crosstalk. Coding across
all available channels obviously allows us to essentially elim-
inate the capacity fluctuations due to fading, by exploiting the
inherent diversity among tones and antennas, and thus achieve
outage capacities that are close to the mean capacities.
In the next step, let us relax the condition that there must
not be crosstalk between the antennas and between the tones:
for different antennas, crosstalk always occurs, while crosstalk
across tones is not likely. In particular, there is no crosstalk be-
tween widely separated tones. Even if tones are overlapping,
as occurs in OFDM, crosstalk is eliminated by the carrier or-
thogonality and, additionally in time-dispersive environments,
by the use of a cyclic prefix. Thus, this is one point where the
antennas and the tones do not have a full analogy. However, such
an analogy can be recovered by two considerations.
1) Viewing the antennas as additional tones: If the channel
is known, a singular value decomposition of the channel
matrix transforms it into isolated (but not identically dis-
tributed) channels. This recovers analogy to the isolated
tones. Note that the transformation from the antenna
space to the singular vector space is linear.
2) Viewing the tones as additional antennas: In this case,
the “additional antennas” just exhibit a special type of
crosstalk description (namely, zero crosstalk). Note that
the crosstalk might be beneficial for diversity purposes.
However, crosstalk between tones could be enforced,
e.g., by using multicarrier-CDMA instead of conven-
tional OFDM.
4
From these considerations it follows that, in principle, any
ST code that is devised for a flat-fading channel can be used in
a frequency selective channel. The size of the ST code
5
is now
determined by the product of the number of antennas and the
number of tones. Thus, codes with a much larger size than those
for flat-fading MIMO systems are required. Nonetheless, the
code construction rules and evaluation methods remain valid;
3
We stress that the analogy between antennas and tones is only valid from the
viewpoint of diversity. In contrast to tones, antennas are also used to provide
antenna gain and interference supression.
4
By similar considerations, one could derive, e.g., an analogy between mul-
tiple antennas and orthogonal ST block coding, where
H
then has a special
(unitary) structure. Other analogies, e.g., with SISO block coding, are possible
as well.
5
We are using “size” of the code in the same way that we define the size of
the coder.
furthermore, code designs for large constellation sizes have re-
cently been hinted at [11].
Summarizing, the transfer function of the MIMO-OFDM
system can be written as
(2)
where each entry block
(3)
is itself a matrix. If a cyclic prefix of sufficient duration (longer
than the maximum excess delay) is used, and no intercarrier in-
terference occurs,
becomes a diagonal matrix. The corre-
lation between the elements of
in (3) depends on the fre-
quency correlation function of the channel. Still, in principle,
all entries
(distinguished by antennas , and/or
tones
, )of are equivalent in the sense that they make up
the total matrix, and thus contribute to the capacity.
Also note that coding in time is possible, thus giving diver-
sity in three dimensions, namely, space, frequency, and time.
The coding in time is completely identical to the time-compo-
nent of any ST codefor flat-fading channels. The “coding across
tones” is more effective in situations where little time diversity
can be achieved.If ample time diversity is available, then full ca-
pacity can be achieved on each tone, and additional frequency
diversity does not have a large advantage. However, in many
practical cases, ample time diversity is not available, as either
TX and RX are stationary (wireless local loop systems, nomadic
mobility), and/or long delays (meaning longer than several co-
herence times) are not allowed for the coder.
III. R
EDUCED-COMPLEXITY SYSTEMS
Equations (2) and (3) show that a full-complexity system re-
quires a coder of size
. We thus propose in the following
to exploit the typical properties of an OFDM system to reduce
the complexity.
The advantage of coding across the tones comes from the
exploitation of the frequency diversity in a delay-dispersive
channel. Due to restrictions on the duration of the cyclic prefix,
the spacing between the tones is usually much smaller than the
coherence bandwidth. Distributing information between two
neighboring tones thus will not enhance the system’s robustness
against fading. Rather, only the distribution of information into
widely separated tones will contribute to system improvement.
As a rule of thumb, the required separation is on the order of
one coherence bandwidth.
We thus suggest to perform coding only across tones that
are separated by approximately the coherence bandwidth, i.e.,
to code across a group of tones (see Fig. 2) that is defined
as
, for
each
. Thus, the information is distributed
onto parallel carriers that are “almost”independently fading, re-
sulting in a high degree of diversity.
372 IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 9, SEPTEMBER 2002
Fig. 2. Grouping frequencies for complexity reduction.
Fig. 3. Cdf of the capacity in a 2
3
2 OFDM MIMO system with 64-QAM.
Lorentzian correlation function with coherence BW
=
eight tone distances.
IV. RESULTS
The capacity of a MIMO-OFDM system coding across the
tones is given by
(4)
where
is the identity matrix and is the mean
signal-to-noise ratio (SNR) per receiver branch. The above for-
mulation assumes that no time diversity is exploited. It would
be easy to include that also, but would add no new insights.
Note that
and exhibit correlation. A simulation
of the capacity just requires generation of (correlated) channel
matrices, and numerical evaluation of (4).
For the reduced-complexity system, each group
of tones
of size
provides capacity (which can be approached by an
ideal code)
and the total capacity is the sum of the capacities of all groups.
In real-world systems, the maximum capacity per tone is lim-
ited by the size of the modulation alphabet, e.g., to
bits/s/Hz for 64-QAM. The capacity for coding across groups
of size
is then
Fig. 3 shows the capacity of a system with 64 tones and a
coherence bandwidth of 8 (measured in units of tone spacings).
The frequency correlation function is assumed to be Lorentzian.
The SNR is 20 dB and the capacity results are normalized to the
bandwidth. We see that using ST coders of size 4 (2 antennas
and 2 tones) the loss in 10%-outage capacity is less than 0.1
bit/s/Hz, while coding on each tone separately results in a loss
of 0.2 bits/s/Hz. When the separation between adjacent tones in
one group is equal to the coherence width, the loss in outage ca-
pacity (compared to the coding across all 64 tones) is negligible.
We have also analyzedsituations with a smaller coherence band-
width. It is intuitively clear that the gain by coding across the
tones is larger in that case, but that
must be larger to fully ex-
ploit that possibility. Specifically, with a coherence bandwidth
of 2 tone spacings, the loss in outage capacity is 0.45, 0.2, 0.1,
and 0.02 for
and , respectively.
V. S
UMMARY AND CONCLUSIONS
We have investigated STF codes for MIMO-OFDM. Starting
from the premise that coding across the tones must be done in
a systematic way, we have pointed out the basic mathematical
analogy between antennas (or spatial eigenmodes) and tones,
and explained how this similarity allowsto reuse the concepts of
ST coding for space-time-frequency (STF) coding required for
OFDM. We then proposed a reduced-complexity scheme that
codes only across tones that are separated by about one coher-
ence bandwidth. A logical next step would be to use real-world
codes on that scheme and investigate performance with full- and
reduced-complexity schemes.
A
CKNOWLEDGMENT
The authors thank Dr. L. Cimini, Dr. S. Mueller-Weinfurtner,
and Dr. Y.-S. Choi for helpful discussions and critical reading
of the manuscript. They also thank the anonymous reviewers
for helpful suggestions, especially for pointing out the analogy
between antennas and block codes.
R
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