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1700 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 10, OCTOBER 2001
A Squaring Method to Simplify the Decoding of Orthogonal
Space–Time Block Codes
Xiangming Li, Tao Luo, Guangxin Yue, and Changchuan Yin, Member, IEEE
Abstract—In this letter, we present a squaring method to sim-
plify the decoding of orthogonal space–time block codes in a wire-
less system with an arbitrary number of transmit- and receive-an-
tennas. Using this squaring method, a closed-form expression of
signal-to-noise ratio after space–time decoding is also derived. It
gives the same decoding performance as the maximum-likelihood
ratio decoding while it shows much lower complexity.
Index Terms—Diversity, multipath channels, multiple antennas,
space–time codes.
I. INTRODUCTION
RECENTLY, a new space–time coding has been proposed
which combines signal processing at the receiver with
coding techniques appropriate to multiple number of transmit-
and receive-antennas wireless system [1]. Alamouti presents
a simple transmit diversity scheme. Using two transmit- and
one receive-antenna, the scheme provides the same diversity
order as maximal-ratio receiver combining with one transmit-
and two receive-antennas [2]. Based on Alamouti’s work,
Tarokh et al. propose orthogonal space–time block codes
(OSTBCs) combining the orthogonal coding method and this
simple diversity technique [3]. Tarokh studies the encoding and
decoding algorithms for various codes and provides simulation
results demonstrating their performance [4]. In this letter, a
squaring method to simplify the decoding of OSTBC is pro-
posed, which is a generalization from real orthogonal designs
in [3] to complex ones. In addition, a closed-form expression
of signal-to-noise ratio (SNR) after OSTBC decoding is also
derived.
II. SQUARING METHOD TO DECODE OSTBC
We consider a wireless communication system with
transmit- and receive-antennas as shown in Fig. 1. The
coefficient ( ) in Fig. 1
is the path gain from transmit antenna to receive antenna .
We assume that these path gains are constant during a frame
and vary from one frame to another (quasi-static flat fading).
Paper approved by R. Raheli, the Editor for Detection, Equalization and
Coding of the IEEE Communications Society. Manuscript received May 10,
2000; revised January 2, 2001. This work was supported in part by the National
Natural Science Fund of China by Grant 69872008.
The authors are with the College of Telecommunications Engineering,
Beijing University of Posts and Telecommunications, Beijing, China (e-mail:
xiang-ming_li@agilent.com).
Publisher Item Identifier S 0090-6778(01)09113-9.
Fig. 1. A wireless system with n-transmit and m-receive antennas.
Let ( ) be the information sequence to be
transmitted. A column orthogonal matrix
.
.
..
.
..
.
.
is used. At each time slot ( ), signals
are transmitted simultaneously from the
transmit antennas. Each signal is a linear combination of
the variables , . Noting the
column orthogonal characteristic of the matrix [3], we have
(1)
where the superscript “ ” denotes transpose conjugate of the
matrix and is the identity matrix. Let
0090–6778/01$10.00 © 2001 IEEE
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 10, OCTOBER 2001 1701
be the signal vector, and let
and
denote the channel vector, complex white Gaussian noise vector
and received vector of the th receive antenna, respectively. The
superscript “ ” means transpose of the vector and is the re-
ceived signal from the th antenna at time slot . Assume
has mean zero and variance per dimension. can be ex-
pressed as
(2)
where is defined as the channel matrix of the th receive
antenna with the following expression:
.
.
..
.
.
(3)
Let
(4)
denote the received vector of the th receive antenna in the ab-
sence of noise. Assuming perfect channel state information is
available, the receiver computes the decision metric [4]
(5)
over all code words
and decides in favor of the code word that minimizes the sum .
Assuming that transmission at the baseband employs a signal
constellation with elements and noting the relationship
between the matrix and the code word, we must compute
metrics for maximum-likelihood decoding using (5), while
we need only compute metrics using the method in [3],
[4] without decoding performance loss. Only several examples
of OSTBC are illustrated in [3] and [4]. Using the squaring
method, we generalize the method in [3] and [4] and simplify
the decoding of the OSTBC.
