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1456 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 5, JULY 1999

Space–Time Block Codes from Orthogonal Designs

Vahid Tarokh, Member, IEEE, Hamid Jafarkhani, and A. R. Calderbank,

Fellow, IEEE

Abstract—We introduce space–time block coding, a new par-

adigm for communication over Rayleigh fading channels using

multiple transmit antennas. Data is encoded using a space–time

block code and the encoded data is split into

streams which

are simultaneously transmitted using

transmit antennas. The

received signal at each receive antenna is a linear superposition

of the

transmitted signals perturbed by noise. Maximum-

likelihood decoding is achieved in a simple way through decou-

pling of the signals transmitted from different antennas rather

than joint detection. This uses the orthogonal structure of the

space–time block code and gives a maximum-likelihood decoding

algorithm which is based only on linear processing at the receiver.

Space–time block codes are designed to achieve the maximum

diversity order for a given number of transmit and receive

antennas subject to the constraint of having a simple decoding

algorithm.

The classical mathematical framework of orthogonal designs

is applied to construct space–time block codes. It is shown

that space–time block codes constructed in this way only exist

for few sporadic values of

. Subsequently, a generalization of

orthogonal designs is shown to provide space–time block codes

for both real and complex constellations for any number of

transmit antennas. These codes achieve the maximum possible

transmission rate for any number of transmit antennas using

any arbitrary real constellation such as PAM. For an arbitrary

complex constellation such as PSK and QAM, space–time block

codes are designed that achieve

of the maximum possible

transmission rate for any number of transmit antennas. For

the speciﬁc cases of two, three, and four transmit antennas,

space–time block codes are designed that achieve, respectively,

all,

, and

of maximum possible transmission rate using

arbitrary complex constellations. The best tradeoff between the

decoding delay and the number of transmit antennas is also

computed and it is shown that many of the codes presented here

are optimal in this sense as well.

Index Terms— Codes, diversity, multipath channels, multiple

antennas, wireless communication.

I. INTRODUCTION

EVERE attenuation in a multipath wireless environment

makes it extremely difﬁcult for the receiver to determine

the transmitted signal unless the receiver is provided with

some form of diversity, i.e., some less-attenuated replica of

the transmitted signal is provided to the receiver.

In some applications, the only practical means of achieving

diversity is deployment of antenna arrays at the transmit-

ter and/or the receiver. However, considering the fact that

Manuscript received June 15, 1998; revised February 1, 1999. The material

in this paper was presented in part at the IEEE Information Theory Workshop,

Killarney, Ireland, 1998.

V. Tarokh and A. R. Caldebank are with AT&T Shannon Laboratories,

Florham Park, NJ 07932 USA.

H. Jafarkhani is with AT&T Labs–Research, Red Bank, NJ 07701 USA.

Communicated by M. Honig, Associate Editor for Communications.

Publisher Item Identiﬁer S 0018-9448(99)04360-6.

receivers are typically required to be small, it may not be

practical to deploy multiple receive antennas at the remote

station. This motivates us to consider transmit diversity.

Transmit diversity has been studied extensively as a method

of combatting impairments in wireless fading channels [2]–[4],

[6], [9]–[15]. It is particularly appealing because of its relative

simplicity of implementation and the feasibility of multiple

antennas at the base station. Moreover, in terms of economics,

the cost of multiple transmit chains at the base can be

amortized over numerous users.

Space–time trellis coding [10] is a recent proposal that

combines signal processing at the receiver with coding tech-

niques appropriate to multiple transmit antennas. Speciﬁc

space–time trellis codes designed for 2–4 transmit antennas

perform extremely well in slow-fading environments (typical

of indoor transmission) and come close to the outage capacity

computed by Telatar [12] and independently by Foschini and

Gans [4]. However, when the number of transmit antennas

is ﬁxed, the decoding complexity of space–time trellis codes

(measured by the number of trellis states in the decoder)

increases exponentially with transmission rate.

