Space–Time Codes for High Data
Rate Wireless Communication: Performance Criterion and Code Construction Vahid Tarokh,Member,IEEE,Nambi Seshadri,Senior Member,IEEE,and A.R.Calderbank,Fellow,IEEE
Abstract—We consider the design of channel codes for im-
proving the data rate and/or the reliability of communications
over fading channels using multiple transmit antennas.Data is
encoded by a channel code and the encoded data is split into n streams that are simultaneously transmitted using n transmit antennas.The received signal at each receive antenna is a linear
superposition of the n transmitted signals perturbed by noise.We
derive performance criteria for designing such codes under the
assumption that the fading is slow and frequency nonselective.
Performance is shown to be determined by matrices constructed
from pairs of distinct code sequences.The minimum rank among
these matrices quanti?es the diversity gain,while the minimum
determinant of these matrices quanti?es the coding gain.The
results are then extended to fast fading channels.The design
criteria are used to design trellis codes for high data rate wireless
communication.The encoding/decoding complexity of these codes
is comparable to trellis codes employed in practice over Gaussian
channels.The codes constructed here provide the best tradeoff
between data rate,diversity advantage,and trellis complexity.
Simulation results are provided for4and8PSK signal sets
with data rates of2and3bits/symbol,demonstrating excellent
performance that is within2–3dB of the outage capacity for these
channels using only64state encoders.
Index Terms—Array processing,diversity,multiple transmit
antennas,space–time codes,wireless communications.
I.I NTRODUCTION
A.Motivation
C URRENT cellular standards support circuit data and fax
services at9.6kb/s and a packet data mode is being standardized.Recently,there has been growing interest in providing a broad range of services including wire-line voice quality and wireless data rates of about64–128kb/s(ISDN) using the cellular(850-MHz)and PCS(1.9-GHz)spectra[2]. Rapid growth in mobile computing is inspiring many proposals for even higher speed data services in the range of144kb/s (for microcellular wide-area high-mobility applications)and up to2Mb/s(for indoor applications)[1].
The majority of the providers of PCS services have further decided to deploy standards that have been developed at cellular frequencies such as CDMA(IS-95),TDMA(IS-54/IS-136),and GSM(DCS-1900).This has led to considerable Manuscript received December15,1996;revised August18,1997.The material in this paper was presented in part at the IEEE International Symposium on Information Theory,Ulm,Germany,June29–July4,1997. The authors are with the AT&T Labs–Research,Florham Park,NJ07932 USA.
Publisher Item Identi?er S0018-9448(98)00933-X.effort in developing techniques to provide the aforementioned new services while maintaining some measure of backward compatibility.Needless to say,the design of these techniques is a challenging task.
Band-limited wireless channels are narrow pipes that do not accommodate rapid?ow of data.Deploying multiple transmit and receive antennas broadens this data https://www.doczj.com/doc/2d9847738.html,rmation the-ory[14],[35]provides measures of capacity,and the standard approach to increasing data?ow is linear processing at the receiver[15],[44].We will show that there is a substantial bene?t in merging signal processing at the receiver with coding technique appropriate to multiple transmit antennas. In particular,the focus of this work is to propose a solution to the problem of designing a physical layer(channel coding, modulation,diversity)that operate at bandwidth ef?ciencies that are twice to four times as high as those of today’s systems using multiple transmit antennas.
B.Diversity
Unlike the Gaussian channel,the wireless channel suffers from attenuation due to destructive addition of multipaths in the propagation media and due to interference from other users. Severe attenuation makes it impossible for the receiver to determine the transmitted signal unless some less-attenuated replica of the transmitted signal is provided to the receiver. This resource is called diversity and it is the single most important contributor to reliable wireless communications. Examples of diversity techniques are(but are not restricted to)?Temporal Diversity:Channel coding in conjunction with time interleaving is used.Thus replicas of the transmit-ted signal are provided to the receiver in the form of redundancy in temporal domain.
?Frequency Diversity:The fact that waves transmitted on different frequencies induce different multipath structure in the propagation media is exploited.Thus replicas of the transmitted signal are provided to the receiver in the form of redundancy in the frequency domain.?Antenna Diversity:Spatially separated or differently po-larized antennas are used.The replicas of transmitted signal are provided to the receiver in the form of redun-dancy in spatial domain.This can be provided with no penalty in bandwidth ef?ciency.