From (5), (2), and (1), the matrix form of (5) can be written
as
(6)
Noting that all the coefficients of the cross terms with respect
to , ( ) are zero in (6), we only consider the
terms comprising of , and . Thus, the minimization
of (6) is equivalent to minimizing all these parts separately
(7)
Let
(8)
(9)
where is complex and is positive. Substitution of (8) and
(9) into (7) yields
(10)
Multiplying the right-hand side of (10) by the real positive
value and adding the positive real value , the following
decision metric is obtained:
(11)
Hence, we have proved that may be used as the combing
scheme, i.e.,
(12)
The maximum-likelihood detection amounts to minimizing
the following decision metrics:
(13)
III. PERFORMANCE ESTIMATE
Summarizing the descriptions in Section II, one can obtain
combining schemes which are independent output branches
at the receiver using squaring method if perfect channel state
information is available. It is difficult to estimate the decoding
performance for the decoding method directly using (5). Since
the decoding performance results of the methods using (5) and
1702 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 10, OCTOBER 2001
(13) are identical, we can estimate the performance using (13).
In this section, we prove that all the output branches have the
same SNR.
Noting (12), (4), and (2), we have
(14)
In the absence of noise, the decision metric is zero when the
correct constellation point is selected at the receiver. As a result,
the signal term of (14) should be
(15)
Assume , where de-
notes expectation. We express the average signal power of the
th output branch at the receiver as
(16)
Considering the influence of noise, we write the noise term
in (14) separately
(17)
For an arbitrary given information sequence
(), we obtain from (4) that
(18)
Expanding and comparing it with (18) yields the
coefficient of
(19)
From (17) and (19), the output noise power of the branch
can be written as
(20)
From (16) and (20), all the output branches give the same
SNR
(21)
Assume the total energy of a block is limited to
( is the transmitting energy for each source symbol).
Using orthogonal matrix to transmit the sequence
(), the total energy can also expressed as
(22)
Thus,
(23)
Substitution of (23) into (16) yields
(24)
Assume the constellation at the receiver satisfies
(25)
where is the minimum distance of the constellation, and
is a constant depending on different constellations. Using
minimum distance sphere bound, the instant symbol error rate
bound is given by
(26)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 10, OCTOBER 2001 1703
We assume are independent samples of zero-mean complex
Gaussian random variables having variance 0.5 per dimension.
Thus are independent Rayleigh distributions with pdf
(27)
Thus, the average symbol error rate bound is given by
(28)
From (28), one can conclude that the symbol error rate de-
creases while the number of transmit- or receiver-antennas in-
creases. When is given, we have
(29)
This means that the increasing number of transmit antennas does
not give an infinite advantage.
IV. CONCLUSION
We propose a general linear maximum-likelihood OSTBC
scheme. The resulting decoding scheme can be applied to gen-
eralized complex orthogonal designs with arbitrary number of
transmit- and receive-antennas, and possesses a very low com-
putational complexity. In addition, a closed-form expression of
SNR after ST decoding is also derived.
ACKNOWLEDGMENT
The authors thank the three anonymous reviewers whose
comments have greatly enhanced the quality of this letter and
Dr. J. Zhao and Dr. X. Tong for helpful discussions.
REFERENCES
[1] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for
high data rate wireless communication: Performance criterion and code
construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Feb.
1998.
[2] S. M. Alamouti, “A simple transmit diversity technique for wireless
communications,” IEEE J. Select. Areas Commun., vol. 16, pp.
1451–1458, Aug. 1998.
[3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block
codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45,
pp. 1456–1467, May 1999.
[4] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block
coding for wireless communications: Performance results,” IEEE J.
Select. Areas Commun., vol. 17, pp. 451–460, Mar. 1999.