In addressing the issue of decoding complexity, Alamouti

[1] recently discovered a remarkable scheme for transmission

using two transmit antennas. This scheme is much less com-

plex than space–time trellis coding for two transmit antennas

but there is a loss in performance compared to space–time

trellis codes. Despite this performance penalty, Alamouti’s

scheme [1] is still appealing in terms of simplicity and

performance and it motivates a search for similar schemes

using more than two transmit antennas. It is a starting point

for the studies in this paper, where we apply the theory of

orthogonal designs to create analogs of Alamouti’s scheme,

namely, space–time block codes, for more than two transmit

antennas.

The theory of orthogonal designs is an arcane branch

of mathematics which was studied by several great number

theorists including Radon and Hurwitz. The encyclopedic work

of Geramita and Seberry [5] is an excellent reference. A

classical result in this area is due to Radon who determined the

set of dimensions for which an orthogonal design exists [8].

Radon’s results are only concerned with real square orthogonal

designs. In this work, we extend the results of Radon to both

nonsquare and complex orthogonal designs and introduce a

theory of generalized orthogonal designs. Using this theory, we

construct space–time block codes for any number of transmit

antennas. Since we approach the theory of orthogonal designs

from a communications perspective, we also study designs

which correspond to combined coding and linear processing

at the transmitter.

0018–9448/99$10.00

 1999 IEEE

TAROKH et al.: SPACE–TIME BLOCK CODES FROM ORTHOGONAL DESIGNS 1457

The outline of the paper is as follows. In Section II, we de-

scribe a mathematical model for multiple-antenna transmission

over a wireless channel. We review the diversity criterion for

code design in this model as established in [10]. In Section III,

we review orthogonal designs and describe their application to

wireless communication systems employing multiple transmit

antennas. It will be proved that the scheme provides maximum

possible spatial diversity order and allows a remarkably simple

decoding strategy based only on linear processing. In Section

IV, we generalize the concept of the orthogonal designs and

develop a theory of generalized orthogonal designs. Using this

mathematical theory, we construct coding schemes for any

arbitrary number of transmit antennas. These schemes achieve

the full diversity order that can be provided by the transmit and

receive antennas. Moreover, they have very simple maximum-

likelihood decoding algorithms based only on linear processing

at the receiver. They provide the maximum possible transmis-

sion rate using totally real constellations as established in the

theory of space–time coding [10]. In Section V, we deﬁne

complex orthogonal designs and study their properties. We will

recover the scheme proposed by Alamouti [1] as a special case,

though it will be proved that generalization to more than two

transmit antennas is not possible. We then develop a theory

of complex generalized orthogonal designs. These designs

exist for any number of transmit antennas and again have

remarkably simple maximum-likelihood decoding algorithms

based only on linear processing at the receiver. They provide

full spatial diversity and

of the maximum possible rate

(as established previously in the theory of space–time coding)

using complex constellations. For complex constellations and

for the speciﬁc cases of two, three, and four transmit antennas,

these diversity schemes are improved to provide, respectively

all,

, and of maximum possible transmission rate.

Section VI presents our conclusions and ﬁnal remarks.

For the reader who is interested only in the code con-

struction but is not concerned with the details, we provide a

summary of the material at the beginning of each subsection.

II. T

HE CHANNEL MODEL AND THE DIVERSITY CRITERION

In this section, we model a multiple-antenna wireless com-

munication system under the assumption that fading is quasi-

static and ﬂat. We review the diversity criterion for code design

assuming this model. This diversity criterion is crucial for our

studies of space–time block codes.

We consider a wireless communication system where the

base station is equipped with

and the remote is equipped

with

antennas. At each time slot , signals

are transmitted simultaneously from the transmit antennas.

The coefﬁcient

is the path gain from transmit antenna to

receive antenna

. The path gains are modeled as samples of

independent complex Gaussian random variables with variance

per real dimension. The wireless channel is assumed to be

quasi-static so that the path gains are constant over a frame of

length

and vary from one frame to another.