When possible,cellular systems should be designed to encom-pass all forms of diversity to ensure adequate performance
0018–9448/98$10.00?1998IEEE
[26].For instance,cellular systems typically use channel coding in combination with time interleaving to obtain some form of temporal diversity[28].In TDMA systems,frequency diversity is obtained using a nonlinear equalizer[4]when multipath delays are a signi?cant fraction of symbol interval. In DS-CDMA,RAKE receivers are used to obtain frequency diversity.Antenna diversity is typically used in the up-link (mobile-to-base)direction to provide the link margin and cochannel interference suppression[40].This is necessary to compensate for the low power transmission from mobiles. Not all forms of diversity can be available at all times.For example,in slow fading channels,temporal diversity is not an option for delay-sensitive applications.When the delay spread is small,frequency(multipath)diversity is not an option.In macrocellular and microcellular environments,respectively, this implies that the data rates should be at least several hundred thousand symbols per second and several million symbols per second,respectively.While antenna diversity at a base-station is used for reception today,antenna diversity at a mobile handset is more dif?cult to implement because of electromagnetic interaction of antenna elements on small platforms and the expense of multiple down-conversion RF paths.Furthermore,the channels corresponding to different antennas are correlated,with the correlation factor determined by the distance as well as the coupling between the antennas. Typically,the second antenna is inside the mobile handset, resulting in signal attenuation at the second antenna.This can cause some loss in diversity bene?t.All these factors motivate the use of multiple antennas at the base-station for transmission.
In this paper,we consider the joint design of coding, modulation,transmit and receive diversity to provide high performance.We can view our work as combined coding and modulation for multi-input(multiple transmit antennas)multi-output(multiple receive antennas)fading channels.There is now a large body of work on coding and modulation for single-input/multi-output channels[5],[10],[11],[29], [30],[38],and[39],and a comparable literature on receive diversity,array processing,and beamforming.In light of these research activities,receive diversity is very well understood. By contrast,transmit diversity is less well understood.We begin by reviewing prior work on transmit diversity.
C.Historical Perspective on Transmit Diversity
Systems employing transmit fall into three general cate-gories.These are
?schemes using feedback,
?those with feedforward or training information but no feedback,and
?blind schemes.
The?rst category uses implicit or explicit feedback of information from the receiver to the transmitter to con?gure the transmitter.For instance,in time-division duplex systems [16],the same antenna weights are used for reception and transmission,so feedback is implicit in the appeal to channel symmetry.These weights are chosen during reception to maximize the signal-to-noise ratio(SNR),and during trans-mission to weight the amplitudes of the transmitted signals. Explicit feedback includes switched diversity with feedback [41]as well as techniques that use spatiotemporal-frequency water pouring[27]based on the feedback of the channel response.However,in practice,vehicle movements or inter-ference causes a mismatch between the state of the channel perceived by the transmitter and that perceived by receiver. Transmit diversity schemes mentioned in the second cat-egory use linear processing at the transmitter to spread the information across the antennas.At the receiver,information is obtained by either linear processing or maximum-likelihood decoding techniques.Feedforward information is required to estimate the channel from the transmitter to the receiver.These estimates are used to compensate for the channel response at the receiver.The?rst scheme of this type was proposed by Wittneben[43]and it includes the delay diversity scheme of Seshadri and Winters[32]as a special case.The linear processing approach was also studied in[15]and[44].It has been shown in[42]that delay diversity schemes are indeed optimal in providing diversity in the sense that the diversity advantage experienced by an optimal receiver is equal to the number of transmit antennas.We can view the linear?lter as a channel code that takes binary data and creates real-valued output.It is shown that there is signi?cant gain to be realized by viewing this problem from a coding perspective rather than purely from the signal processing point of view.
The third category does not require feedback or feedforward information.Instead,it uses multiple transmit antennas com-bined with channel coding to provide diversity.An example of this approach is to combine phase sweeping transmitter diver-sity of[18]with channel coding[19].Here a small frequency offset is introduced on one of the antennas to create fast fading. An appropriately designed channel code/interleaver pair is used to provide diversity bene?t.Another scheme is to encode information by a channel code and transmit the code symbols using different antennas in an orthogonal manner.This can be done either by frequency multiplexing[9],time multiplexing [32],or by using orthogonal spreading sequences for different antennas[37].A disadvantage of these schemes over the previous two categories is the loss in bandwidth ef?ciency due to the use of the channel https://www.doczj.com/doc/2d9847738.html,ing appropriate coding,it is possible to relax the orthogonality requirement needed in these schemes and obtain the diversity as well as coding advantage offer without sacri?cing bandwidth.This is possible when the whole system is viewed as a multiple-input/multiple-output system and suitable codes are used.
Information-theoretic aspects of transmit diversity were addressed in[14],[25],and[35].We believe that Telatar [35]was the?rst to obtain expressions for capacity and error exponents for multiple transmit antenna system in the presence of Gaussian noise.Here,capacity is derived under the assumption that fading is independent from one channel use to the other.At about the same time,Foschini and Gans[14] derived the outage capacity under the assumption that fading is quasistatic;i.e.,constant over a long period of time,and then changes in an independent manner.A particular layered space–time architecture was shown to have the potential to achieve a substantial fraction of capacity.A major conclusion
of these works is that the capacity of a multi-antenna systems
A comprehensive information-theoretic treatment for many of the transmit diversity schemes that have been studied before is presented by Narula,Trott,and Wornell[25].