At time

the signal received at antenna is given by

(1)

where

are independent samples of a zero-mean complex

Gaussian random variable with variance

SNR per com-

plex dimension. The average energy of the symbols transmitted

from each antenna is normalized to be

Assuming perfect channel state information is available, the

receiver computes the decision metric

(2)

over all codewords

and decides in favor of the codeword that minimizes this sum.

Given perfect channel state information at the receiver,

we may approximate the probability that the receiver decides

erroneously in favor of a signal

assuming that

was transmitted. (For details see [6], [10].) This analysis leads

to the following diversity criterion.

• Diversity Criterion For Rayleigh Space–Time Code: In

order to achieve the maximum diversity

, the matrix

has to be full rank for any pair of distinct codewords

and .If has minimum rank over the set of pairs

of distinct codewords, then a diversity of

is achieved.

Subsequent analysis and simulations have shown that codes

designed using the above criterion continue to perform well

in Rician environments in the absence of perfect channel state

information and under a variety of mobility conditions and

environmental effects [11].

III. O

RTHOGONAL DESIGNS AS

CODES FOR WIRELESS CHANNELS

In this section, we consider the application of real orthog-

onal designs (Section III-A) to coding for multiple-antenna

wireless communication systems. Unfortunately, these designs

only exist in a small number of dimensions. Encoding using

orthogonal designs is shown to be trivial in Section III-B.

Maximum-likelihood decoding is shown to be achieved by

decoupling of the signals transmitted from different antennas

and is proved to be based only on linear processing at the

receiver (Section III-C). The possibility of linear processing

at the transmitter, leads to the concept of linear processing

orthogonal designs developed in Section III-D. We then prove

a normalization result (Theorem 3.4.1) which allows us to

1458 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 5, JULY 1999

focus on a speciﬁc class of linear processing orthogonal

designs. To study the set of dimensions for which linear

processing orthogonal designs exist, we need a brief review of

the Hurwitz–Radon theory which is provided in Section III-E.

Using this theory, we prove that allowing linear processing

at the transmitter only increases the hardware complexity at

the transmitter and does not expand the set of dimensions for

which a real orthogonal design exists.

A reader who is only interested in code construction and

applications of space–time block codes may choose to focus

attention on Sections III-A, III-B, and III-C as well as Theo-

rem 3.5.1, Deﬁnition 3.5.2, and Lemma 3.5.1.

A. Real Orthogonal Designs

A real orthogonal design of size

is an orthogonal

matrix with entries the indeterminates

The existence problem for orthogonal designs is known as

the Hurwitz–Radon problem in the mathematics literature [5],

and was completely settled by Radon in another context at the

beginning of this century. In fact, an orthogonal design exists

if and only if

or .

Given an orthogonal design

, one can negate certain

columns of

to arrive at another orthogonal design where all

the entries of the ﬁrst row have positive signs. By permuting

the columns, we can make sure that the ﬁrst row of

. Thus we may assume without loss of generality

that

has this property.

Examples of orthogonal designs are the

design

(3)

the

design

(4)

and the

design

(5)

The matrices (3) and (4) can be identiﬁed, respectively,

with complex number

and the quaternionic number

B. The Coding Scheme

In this section, we apply orthogonal designs to construct

space–time block codes that achieve diversity. We assume that

transmission at the baseband employs a real signal constella-

tion

with elements. We focus on providing a diversity

order of

. Corollary 3.3.1 of [10] implies that the maximum

transmission rate is

bits per second per hertz (bits/s/Hz).