D.Space–Time Codes
We consider the delay diversity scheme as proposed by Wittneben[44].This scheme transmits the same information from both antennas simultaneously but with a delay of one symbol interval.We can view this as a special case of the arrangement in Fig.1,where the information is encoded by a channel code(here the channel code is a repetition code of
length
Fig.2.The block diagram of the transmitter.
on the data rate as a function of the constellation size and diversity advantage.For a given diversity,we provide explicit constructions of trellis codes that achieve the minimum trellis complexity as well as the maximum data rate.Then,we revisit delay diversity and show that some of the codes constructed before have equivalent delay diversity representations.This section also includes multilevel constructions which provide an ef?cient way to construct and decode codes when the number of antennas is large(4–8).It is further shown that it is not possible for block-coded modulation schemes to outperform trellis codes constructed here at a given diversity advantage and data rate.Simulation results for many of the codes that we have constructed and comparisons to outage capacity for these channels are also presented.We then consider design
of space–time codes that guarantee a diversity advantage
of
when there is no mobility and a diversity advantage
of
antennas and the mobile is
equipped
with
streams of data.Each stream
of data is used as the input to a pulse shaper.The output
of each shaper is then modulated.At each time slot
is a
signal
for
signals are transmitted simultaneously each from a
different transmit antenna and that all these signals have the
same transmission
period
transmitted signals corrupted
by Rayleigh or Rician fading(see Fig.2).We assume that the
elements of the signal constellation are contracted by a factor
of
.
At the receiver,the demodulator computes a decision statis-
tic based on the received signals arriving at each receive
antenna.The
signal received by
antenna
(1)
where the
noise at
time
to receive
antenna
and
variance per dimension.This is equivalent to the
assumption that signals transmitted from different antennas undergo independent fades.
We shall derive a design criterion for constructing codes under this transmission scenario.We begin by establishing the notation and by reviewing the results from linear algebra that we will employ.This notation will also be used in the sequel to this paper [34].
Let
be complex vectors in
the .The inner product
of is given
by
where ,
let .
Recall from linear algebra that
an
is Hermitian
if is nonnegative de?nite
if
complex
vector
matrix
where
matrix
if .We shall
make use of the following results from linear algebra [20].?An
eigenvector
matrix
is
a vector of unit length such
that
.The vector space
spanned by the eigenvectors
of
,
where
.
?Any
matrix is nonnegative de?nite.
?For any nonnegative-de?nite Hermitian
matrix
such
that
.
?Given a Hermitian
matrix span
C dimensions and it is easy to construct an orthonormal basis of
C
.Furthermore,there exists a unitary
matrix
such
that
are an orthonormal basis of
C
.The diagonal elements
of
,
of
.Thus our design
criterion is not constellation-dependent and applies equally well to 4-PSK,8-PSK,and 16-QAM.
We consider the probability that a maximum-likelihood receiver decides erroneously in favor of a
signal
was transmitted.
Assuming ideal channel state information (CSI),the proba-bility of
transmitting at the decoder
is well approximated
by
(2)
where
,we rewrite (3)
as
(4)
where
for
(5)
where
and a real diagonal
matrix are a complete orthonormal basis
of
C
.Furthermore,the diagonal
elements
of
of
..
.
..
.
(7)
Next,recall
that
are samples of a complex Gaussian random variable with
mean
Since is an orthonormal basis of
C
per dimension and
mean
is the zero-order modi?ed Bessel
function of the ?rst kind.
Thus to compute an upper bound on the average probability of error,we simply
average
and
as a
fortiori
for all
.Then the inequality (8)
can be written
as
denote the rank of
matrix
has
dimension
and
exactly eigenvalues
of
are
cofactors
of
and
over the set of two tuples of distinct codewords,then
a diversity
of
is achieved.This criterion was also derived in [15].
?The Determinant Criterion:Suppose that a diversity ben-e?t
of
is our target.The minimum
of principal cofactors
of
and
is the rank
of
and
and
is achieved.Thus the following design criteria is valid for the Rician space–time codes for large signal-to-noise ratios.Design Criteria for The Rician Space–Time Codes:
?The Rank Criterion:This criterion is the same as that given for the Rayleigh channel.
?The Coding Advantage Criterion:
Let
principal cofactors
of is the rank
of
taken over distinct
codewords has to be maxi-mized.
Note that one could still use the coding advantage criterion,since the performance will be at least as good as the right-hand side of inequality (9).C.The Case of Dependent Fade Coef?cients
In this subsection,we assume that the
coef?cients are samples of possibly dependent zero-mean complex Gaussian
random variables having
variance
per dimension.This is the Rayleigh fading,but the extension to the Rician case is straightforward.