We provide this transmission rate using an

orthogonal

design. At time slot 1,

bits arrive at the encoder and

select constellation signals

. Setting for

, we arrive at a matrix with

entries

. At each time slot

the entries are transmitted simultaneously

from transmit antennas

Clearly, the rate of transmission is

bits/s/Hz. We now

demonstrate that the diversity order of such a space–time block

code is

Theorem 3.2.1: The diversity order of the above coding

scheme is

Proof: The rank criterion requires that the matrix

be nonsingular for any two

distinct code sequences

. Clearly,

where is the matrix constructed from

by replacing with for all . The

determinant of the orthogonal matrix

is easily seen to be

where is the transpose of . Hence

which is nonzero. It follows that

is nonsingular and the maximum diversity

order

is achieved.

C. The Decoding Algorithm

Next, we consider the decoding algorithm. Clearly, the

rows of

are all permutations of the ﬁrst row of with

possibly different signs. Let

denote the permutations

corresponding to these rows and let

denote the sign of

in the th row of . Then means that is up

to a sign change the

th element of . Since the columns

are pairwise-orthogonal, it turns out that minimizing the

metric of (2) amounts to minimizing

(6)

where

(7)

TAROKH et al.: SPACE–TIME BLOCK CODES FROM ORTHOGONAL DESIGNS 1459

and where denotes the complex conjugate of .

The value of

only depends on the code symbol , the

received symbols

, the path coefﬁcients , and the

structure of the orthogonal design

. It follows that minimiz-

ing the sum given in (6) amounts to minimizing (7) for all

. Thus the maximum-likelihood detection rule is to

form the decision variables

for all and decide in favor of among all the

constellation symbols

(8)

This is a very simple decoding strategy that provides diversity.

D. Linear Processing Orthogonal Designs

There are two attractions in providing transmit diversity via

orthogonal designs.

• There is no loss in bandwidth, in the sense that orthogonal

designs provide the maximum possible transmission rate

at full diversity.

• There is an extremely simple maximum-likelihood decod-

ing algorithm which only uses linear combining at the

receiver. The simplicity of the algorithm comes from the

orthogonality of the columns of the orthogonal design.

The above properties are preserved even if we allow linear

processing at the transmitter. Therefore, we relax the deﬁni-

tion of orthogonal designs to allow linear processing at the

transmitter. Signals transmitted from different antennas will

now be linear combinations of constellation symbols.

Deﬁnition 3.4.1: A linear processing orthogonal design in

variables

is an matrix such that:

• The entries of

are real linear combinations of variables

•

, where is a diagonal matrix with th

diagonal element of the form

with the coefﬁcients

all strictly positive

numbers.

It is easy to show that transmission using a linear processing

orthogonal design provides full diversity and a simpliﬁed

decoding algorithm as above. The next theorem shows that

we may, with no loss of generality, constrain the matrix

Deﬁnition 3.4.1 to be a scaled identity matrix.

Theorem 3.4.1: A linear processing orthogonal design

variables

exists if and only if there exists a

linear processing orthogonal design

such that

Proof: Let be a linear processing

orthogonal design, and let

where the matrices are diagonal and full-rank (since the

coefﬁcients

are strictly positive).

Then it follows that

(9)

(10)

and

is a full-rank diagonal matrix with positive diagonal

entries. Let

denote the diagonal matrix having the

property that

. We deﬁne .

Then the matrices

satisfy the following properties:

(11)

(12)

It follows that

is a linear processing

orthogonal array having the property

In view of the above theorem, we may, without any loss of

generality, assume that a linear processing orthogonal design

satisﬁes

E. The Hurwitz–Radon Theory

In this section, we deﬁne a Hurwitz–Radon family of

matrices. These matrices encode the interactions between

variables in an orthogonal design.

Deﬁnition 3.5.1: A set of

real matrices

is called a size Hurwitz–Radon family

of matrices if

and

We next recall the following theorem of Radon [8].

Theorem 3.5.1: Let

, where is odd and

with and . Any Hurwitz–Radon family of

matrices contains strictly less than

matrices. Furthermore . A Hurwitz–Radon family

containing

matrices exists if and only if or .

Deﬁnition 3.5.2: Let

be a matrix and let

be any arbitrary matrix. The tensor product is the

matrix given by

(13)

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